time-splitting spectral methods for nonlinear schrödinger...

Lecture on

Time-Splitting Spectral Methodsfor Nonlinear Schrödinger Equations

Mechthild ThalhammerLeopold–Franzens Universität Innsbruck, Austria

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Spring School 2009

Analytical and Numerical Aspects of Evolution Equations

Organisers. Etienne Emmrich, Petra WittboldTechnische Universität Berlin, Germany

Summary

In this lecture, I address the issue of efficient numerical methods for the time integration ofnonlinear Schrödinger equations. As model problems, I consider systems of coupled Gross–Pitaevskii equations that arise in quantum physics for the description of multi-componentBose–Einstein condensates. My concern is to study the quantitative and qualitative behaviourof high-accuracy space and time discretisations that rely on time-splitting Fourier and Hermitespectral methods. In particular, this includes a stability and convergence analysis of high-orderexponential operator splitting methods for evolutionary Schrödinger equations. Numericalexamples illustrate the theoretical results.

i

Contents

Summary i

Preface v

I. Exponential operator splitting methods 1

1. Linear problems 51.1. Splitting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2. Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2. Local error expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.3. Convergence result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2. Alternative local error expansions 132.1. Baker–Campbell–Hausdorff formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2. Quadrature formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3. Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3. Nonlinear problems 193.1. Splitting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2. Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1. Local error expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

II. Fourier and Hermite spectral methods 25

4. Fourier spectral method 294.1. Approach in one space dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.1. Basic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1.2. Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.3. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2. Approach in several space dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.1. Basic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.2. Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.3. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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5. Hermite spectral method 375.1. Approach in one space dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1.1. Hermite basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1.2. Basic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.1.3. Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.1.4. Gauß–Hermite quadrature formula . . . . . . . . . . . . . . . . . . . . . . . 425.1.5. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2. Approach in several space dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 455.2.1. Basic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2.2. Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2.3. Approximation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2.4. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

III. Time integration of Gross–Pitaevskii systems 49

6. Gross–Pitaevskii systems 516.1. Original formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2. Normalised formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3. Special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7. Ground state solution 557.1. Energy functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.2. Ground state solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.3. A single Gross–Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.3.1. Groundstate solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8. Time-splitting pseudo-spectral methods 598.1. Abstract formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2. Time-splitting Fourier pseudo-spectral method . . . . . . . . . . . . . . . . . . . . 59

8.2.1. First part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2.2. Second part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.3. Time-splitting Hermite pseudo-spectral method . . . . . . . . . . . . . . . . . . . 618.3.1. First part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618.3.2. Second part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8.4. Numerical illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.4.1. Computation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.4.2. Spatial error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638.4.3. Temporal convergence order . . . . . . . . . . . . . . . . . . . . . . . . . . . 638.4.4. Long-term integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Bibliography 67

Appendix 71

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Figure 1.: Ground state solution of the Gross–Pitaevskii equation (1) with κ= 25 and ϑ= 400.

Preface

The actual research activities on efficient space and time discretisations for time-independentas well as time-dependent nonlinear Schrödinger equations is reflected in various contribu-tions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 18, 22, 23, 24, 26, 27, 28, 31, 33, 38, 42, 43, 45, 48, 49].The present manuscript shall provide an introduction to advanced integration methods fornonlinear Schrödinger equations that rely on high-order time-splitting Hermite and Fourierspectral methods.

Part I is dedicated to exponential operator splitting methods [11, 30, 32, 34, 35, 41, 50] forordinary differential equations. In particular, a result on the convergence behaviour of splittingmethods is deduced. To avoid technicalities, the focus is on a splitting scheme involving twocompositions applied to non-stiff linear differential equations. Extensions to splitting methodsof arbitrarily high order and nonlinear evolutionary problems of parabolic or Schrödingertype, respectively, are indicated, see also [10, 17, 19, 20, 21, 25, 26, 31, 33, 36, 37, 44]. InPart II, Fourier and Hermite spectral methods and their numerical realisations are discussed,see [12, 46]. Part III is concerned with high-order time-splitting Fourier and Hermite pseudo-spectral methods for the space and time discretisation of Gross–Pitaevskii systems [29, 39] thatarise in the description of multi-component Bose–Einstein condensates. In an appendix, theworks [13, 14, 36, 44] are included.

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As an illustration, the ground state of the two-dimensional Gross–Pitaevskii equation

i ∂tψ(ξ, t ) =(− 1

2 ∆+U (ξ)+ϑ ∣∣ψ(ξ, t )∣∣2

)ψ(ξ, t ) , ξ ∈R2 , t ≥ 0,∥∥ψ(·,0)

∥∥L2 = 1, U (ξ) =U (ξ1,ξ2) = 1

2

2∑i=1

(ξ2

i +κsin2(π4 ξi

)),

(1)

describing a Bose-Einstein condensate in a lattice under an external harmonic potential isdisplayed in Figure 1. The ground state solution is computed by means of the imaginarytime method; hereby, the space and time discretisation relies on the Fourier spectral methodwith 256 basis functions in each space direction and a linearly implicit Euler method withconstant time step 10−3. A MATLAB code for the ground state computation and the timeevolution of Gross–Pitaevskii systems in one, two, and three space dimensions is available onrequest.

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Part I.

Exponential operator splitting methods

1

Situation. We consider the following initial value problem for a function y : [t0,T ] → Rd

involving a nonlinear autonomous differential equation

y ′(t ) = F(y(t )

), t0 ≤ t ≤ T , y(t0) given. (0.2)

Assumption. Throughout, we suppose the function F : D ⊂Rd →Rd defining the right-handside of the differential equation in (0.2) to be sufficiently often differentiable with boundedderivatives. For simplicity, we further assume D =Rd .

Generalisation. It is straightforward to extend our considerations to the case where theEuclidian space Rd is replaced with a Banach space

(X ,‖ ·‖X

).

Exact solution. For the following, it is useful to introduce the exact solution operator EF

associated with the initial value problem (0.2) through

y(t0 +τ) = EF(τ, t0, y(t0)

), 0 ≤ τ≤ T − t0 . (0.3a)

A standard existence and uniqueness result for (0.2) implies the identity

EF(σ+τ, t0, y(t0)

)= EF(τ, t0 +σ, y(t0 +σ)

), 0 ≤σ+τ≤ T − t0 . (0.3b)

Numerical approximation. For an initial approximation y0 ≈ y(t0) and a time grid withassociated time stepsizes

t0 < t1 < ·· · < tN = T , hn−1 = tn − tn−1 , 1 ≤ n ≤ N ,

numerical approximations yn to the exact solution values at time tn are determined through areccurrence relation of the form

yn =ΦF (hn−1, tn−1, yn−1) , 1 ≤ n ≤ N , y0 given;

this is in accordance with the identity

y(tn) = EF(hn−1, tn−1, y(tn−1)

), 1 ≤ n ≤ N , y(t0) given,

see also (0.3). A numerical method is said to be consistent of order p iff the local error fulfills

dn =ΦF(hn−1, tn−1, y(tn−1)

)−EF(hn−1, tn−1, y(tn−1)

)=O(hp+1

n−1

), (0.4)

provided that the exact solution of (0.2) and the nonlinear function F defining the differentialequation are sufficiently regular. It suffices to specify the first step of the numerical scheme

y1 =ΦF (h, t0, y0) ≈ y(t1) = EF(h, t0, y(t0)

), h = h0 .

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Formal calculus. In view of Chapter 3, it is convenient to employ a formal calculus of Lie-derivatives, which allows to extend results that were deduced for linear differential equationsto nonlinear problems. For any matrix L ∈Rd×d or bounded linear operator L : X → X , respec-tively, the exponential function is well defined through

eτL =∞∑

j=0

1j ! τ

j L j , τ ∈R . (0.5)

In accordance with the identity L = ddτ

∣∣∣τ=0

eτL , we set

eτDF y(t0) = EF(τ, t0, y(t0)

)= y(t0 +τ) , 0 ≤ τ≤ T − t0 ,

DF y(t0) = ddτ

∣∣∣τ=0

EF(τ, t0, y(t0)

)= ddτ

∣∣∣τ=0

y(t0 +τ) = y ′(t0) = F(y(t0)

).

More generally, for a function G : X → X we let

eτDF G y(t0) =G(EF

(τ, t0, y(t0)

))=G(y(t0 +τ)

), 0 ≤ τ≤ T − t0 ,

DF G y(t0) = ddτ

∣∣∣τ=0

G(EF

(τ, t0, y(t0)

))= ddτ

∣∣∣τ=0

G(y(t0 +τ)

)=G ′(y(t0))

F(y(t0)

).

To summarise, the basic relations for the Lie-derivative DF and the flow operator eτDF are

DF y(t0) = F(y(t0)

), DF G y(t0) =G ′(y(t0)

)F

(y(t0)

),

eτDF y(t0) = EF(τ, t0, y(t0)

), eτDF G y(t0) =G

(EF

(τ, t0, y(t0)

)), 0 ≤ τ≤ T − t0 .

(0.6)

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1. Linear problems

Situation. Henceforth, we consider the following initial value problem for y : [t0,T ] → Rd

involving a linear differential equation of the form

y ′(t ) = A y(t )+B y(t ) , t0 ≤ t ≤ T , y(t0) given. (1.1)

Generalisation. It is straightforward to extend our considerations to the case where theEuclidian space Rd is replaced with a Banach space

(X ,‖ ·‖X

).

Assumption. We suppose the matrices A,B ∈Rd×d or linear operators A,B : X → X , respec-tively, to be bounded, that is, there exist (moderate) constants CA,CB ≥ 0 such that∥∥A

∥∥X←X ≤CA ,

∥∥B∥∥

X←X ≤CB ; (1.2)

we tacitly assume CA,CB ≥ 1. Consequently, also A+B is bounded∥∥A+B∥∥

X←X ≤CA +CB .

In general, the linear operators A and B do not commute, that is, it holds AB 6= B A.

Exact solution. In the present situation, the exact solution of the initial value problem (1.1)is given by

y(t0 +τ) = eτ (A+B) y(t0) , 0 ≤ τ≤ T − t0 ,

see also (0.5). As the exact solution operator EA+B is linear with respect to the initial value, wewrite EA+B (τ) y(t0) = EA+B

(τ, t0, y(t0)

)for short, that is, we have

y(t0 +τ) = EA+B (τ) y(t0) = eτ (A+B) y(t0) , 0 ≤ τ≤ T − t0 . (1.3)

Numerical approximation. Regarding (1.3), we require the numerical solution operatorΦA+B

to be linear with respect to the initial value

y1 =ΦA+B (h) y0 ≈ y(t1) = EA+B (h) y(t0) , h = h0 .

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1.1. Splitting methods

Approach. Exponential operator splitting methods rely on a decomposition of the right-handside of the differential equation in (1.1) into two (or more) parts and the presumption that theinitial value problems

z ′(t ) = A z(t ) , t0 ≤ t ≤ T , z(t0) given,

z ′(t ) = B z(t ) , t0 ≤ t ≤ T , z(t0) given,(1.4)

are solvable numerically in an accurate and efficient manner. The (approximate) solutions ofthe initial value problems (1.4) are then composed in a suitable way; this yields an approxima-tionΦA+B ≈ EA+B to the exact solution operator. For simplicity and in view of Gross–Pitaevskiisystems, we may assume that the exact solutions of (1.4)

z(t0 +τ) = EA(τ) z(t0) = eτA z(t0) , 0 ≤ τ≤ T − t0 ,

z(t0 +τ) = EB (τ) z(t0) = eτB z(t0) , 0 ≤ τ≤ T − t0 ,

are available.

General form of splitting methods. Any exponential operator splitting method involvingseveral compositions can be cast into the following form

y1 =ΦA+B (h) y0 =s∏

i=1EB (bi h)EA(ai h) y0 =

s∏i=1

ebi hB eai h A y0 , h = h0 , (1.5)

yielding an approximation to the exact solution value

y(t1) = EA+B (h) y(t0) = eh (A+B) y(t0) , h = h0 .

In (1.5), the product is defined downwards, i.e., for linear operators (L i ) j≤i≤k we set

k∏i= j

L i = Lk · · · Lj+1 Lj , j ≤ k ,k∏

i= jL i = I , j > k .

Example (Lie–Trotter splitting method). The Lie–Trotter splitting method for (1.1) can becast into the general form (1.5) with

s = 1, a1 = 1, b1 = 1, or s = 2, a1 = 0, a 2 = 1 b1 = 1, b 2 = 0, (1.6a)

respectively, that is, the first numerical solution value is given by

y1 = ehB eh A y0 , or y1 = eh A ehB y0 , (1.6b)

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method order #comp.

McLachlan MCLACHLAN [30, V.3.1, (3.3), pp. 138–139] p = 2 s = 3

Strang STRANG (1.7) p = 2 s = 2

BM4-1 BLANES & MOAN [11, Table 2, PRKS6] p = 4 s = 7

BM4-2 BLANES & MOAN [11, Table 3, SRKNb6 ] p = 4 s = 7

M4 MCLACHLAN [30, V.3.1, (3.6), pp. 140] p = 4 s = 6

S4 SUZUKI [30, II.4, (4.5), pp. 41] p = 4 s = 6

Y4 YOSHIDA [30, II.4, (4.4), pp. 40] p = 4 s = 4

BM6-1 BLANES & MOAN [11, Table 2, PRKS10] p = 6 s = 11

BM6-2 BLANES & MOAN [11, Table 3, SRKNb11] p = 6 s = 12

BM6-3 BLANES & MOAN [11, Table 3, SRKNa14] p = 6 s = 15

KL6 KAHAN & LI [30, V.3.2, (3.12), pp. 144] p = 6 s = 10

S6 SUZUKI [30, II.4, (4.5), pp. 41] p = 6 s = 26

Y6 YOSHIDA [30, V.3.2, (3.11), pp. 144] p = 6 s = 8

Table 1.: Exponential operator splitting methods of order p involving s compositions.

respectively. In Section 1.2 it is verified that the Lie–Trotter splitting method is of (classical)order one.

Example (Strang splitting method). The symmetric Lie–Trotter splitting method or Strangsplitting method [40, 47] can be cast into the general form (1.5) with

s = 2, a1 = a 2 = 12 , b1 = 1, b 2 = 0, or s = 2, a1 = 0, a 2 = 1, b1 = b 2 = 1

2 , (1.7a)

respectively, that is, the first numerical solution value is given by

y1 = e12h A ehB e

12h A y0 , or y1 = e

12hB eh A e

12hB y0 , (1.7b)

respectively. The computational effort of the Strang splitting method is essentially that of theLie–Trotter splitting method. In Section 1.2 it is verified that the Strang splitting method is of(classical) order two.

Higher order splitting methods. Exponential operator splitting methods of order four andsix are given in [11, 30], e.g., see also Table 1.

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1.2. Convergence analysis

Objective. In the following, we are concerned with deducing an estimate for the global er-ror yN −y(T ) of an exponential operator splitting method (1.5) when applied to the initial valueproblem (1.1); to this purpose, we follow a standard approach based on a Lady Windermere’sFan argument.

Local error and order. In the present situation, the local error equals

dn = D(hn−1) y(tn−1) = (ΦA+B (hn−1)−EA+B (hn−1)

)y(tn−1) , 1 ≤ n ≤ N ,

see also (0.4). Therefore, the numerical method (1.5) is consistent of order p whenever thedefect operator D fulfills

D(h) =O(hp+1) . (1.8)

Lady Windermere’s Fan. In order to relate the global and the local error, we employ thetelescopic identity

yN − y(tN ) =N−1∏j=0ΦA+B (h j )

(y0 − y(t0)

)+ N∑n=1

N−1∏j=n

ΦA+B (h j )dn . (1.9)

In Sections 1.2.1 and 1.2.2, we are concerned with deriving a bound for the splitting opera-torΦA+B and a suitable expansion of the defect operator D .

Explanation. The validity of relation (1.9) is verified by a short calculation

N−1∏j=0ΦA+B (h j )

(y0 − y(t0)

)+ N∑n=1

N−1∏j=n

ΦA+B (h j )dn

=N−1∏j=0ΦA+B (h j )

(y0 − y(t0)

)+ N∑n=1

N−1∏j=n

ΦA+B (h j )(ΦA+B (hn−1)−EA+B (hn−1)

)y(tn−1)

=N−1∏j=0ΦA+B (h j ) y0 −

N−1∏j=0ΦA+B (h j ) y(t0)

+N∑

n=1

N−1∏j=n−1

ΦA+B (h j ) y(tn−1)−N∑

n=1

N−1∏j=n

ΦA+B (h j ) y(tn)

= yN −N−1∏j=0ΦA+B (h j ) y(t0)+

N−1∑n=0

N−1∏j=n

ΦA+B (h j ) y(tn)−N∑

n=1

N−1∏j=n

ΦA+B (h j ) y(tn)

= yN −N−1∏j=0ΦA+B (h j ) y(t0)+

N−1∏j=0ΦA+B (h j ) y(t0)− y(tN )

= yN − y(tN ) .

8

1.2.1. Stability

Assumption. In order to prove the desired stability result for exponential operator splittingmethods, we employ the bounds∥∥eτA

∥∥X←X ≤ eMA |τ| ,

∥∥eτB∥∥

X←X ≤ eMB |τ| , τ ∈R , (1.10)

involving certain positive constants MA, MB ≥ 0.

Remark. In the present situation, for bounded linear operators A,B : X → X relation (1.11)holds with MA = CA and MB = CB , see (1.2). Namely, a straightforward estimation of theexponential series (0.5) yields∥∥eτL

∥∥X←X ≤ e‖L‖X←X |τ| , τ ∈R .

Stability result. Under assumption (1.10), the estimate

∥∥ΦA+B (h j )∥∥

X←X ≤s∏

i=1

∥∥ebi h j B∥∥

X←X

∥∥eai h j A∥∥

X←X ≤ eCΦh j ,

follows, see also (1.5), which further implies the stability bound

∥∥∥m−1∏j=k

ΦA+B (h j )∥∥∥

X←X≤ eCΦ(tm−tk ) , CΦ = MA

s∑i=1

|ai |+MB

s∑i=1

|bi | , m > k ≥ 0. (1.11)

Extension (Evolutionary Schrödinger equations) . The stability estimate (1.11) can also beestablished in the context of abstract evolution problems. For instance, for evolutionarySchrödinger equations of the form (1.1), we require that the unbounded linear operatorsA : D(A) ⊂ X → X and B : D(B) ⊂ X → X generate C0-groups

(eτA

)τ∈R and

(eτB

)τ∈R such that∥∥eτA

∥∥X←X ≤ eMA |τ| ,

∥∥eτB∥∥

X←X ≤ eMB |τ| , τ ∈R .

In this case, exponential operator splitting methods that involve negative coefficients arepermitted.

Extension (Parabolic evolution equations) . For evolution equations (1.1) of parabolic type,we require the unbounded linear operators A : D(A) ⊂ X → X and B : D(B) ⊂ X → X to generateC0-semigroups

(eτA

)τ≥0 and

(eτB

)τ≥0 such that∥∥eτA

∥∥X←X ≤ eMA τ ,

∥∥eτB∥∥

X←X ≤ eMB τ , τ≥ 0.

In this case, exponential operator splitting methods that involve complex coefficients withpositive real part are permitted.

9

1.2.2. Local error expansion

Situation. For the following, to avoid technicalities, we consider exponential operator split-ting methods (1.5) that involve two compositions only

y1 = eb 2hB ea 2h A eb1hB ea1h A y0 , h = h0 . (1.12)

Method examples that can be cast into this form are the Lie–Trotter splitting method (1.6) andthe Strang splitting method (1.7).

Objective. We are concerned with deducing a suitable expansion of the defect operator

D(h) =ΦA+B (h)−EA+B (h) = eb 2hB ea 2h A eb1hB ea1h A − eh (A+B) , (1.13)

with respect to h, see also (1.3) and (1.5).

Approach. We employ the power series expansion (0.5) for the matrix exponential; moreprecisely, performing a stepwise Taylor series expansion of eτL , we obtain

eτL = I +eστL∣∣∣1

σ=0

= I +τL∫ 1

0eστL dσ

= I +τL+τ2L2∫ 1

0(1−σ) eστL dσ

= I +τL+ 12 τ

2L2 +τ3L3∫ 1

0

12 (1−σ)2 eστL dσ , τ≥ 0.

With the help of the bounds (1.2) and (1.10), we thus have

eτL = I +O (τ,CL , ML)

= I +τL+O(τ2,C 2

L , ML)

= I +τL+ 12 τ

2 L2 +O(τ3,C 3

L , ML)

, τ≥ 0.

(1.14)

Expansion (Exact solution operator). By means of (1.14), we obtain the following expansionof the exact solution operator

EA+B (h) = eh (A+B)

= I +h (A+B)+ 12 h2(A2 + AB +B A+B 2)+O

(h3,C 3

A+B , MA+B)

;(1.15)

here, we assume that the following estimate is valid∥∥eτ (A+B)∥∥

X←X ≤ eMA+B |τ| , τ ∈R .

10

Expansion (Splitting operator). By means of relation (1.14), we next employ a stepwiseexpansion of the splitting operator

ΦA+B (h) = eb 2hB ea 2h A eb1hB ea1h A

= (I +b 2hB + 1

2 b22h2B 2) ea 2h A eb1hB ea1h A +O

(h3,C 3

B , MA, MB)

= ea 2h A eb1hB ea1h A +b 2hB ea 2h A eb1hB ea1h A + 12 b2

2h2B 2 ea 2h A eb1hB ea1h A

+O(h3,C 3

B , MA, MB)

= (I +a 2h A+ 1

2 a22h2 A2) eb1hB ea1h A +b 2hB

(I +a 2h A

)eb1hB ea1h A

+ 12 b2

2h2B 2 eb1hB ea1h A +O(h3,C 3

A,C 3B , MA, MB

)= eb1hB ea1h A +h

(a 2 A+b 2B

)eb1hB ea1h A

+h2 (12 a2

2 A2 +a 2 b 2B A+ 12 b2

2B 2) eb1hB ea1h A +O(h3,C 3

A,C 3B , MA, MB

)and, furthermore, we have

ΦA+B (h) = (I +b1hB + 1

2 b21h2B 2) ea1h A +h

(a 2 A+b 2B

)(I +b1hB

)ea1h A

+h2 (12 a2

2 A2 +a 2 b 2B A+ 12 b2

2B 2) ea1h A +O(h3,C 3

A,C 3B , MA, MB

)= ea1h A +h

(a 2 A+ (b1 +b 2)B

)ea1h A

+h2 (12 a2

2 A2 +a 2b1 A B +a 2 b 2B A+ 12 (b1 +b 2)2B 2) ea1h A

+O(h3,C 3

A,C 3B , MA, MB

)= I +a1h A+ 1

2 a21h2 A2 +h

(a 2 A+ (b1 +b 2)B

)(I +a1h A

)+h2 (1

2 a22 A2 +a 2b1 A B +a 2 b 2B A+ 1

2 (b1 +b 2)2B 2)+O

(h3,C 3

A,C 3B , MA, MB

).

This finally yields the following expansion

ΦA+B (h) = I +h((a1 +a 2)A+ (b1 +b 2)B

)+h2 (1

2 (a1 +a 2)2 A2 +a 2b1 A B + (a1(b1 +b 2)+a 2 b 2

)B A

+ 12 (b1 +b 2)2B 2)+O

(h3,C 3

A,C 3B , MA, MB

).

(1.16)

Expansion (Defect operator). Altogether, the above relations (1.15) and (1.16) imply thefollowing expansion of the defect operator D =ΦA+B −EA+B with respect to h

D(h) = h((a1 +a 2 −1)A+ (b1 +b 2 −1)B

)+h2

(12

((a1 +a 2)2 −1

)A2 + (

a 2b1 − 12

)A B + (

a1(b1 +b 2)+a 2 b 2 − 12

)B A

+ 12

((b1 +b 2)2 −1

)B 2

)+O

(h3,C 3

A,C 3B , MA, MB , MA+B

).

(1.17)

11

Order conditions. Employing the above expansion (1.17) of the defect operator D and re-quiring (1.8) to be valid with p = 1 for arbitrary matrices or bounded linear operators A and B ,respectively, the (classical) first order conditions

a1 +a 2 = 1, b1 +b 2 = 1, (1.18a)

follow. For (classical) order two, that is, setting p = 2, the additional conditions are a 2b1 = 12

and a1 +a 2 b 2 = 12 , or, equivalently,

(1−a1)b1 = 12 . (1.18b)

Obviously, the Lie–Trotter splitting method (1.6) has (classical) order one; the second-orderStrang splitting method (1.7) is retained from the order conditions (1.18) under the symmetryrequirement b 2 = 0 and a1 = a 2 or a1 = 0 and b1 = b 2, respectively.

Local error estimate. The above consideration imply the local error estimate∥∥D(h)∥∥

X←X ≤C hp+1 (1.19)

with constant C depending on C pA ,C p

B , MA, MB , MA+B , and further on the method coefficients.In particular, the above bound holds true with p = 1 for the Lie–Trotter splitting method (1.6)and with p = 2 for the Strang splitting method (1.7).

1.2.3. Convergence result

Convergence estimate. Assume that the exponential operator splitting method (1.5) appliedto the linear initial value problem (1.1) fulfills the (classical) order conditions for order p ≥ 1.Then, the following global error estimate

∥∥yN − y(tN )∥∥

X ≤C(∥∥y0 − y(t0)

∥∥X +

N−1∑n=0

hp+1n

)holds with constant C depending in particular on CA,CB , MA+B , MA, MB ,T , and y(t0). Namely,estimating the global error relation (1.9) by means of the stability bound (1.11) and the localerror estimate (1.19), the desired result follows. Especially, for constant stepsizes, that is, itholds hn = h for 0 ≤ n ≤ N −1, the expected convergence bound∥∥yn − y(tn)

∥∥X ≤C

(∥∥y0 − y(0)∥∥

X +hp)

, 0 ≤ n ≤ N ,

follows.

12

2. Alternative local error expansions

Situation. As before, we focus on exponential operator splitting methods (1.5) for linearinitial value problems of the form (1.1) that involve two compositions only, see (1.12).

Objective. Regarding possible extensions of the convergence analysis given in Section 1.2 toevolutionary Schrödinger equations or parabolic evolution equations, respectively, we next in-vestigate alternative approaches for deducing a suitable expansion of the defect operator (1.13)with respect to h.

Notation. For matrices L1,L 2 ∈Rd×d or bounded linear operators L1,L 2 : X → X , respectively,the iterated commutators are defined by

ad j+1L1

(L 2) = [L1,ad j

L1(L 2)

]= L1 ad jL1

(L 2)−ad jL1

(L 2)L1 , j ≥ 0, (2.1)

where ad0L1

(L 2) = L 2, see [30]. Note that for the first commutator adL1(L 2) = L1L 2 −L 2L1 itfollows adL1(L1) = 0 and adL 2(L1) =− adL1(L 2).

2.1. Baker–Campbell–Hausdorff formula

Approach. The Baker–Campbell–Hausdorff formula considerably facilitates the expansion ofcompositions involving the matrix exponential, see for example [30]. However, as for the powerseries expansion (0.5), it is not evident to extend this approach to evolutionary equations (1.1)with unbounded linear operators A and B ; in particular, it is difficult to obtain error estimatesthat are optimal with respect to the regularity properties of the exact solution.

Baker–Campbell–Hausdorff formula. The Baker–Campbell–Hausdorff formula implies theexpansion

eh L 2 eh L1 = eh L , L = L1 +L 2 − 12 h adL1(L 2)+O

(h2) . (2.2)

Local error expansion. An application of the above relation (2.2) to (1.12) yields

ebi hB eai h A = eh L i , L i = ai A+bi B − 12 h ai bi adA(B)+O

(h2) , i = 1,2;

13

moreover, we obtain the identity

ΦA+B (h) = eb 2hB ea 2h A eb1hB ea1h A = eh L 2 eh L1 = eh L ,

L = (a1 +a 2)A+ (b1 +b 2)B − 12 h

(a1(b1 +b 2)+a 2(b 2 −b1)

)adA(B)+O

(h2) .

As a consequence, the requirement

D(h) =ΦA+B (h)−EA+B (h) = eh L − eh (A+B) =O(hp+1)

implies L − (A+B) =O(hp

), that is, we have

(a1 +a 2 −1)A+ (b1 +b 2 −1)B − 12 h

(a1(b1 +b 2)+a 2(b 2 −b1)

)adA(B) =O

(hp)

.

In particular, for p = 1 or p = 2, respectively, we retain the first and second order condi-tions (1.18).

2.2. Quadrature formulas

Approach. In the following, we present an approach that is more involved than the Baker–Campbell–Hausdorff formula but well suited for an extension to linear and nonlinear evo-lutionary equations, see [26, 31, 33, 36, 44]. The basic idea is to expand the exact solutionoperator by means of the variation-of-constants formula and to deduce a similar expansionof the splitting operator by employing the standard exponential power series for terms of theform ebi hB ; the expansion of the splitting operator is then considered as a quadrature formulaapproximation of a multiple integral.

Variation-of-constants formula. The exact solution of the initial value problem (1.1) canalso be represented by means of the variation-of-constants formula

y(t0 +τ) =(eτA +

∫ τ

0e(τ−σ)A B eσ (A+B) dσ

)y(t0) , 0 ≤ τ≤ T − t0 ;

we thus obtain the following representation of the exact solution operator

EA+B (τ) = eτA +∫ τ

0e(τ−σ)A B eσ (A+B) dσ , (2.3)

see also (1.3).

14

Expansion (Exact solution). A repeated application of the above relation (2.3) yields

EA+B (h) = eh A +∫ h

0e(h−σ1)A B eσ1(A+B) dσ1

= eh A +∫ h

0e(h−σ1)A B

(eσ1A +

∫ σ1

0e(σ1−σ2)A B eσ2(A+B) dσ2

)dσ1

= eh A +∫ h

0e(h−σ1)A B eσ1A dσ1+

∫ h

0

∫ σ1

0e(h−σ1)A B e(σ1−σ2)A B eσ2(A+B) dσ2 dσ1,

and, moreover, we have


0e(h−σ1)A B eσ1A dσ1

+∫ h

0

∫ σ1

0e(h−σ1)A B e(σ1−σ2)A B

(eσ2A +

∫ σ2

0e(σ2−σ3)A B eσ3(A+B) dσ3

)dσ2 dσ1

= eh A +∫ h


∫ h

0

∫ σ1

0e(h−σ1)A B e(σ1−σ2)A B eσ2A dσ2 dσ1

+∫ h

0

∫ σ1

0

∫ σ2

0e(h−σ1)A B e(σ1−σ2)A B e(σ2−σ3)A B eσ3(A+B) dσ3 dσ2 dσ1.

We thus obtain the following expansion of the exact solution operator with respect to h = h0



∫ h

0

∫ σ1

0e(h−σ1)A B e(σ1−σ2)A B eσ2A dσ2 dσ1

+O(h3,C 3

B , MA)

.

Expansion of exponential. Regarding a stepwise expansion of the splitting operator, it isconvenient to employ the following stepwise expansion of the exponential function; to capturethe remainder, we introduce the complex functions ϕj : C→ C : z 7→ ϕj (z), j ≥ 0, definedthrough

ϕ0(z) = ez , ϕj (z) = 1( j−1)!

∫ 1

0σ j−1 e(1−σ)z dσ , j ≥ 1, z ∈C .

Relation (1.10) for the exponential implies the bound∥∥ϕj (τB)∥∥

X←X ≤ 1j ! eMB |τ| , j ≥ 0, τ ∈R . (2.4)

By a partial integration, it is seen that the ϕ-functions fulfill the recurrence relation

ϕj (z) = 1j ! + zϕj+1(z) , j ≥ 0, z ∈C . (2.5)

For instance, we obtain the following expansion

ez = 1+ zϕ1(z) = 1+ z + z2ϕ2(z) = 1+ z + 12 z2 + z3ϕ3(z) ,

15

which correspond to a standard Taylor series expansion

ez = 1+ z∫ 1

0eσz dσ= 1+ z + z2

∫ 1

0(1−σ) eσz dσ= 1+ z + 1

2 z2 + z3∫ 1

0

12 (1−σ)2 eσz dσ .

Expansion (Splitting operator). As a first step, we expand the splitting operatorΦA+B (h) bymeans of the identity eτB = I +τBϕ1(τB), see also (2.5); more precisely, replacing ebi hB eai h A

with eai h A +bi hBϕ1(bi hB)eai h A, i = 1,2, we obtain

ΦA+B (h) = eb 2hB ea 2h A eb1hB ea1h A

= e(a1+a 2)h A +h(b1ea 2h ABϕ1(b1hB)ea1h A +b 2Bϕ1(b 2hB)e(a1+a 2)h A)

+h2b1b 2Bϕ1(b 2hB)ea 2h A Bϕ1(b1hB)ea1h A .

In order to discover the similarities between the expansion of the exact solution operator andthe expansion of the splitting operator, we henceforth denote

c1 = a1 , c2 = a1 +a 2 ,

and further require the order condition c2 = a1 +a 2 = 1 to be fulfilled. Consequently, it follows

ΦA+B (h) = eh A +h(b1e(1−c1)h ABϕ1(b1hB)ec1h A +b 2Bϕ1(b 2hB)eh A)

+h2b1b 2Bϕ1(b 2hB)e(1−c1)h A Bϕ1(b1hB)ec1h A .

Inserting the identity ϕ1(τB) = I +τBϕ2(τB), we further obtain

ΦA+B (h) = eh A +h(b1e(1−c1)h AB

(I +b1hBϕ2(b1hB)

)ec1h A +b 2B

(I +b 2hBϕ2(b 2hB)

)eh A)

+h2b1b 2B(I +b 2hBϕ2(b 2hB)

)e(1−c1)h A Bϕ1(b1hB)ec1h A

= eh A +h(b1e(1−c1)h AB ec1h A +b 2B eh A)+h2 (

b21 e(1−c1)h AB 2ϕ2(b1hB)ec1h A

+b1b 2B e(1−c1)h A Bϕ1(b1hB)ec1h A +b22B 2ϕ2(b 2hB)eh A)+O

(h3,C 3

B , MA, MB)

,

see also (2.5) and (2.4). We finally expand all terms involving h2 by means of the recurrencerelation (2.5); in particular, inserting the identity ϕ2(τB) = 1

2 I +τBϕ3(τB), it follows

ΦA+B (h) = eh A +h(b1e(1−c1)h AB ec1h A +b 2B eh A)

+h2 (12 b2

1 e(1−c1)h AB 2ec1h A +b1b 2B e(1−c1)h A B ec1h A + 12 b2

2B 2 eh A)+O

(h3,C 3

B , MA, MB)

.

16

Expansion (Defect operator). Altogether, the above expansions yield the following relationfor the defect operator D =ΦA+B −EA+B

D(h) =Q1 − I1 +Q 2 − I 2 +O(h3,C 3

B , MA, MB)

,

Q1 = h(b1e(1−c1)h AB ec1h A +b 2B eh A)

, I1 =∫ h

0e(h−σ1)A B eσ1A dσ1,

Q 2 = h2 (12 b2

1 e(1−c1)h AB 2ec1h A +b1b 2B e(1−c1)h A B ec1h A + 12 b2

2B 2 eh A),

I 2 =∫ h

0

∫ σ1

0e(h−σ1)A B e(σ1−σ2)A B eσ2A dσ2 dσ1.

We next relate Q1 and Q 2 to the integrals I1 and I 2. More precisely, we consider Q1 as anapproximation to the single integral I1

g (σ1) = e(h−σ1)A B eσ1A , 0 ≤σ1≤ h ,

Q1 = h(b1 g (c1h)+b 2 g (c2h)

), I1 =

∫ h

0g (σ1) dσ1,

resulting from the application of a quadrature formula with weights and nodes (bi ,ci )2i=1,

where s = 2. A standard Taylor series expansion of the integrand g about zero yields

g ′(σ1) =− e(h−σ1)A adA(B) eσ1A , g ′′(σ1) = e(h−σ1)A ad2A(B) eσ1A , 0 ≤σ1≤ h ,

Q1 = h (b1 +b 2) g (0)+h2 (b1c1 +b 2c2) g ′(0)+O(h3, g ′′) ,

I1 = h g (0)+ 12 h2 g ′(0)+O

(h3, g ′′) .

In a similar manner, we interprete Q 2 as quadrature formula approximation to the doubleintegral I 2

G(σ1,σ2) = e(h−σ1)A B e(σ1−σ2)A B eσ2A , 0 ≤σ2≤σ1≤ h ,

Q 2 = h2(12 b2

1 G(c1h,c1h)+b1b 2 G(c2h,c1h)+ 12 b2

2 G(c2h,c2h))

,

I 2 =∫ h

0

∫ σ1

0G(σ1,σ2) dσ2dσ1.

Here, by a Taylor series expansion of the function G , it follows

G ′(σ1,σ2) =− e(h−σ1)A (adA(B) e(σ1−σ2)A B ,B e(σ1−σ2)A adA(B)

)eσ2A , 0 ≤σ2≤σ1≤ h ,

Q 2 = 12 h2(b1 +b 2)2 G(0,0)+O

(h3,G ′) , I 2 = 1

2 h2 G(0,0)+O(h3,G ′) ,

with G ′ denoting the Jacobian of G . Provided that the bound∥∥adA(B)∥∥

X←X +∥∥ad2A(B)

∥∥X←X ≤Cad

17

holds with some constant Cad > 0, see also (2.1) for the definition of the iterated commutators,we finally have

D(h) = h (b1 +b 2 −1) g (0)+h2((

b1c1 +b 2c2 − 12

)g ′(0)+ 1

2

((b1 +b 2)2 −1

)G(0,0)

+O(h3,C 3

B , MA, MB ,Cad)

= h (b1 +b 2 −1)eh AB −h2((

b1c1 +b 2c2 − 12

)eh A adA(B)+ 1

2

((b1 +b 2)2 −1

)eh A B 2

)+O

(h3,C 3

B , MA, MB ,Cad)

.

As before, the requirement D(h) =O(hp+1

)for p = 1 or p = 2, respectively, yields the first and

second order conditions (1.18).

2.3. Differential equations

Approach. Another approach that is particularly suited for evolutionary equations involvingcritical parameters relies on the deduction of a differential equation for the splitting operator;however, as several rather technical arguments are needed, we do not present this approachhere and refer to [20, 21].

18

3. Nonlinear problems

Situation. Henceforth, we consider the following initial value problem for y : [t0,T ] → Rd

involving a nonlinear differential equation of the form

y ′(t ) = A(y(t )

)+B(y(t )

), t0 ≤ t ≤ T , y(t0) given. (3.1)

Generalisation. It is straightforward to extend our considerations to the case where theEuclidian space Rd is replaced with a Banach space (X ,‖ ·‖X ).

Assumption. We suppose the functions A : X → X and B : X → X to be sufficiently oftendifferentiable with bounded derivatives.

Exact solution. The exact solution of the initial value problem (3.1) is (formally) given by thenonlinear exact solution operator EA+B , that is, it holds

y(t0 +τ) = EA+B(τ, t0, y(t0)

), 0 ≤ τ≤ T − t0 , (3.2a)

see also (0.3). Employing the compact, formally linear notation of Lie-derivatives, we have

y(t0 +τ) = eτD A+B y(t0) , 0 ≤ τ≤ T − t0 , (3.2b)

see (0.6).

Numerical approximation. In accordance with the above relation for the exact solution, thenumerical approximation at time t1 is given by

y1 =ΦA+B (h, t0, y0) ≈ y(t1) = EA+B(h, t0, y(t0)

), h = h0 ,

with numerical solution operatorΦA+B .

3.1. Splitting methods

Approach. Exponential operator splitting methods rely on a decomposition of the right-handside of the differential equation (3.1) into two (or more) parts and the presumption that the

19

initial value problems

z ′(t ) = A(z(t )

), t0 ≤ t ≤ T , z(t0) given,

z ′(t ) = B(z(t )

), t0 ≤ t ≤ T , z(t0) given,

(3.3)

are solvable numerically in an accurate and efficient manner. The (approximate) solutions ofthe initial value problems (3.3) are then composed in a suitable way; this yields an approxima-tionΦA+B ≈ EA+B to the exact solution operator. For simplicity and in view of Gross–Pitaevskiisystems, we may assume that the exact solutions of (1.4)

z(t0 +τ) = EA(τ, t0, z(t0)

), 0 ≤ τ≤ T − t0 ,

z(t0 +τ) = EB(τ, t0, z(t0)

), 0 ≤ τ≤ T − t0 ,

are available.

General form of splitting method. Any exponential operator splitting method involvingseveral compositions can be cast into the following form

τ0 = t0 , Y0 = y0 ,

Yi = EB(bi h,τi−1 +ai h,EA

(ai h,τi−1,Yi−1

)), τi = τi−1 + (ai +bi )h , 1 ≤ i ≤ s ,

y1 = Ys ,

(3.4a)

yielding an approximation to the exact solution value

y(t1) = EA+B(h, t0, y(t0)

), h = h0 .

Employing the compact notation of Lie-derivative, we have

y1 =ΦA+B (h, t0, y0) =s∏

i=1eas+1−i h DA ebs+1−i h DB y0 , h = h0 ; (3.4b)

as before, the product is defined downwards. In comparison with (1.5), the order of thecompositions is reversed.

Explanation. Recall that by definition (0.6) it holds eτDF G y(t0) =G(EF

(τ, t0, y(t0)

)). Conse-

quently, setting G = ebi h DB , i.e., G(z(τ)

)= EB(bi h,τ, z(τ)

), it follows

eai h DA ebi h DB Yi−1 =G(EA(ai h,τi−1,Yi−1)

)= EB(bi h,τi−1 +ai h,EA(ai h,τi−1,Yi−1)

)= Yi .

By repetition we obtain (3.4b).

Examples. The first-order Lie–Trotter splitting method (1.6) and the second-order Strangsplitting method (1.7) can be cast into the general form (3.4). As well, methods of higher orderare included, see also Table 1.

20

3.2. Convergence analysis

Objective. In the following, we are concerned with deducing an estimate for the globalerror yN − y(T ) of an exponential operator splitting method (3.4) when applied to the initialvalue problem (3.1) To this purpose, as in the linear case, we follow a standard approach basedon a Lady Windermere’s Fan argument.

Local error and order. In the present situation, the local error equals

dn = D(hn−1, tn−1, y(tn−1)

)=ΦA+B

(hn−1, tn−1, y(tn−1)

)−EA+B(hn−1, tn−1, y(tn−1)

), 1 ≤ n ≤ N ,

see also (0.4). Thus, the numerical method (3.4) is consistent of order p iff

dn =O(hp+1

n−1

).

Again, it suffices to consider the case n = 1.

Lady Windermere’s Fan. For nonlinear differential equations, similarly as in the linear case,the global error fulfills the telescopic identity

yN − y(tN ) =N−1∏j=0ΦA+B (h j )

(y0 − y(t0)

)+ N∑n=1

N−1∏j=n

ΦA+B (h j )dn , (3.5)

see also (1.9); here, we employ the short notation

m−1∏j=k

ΦA+B (h j ) z(tk ) =ΦA+B

(hm−1, tm−1,ΦA+B

(. . . ,ΦA+B

(hk , tk , z(tk )

))), m > k ≥ 0.

In Section 3.2.1, we are concerned with extending the local error expansion of Section 1.2.2 tononlinear problems.

3.2.1. Local error expansion

Situation. For the following, to avoid technicalities, we consider exponential operator split-ting methods (3.4) that involve two compositions only

y1 =ΦA+B(h, t0, y(t0)

)= ea1h DA eb1h DB ea 2h DA eb 2 h DB y0 , h = h0 .

Method examples that can be cast into this form are the Lie–Trotter splitting method (1.6) andthe Strang splitting method (1.7).

21

Objective. We are concerned with deducing a suitable expansion of the defect

d1 =ΦA+B(h, t0, y(t0)

)−EA+B(h, t0, y(t0)

)= ea1h DA eb1h DB ea 2h DA eb 2 h DB y(t0) − eh D A+B y(t0) , h = h0 ,

with respect to h, see also (3.2) and (3.4).

Approach. For the solution of an initial value problem of the form (0.2), we employ a Taylorseries expansion and further express the arising derivatives of y by means of the function Fdefining the right-hand side of the differential equation; more precisely, using that y ′= F (y)and thus by the chain rule y ′′= F ′(y) y ′= F ′(y)F (y), we obtain

EF(τ, t0, y(t0)

)= y(t0)+τ y ′(t0)+ 12 τ

2 y ′′(t0)+O(τ3)

= y(t0)+τF(y(t0)

)+ 12 τ

2 F ′(y(t0))

F(y(t0)

)+O(τ3) , τ≥ 0,

(3.6)

with remainder depending on y ′′′.

Remark. The above relation (3.6) corresponds to the formal expansion

eτDF y(t0) =(I +τDF + 1

2 τ2D 2

F +O(τ3)) y(t0) ,

= y(t0)+τDF y(t0)+ 12 τ

2D 2F y(t0)+O

(τ3) .

Namely, applying definition (0.6), it follows

G(z) = DF z = F (z) , G ′(z) = F ′(z) ,

H(z) = D 2F z = (

DF G)(z) =G ′(z)F (z) = F ′(z)F (z) .

Expansion (Exact solution). Expanding the exact solution value by means of (3.6), yields

EA+B(h, t0, y(t0)

)= y(t0)+h(

A(y(t0)

)+B(y(t0)

))+ 1

2 h2(

A ′(y(t0))+B ′(y(t0)

))(A

(y(t0)

)+B(y(t0)

))+O(h3)

= y(t0)+h(

A(y(t0)

)+B(y(t0)

))+ 1

2 h2(

A ′(y(t0))

A(y(t0)

)+ A ′(y(t0))B

(y(t0)

)+B ′(y(t0))

A(y(t0)

)+B ′(y(t0)

)B

(y(t0)

))+O(h3) ,

which corresponds to the formal expansion

ehD A+B y(t0) = y(t0)+h DA+B y(t0)+ 12 h2D 2

A+B y(t0)+O(h3) .

22

Expansion (Splitting solution, single composition). We first consider a single composition

eai h DA ebi h DB z(τ) = EB(bi h, τ, z(τ)

), τ= τ+ai h , z(τ) = EA

(ai h,τ, z(τ)

).

The above relation (3.6) implies

EB(bi h, τ, z(τ)

)= z(τ)+O (h)

= z(τ)+bi h B(z(τ)

)+O(h2)

= z(τ)+bi h B(z(τ)

)+ 12 b2

i h2B ′(z(τ))

B(z(τ)

)+O(h3) .

In a similar manner, we have

z(τ) = EA(ai h,τ, z(τ)

)= z(τ)+O (h)

= z(τ)+ai h A(z(τ)

)+O(h2)

= z(τ)+ai h A(z(τ)

)+ 12 a2

i h2 A ′(z(τ))

A(z(τ)

)+O(h3) .

Consequently, by additional Taylor series expansions, it follows

B(z(τ)

)= B(z(τ)

)+O (h) = B(z(τ)

)+ai h B ′(z(τ))

A(z(τ)

)+O(h2) ,

B ′(z(τ))= B ′(z(τ)

)+O (h) , B ′(z(τ))

B(z(τ)

)= B ′(z(τ))

B(z(τ)

)+O (h) ,

wherefore we finally obtain

EB(bi h, τ, z(τ)

)= z(τ)+h(ai A

(z(τ)

)+bi B(z(τ)

))+h2

(12 a2

i A ′(z(τ))

A(z(τ)

)+ai bi B ′(z(τ))

A(z(τ)

)+ 12 b2

i B ′(z(τ))

B(z(τ)

))+O

(h3) .

Note that DF (z) = F (z), D2F z = F ′(z)F (z), and further

G(z) = DB z = B(z) , G ′(z) = B ′(z) , DADB z =G ′(z) A(z) = B ′(z) A(z) ;

we thus conclude that the formal expansion

eai hDA ebi hDB z(τ) = (I +ai h DA + 1

2 a2i h2D2

A

)(I +bi h DB + 1

2 b2i h2D2

B

)z(τ)+O

(h3)

=(I +h

(ai DA +bi DB

)+h2 (12 a2

i D2A +ai bi DADB + 1

2 b2i D2

B

))z(τ)+O

(h3)

= z(τ)+h(ai DA +bi DB

)z(τ)+h2(1

2 a2i D2

A +ai bi DADB + 12 b2

i D2B

)z(τ)

+O(h3)

= z(τ)+h(ai A

(z(τ)

)+bi B(z(τ)

))+h2

(12 a2

i A ′(z(τ))

A(z(τ)

)+ai bi B ′(z(τ))

A(z(τ)

)+ 12 b2

i B ′(z(τ))

B(z(τ)

))+O

(h3)

23

is in accordance with the above relation deduced by Taylor series expansions.

Expansion (Splitting solution). We next apply the previously verified formal expansion to asplitting method involving two compositions; this yields

ΦA+B(h, t0, y(t0)

)= ea1h DA eb1h DB ea 2h DA eb 2 h DB y(t0)

=(I +h

(a1DA +b1DB

)+h2 (12 a2

1D 2A +a1b1DADB + 1

2 b21D 2

B

))(I +h

(a 2DA +b 2DB

)+h2 (12 a2

2D 2A +a 2b 2DADB + 1

2 b22D 2

B

))y(t0)+O

(h3)

= y(t0)+h((a1 +a 2)DA + (b1 +b 2)DB

)y(t0)

+h2(

12

(a2

1 +a22

)D 2

A + (a1b1 +a 2b 2)DADB + 12

(b2

1 +b22

)D 2

B

+ (a1DA +b1DB

)(a 2DA +b 2DB

))y(t0)+O

(h3)

= y(t0)+h((a1 +a 2)DA + (b1 +b 2)DB

)y(t0)

+h2(

12 (a1 +a 2)2D 2

A + (a1(b1 +b 2)+a 2b 2

)DADB +b1a 2DB DA

+ 12 (b1 +b 2)2D 2

B

)y(t0)+O

(h3) .

Recalling the identities DF z = F (z), D 2F z = F ′(z)F (z), and DF DG z =G ′(z)F (z), we finally have

ΦA+B(h, t0, y(t0)

)= y(t0)+h((a1 +a 2)A

(y(t0)

)+ (b1 +b 2)B(y(t0)

))+h2

(12 (a1 +a 2)2 A ′(y(t0)

)A

(y(t0)

)+ 12 (b1 +b 2)2B ′(y(t0)

)B

(y(t0)

)+b1a 2 A ′(y(t0)

)B

(y(t0)

)+ (a1(b1 +b 2)+a 2b 2

)B ′(y(t0)

)A

(y(t0)

))+O

(h3) .

Expansion (Local error). Altogether, the above expansions of the exact and numerical solu-tion value imply

d1 =ΦA+B(h, t0, y(t0)

)−EA+B(h, t0, y(t0)

)= h

((a1 +a 2 −1)A

(y(t0)

)+ (b1 +b 2 −1)B(y(t0)

))+h2

(12

((a1 +a 2)2 −1

)A ′(y(t0)

)A

(y(t0)

)+ 12

((b1 +b 2)2 −1

)B ′(y(t0)

)B

(y(t0)

)+ (

b1a 2 − 12

)A ′(y(t0)

)B

(y(t0)

)+ (a1(b1 +b 2)+a 2b 2 − 1

2

)B ′(y(t0)

)A

(y(t0)

))+O

(h3) ,

and, as a consequence, we retain the first and second order conditions (1.18) from requiringd1 =O

(hp+1

)for p = 1 or p = 2, respectively.

24

Part II.

Fourier and Hermite spectral methods

25

Notation. Henceforth, we let N = m ∈ Z : m ≥ 0

. Further, we employ the multi-index

notation m = (m1, . . . ,md ) ∈ Zd and the compact vector notation x = (x1, . . . , xd ) ∈ Rd . Wedenote by ∂k

xi, 1 ≤ i ≤ d , the partial derivatives of order k and by ∆ = ∆x = ∂2

x1+ ·· ·+∂2

xdthe

d-dimensional Laplace operator. For a domainΩ⊂Rd , the Lebesgue space L2(Ω) = L2(Ω,C) ofsquare integrable complex-valued functions is endowed with standard scalar product (· | ·)L2

and corresponding norm ‖ ·‖L2 , defined by

(f∣∣g

)L2 =

∫Ω

f (x ) g (x ) dx ,∥∥ f

∥∥L2 =

√(f∣∣ f

)L2 , f , g ∈ L2(Ω) .

Objective. We are concerned with the efficient numerical solution of the linear partial differ-ential equation

i ∂tψ(x, t ) =A(x )ψ(x, t ) , x ∈Ω , t ≥ 0, (3.7)

involving a second order differential operator A . Regarding the spatial discretisation of non-linear Schrödiger equations by Fourier and Hermite spectral methods, we focus on the casesA =−∆ and A =−∆+Uγ, where Uγ denotes a scaled harmonic potential.

Approach. For solving (3.7), we make use of the fact that there exists a family(Bm

)m∈M

which forms a complete orthonormal system of the function space L2(Ω), i.e., for any elementϕ ∈ L2(Ω) the representation

ϕ= ∑m∈M

ϕm Bm , ϕm = (ϕ

∣∣Bm)L2 ,

holds. Moreover, the basis functions(Bm

)m∈M are eigenfunctions of the linear operator A ;

more precisely, the eigenvalue relation

A Bm =λm Bm , m ∈M ,

is valid with real eigenvalues (λm)m∈M . The above identity motivates the definition of a linearoperator F (A ) through

F (A )ϕ= ∑m∈M

ϕm F (A ) Bm = ∑m∈M

ϕm F (λm)Bm ;

for instance, the linear operator ecA is given by

ecAϕ= ∑m∈M

ϕm ecλm Bm .

By Parseval’s identity, provided that the sequence(ϕm F (λm)

)m∈M is square-summable, the

above definition is well-defined; in particular, for c ∈ iR it follows∥∥ecAϕ∥∥

L2 =∥∥ϕ∥∥

L2 , ϕ ∈ L2(Ω) .

27

As a consequence, the solution of (3.7) possesses the following representation

ψ(·, t ) = e− i t Aψ(·,0) = ∑m∈M

ψm(0) e− i t λm Bm , t ≥ 0, ψ(·,0) = ∑m∈M

ψm(0) Bm .

For the numerical realisation of the above relation, the infinite sum is truncated and thespectral coefficients ψm(0) are approximated by means of a quadrature formula.

28

4. Fourier spectral method

Objective. In the following, we are concerned with the numerical solution of the linear partialdifferential equation (3.7) involving the d-dimensional Laplace operator

A =−∆

on a bounded (symmetric) domainΩ= (−a1, a1)×·· ·× (−ad , ad ) ⊂Rd with ai > 0 (sufficientlylarge), 1 ≤ i ≤ d ; furthermore, we impose periodic boundary conditions. We first restrictourselves to the case d = 1 and then extend our considerations to arbitrary space dimensions.

4.1. Approach in one space dimension

Notation. For a > 0 we setΩ= (−a, a) ⊂R and further M =Z.

Approach. For the construction of the Fourier basis functions (Fm)m∈M and the derivation ofbasic relations we refer to [46]. Combining the theories of Sobolev spaces and selfadjoint linearoperators on Hilbert spaces, it is shown that the linear differential operator A =−∂ 2

x , subjectto periodic boundary conditions, is selfadjoint on a suitably chosen domain D(A ) ⊂ L2(Ω).Further, the corresponding eigenfunctions (Fm)m∈M , which form a complete orthonormalsystem of the function space L2(Ω), and the eigenvalues (λm)m∈M are determined.

4.1.1. Basic relations

Fourier basis functions. The Fourier basis functions (Fm)m∈M are given by

Fm(x ) = 1p2a

e iπm(

1a x+1

), x ∈Ω , m ∈M .

In particular, they fulfill the orthonormality relation(Fk

∣∣Fm)L2 = δkm , k,m ∈M . (4.1)

Fourier series expansion. The family (Fm)m∈M is complete in L2(Ω), i.e., for any functionϕ ∈ L2(Ω) the representation

ϕ= ∑m∈M

ϕm Fm , ϕm = (ϕ

∣∣Fm)L2 , (4.2)

29

holds with spectral coefficients (ϕm)m∈M ; the convergence of the infinite sum versus ϕ isensured in L2(Ω).

Parseval’s identity. The above relations (4.1) and (4.2) imply∥∥ϕ∥∥2L2 =

∑m∈M

∣∣ϕm∣∣2 , ϕ ∈ L2(Ω) . (4.3)

Eigenvalue relation. The differential operator A = −∂ 2x fulfills the following eigenvalue

relation with eigenfunctions (Fm)m∈M and associated eigenvalues (λm)m∈M

−∂ 2x Fm =λm Fm , λm = 1

a 2 m 2π2 , m ∈M . (4.4)

Explanations. The orthononality relation (4.1) also follows from a straightforward calculation

(Fk

∣∣Fm)L2 =

∫ a

−aFk (x ) Fm(x ) dx = 1

2a

∫ a

−ae iπ(k−m)

(1a x+1

)dx =

1, k = m ,

0 , k 6= m ,k,m ∈M .

Moreover, in an easy manner, the eigenvalues are obtained by differentiation

∂x Fm = i 1a mπFm , ∂ 2

x Fm =− 1a 2 m 2π2 Fm , m ∈M .

4.1.2. Discretisation

Notations. For an even integer number M > 0 we set MM = m ∈Z : −1

2 M ≤ m ≤ 12 M −1

and

further J = j ∈Z : 0 ≤ j ≤ M −1

.

Approach. We first consider a real-valued regular periodic function f :Ω→Rwith continuousextension toΩ; in particular, it holds f (−a) = f (a). For the quadrature approximation of theintegral ∫

Ωf (x ) dx

we apply the trapezoidal rule with equidistant nodes and corresponding weights (xj ,ωj )j∈J

∑j∈J

ωj f (xj ) ≈∫Ω

f (x ) dx , xj =−a + 2aM j , ωj = 2a

M , j ∈J . (4.5)

The above quadrature approximation extends to complex-valued function f :Ω→C by consid-ering the real and imaginary part of f .

Approximation of spectral coefficients. Note that

Fm(xj ) = 1p2a

e i 2π j mM , m ∈MM , j ∈J . (4.6)

30

An application of the trapezoidal rule (4.5) yields

ϕm = (ϕ

∣∣Fm)L2 =

∫Ωϕ(x ) Fm(x ) dx ≈ 2a

M

∑j∈J

ϕ(xj ) Fm(xj ) , m ∈MM ,

see also (4.2); we thus obtain the following approximations to the Fourier spectral coefficients

ϕm ≈p

2aM

∑j∈J

ϕ(xj ) e− i 2π j mM , m ∈MM . (4.7)

Approximation of function values. On the other hand, from the Fourier spectral coeffi-cients (ϕm)m∈MM approximations to the values of ϕ at the grid points (xj )j∈J are retainedthrough

ϕ(xj ) ≈ 1p2a

∑m∈MM

ϕm e i 2π j mM , j ∈J , (4.8)

see also (4.2) and (4.6).

4.1.3. Implementation

Notations. As before, for an even integer M > 0 we set MM = m ∈Z : −1

2 M ≤ m ≤ 12 M −1

and further J =

j ∈Z : 0 ≤ j ≤ M −1.

Implementation. The efficient implementation of the Fourier spectral method relies onFast Fourier Techniques. In the following, we discuss the realisation of the pseudo-spectraltransformations (4.7) and (4.8) in MATLAB. For notational simplicity, we do not employ differ-ent notations for the exact spectral coefficients and the numerical approximations obtainedthrough (4.7); similarly, we do not distinguish between the function values and the numericalapproximations (4.8). Tilded letters correspond to quantities in MATLAB.

Grid points. For the Fourier pseudo-spectral transformations, we employ a collocation at thetrapezoidal quadrature nodes

xj+1 = xj , j ∈J .

Real to spectral. For given function values

ϕj+1 =ϕ(xj ) , j ∈J ,

31

approximations to the spectral coefficients (ϕm)m∈MM are computed through (4.7)

ϕm =p

2aM

∑j∈J

ϕ(xj ) e− i 2π j mM =

p2a

M

M∑j=1

ϕ(xj−1) e− i 2π ( j−1)mM

=p

2aM

M∑j=1

ϕj e− i 2π ( j−1)mM , m ∈MM .

Note that the periodicity of the Fourier basis functions and the (tacitly assumed) periodicityof ϕ implies ϕm+`M =ϕm for any ` ∈Z. In MATLAB, an application of the command fft resultsin

fft(ϕ1, . . . ,ϕM

)= (ϕ(s)

1 , . . . ,ϕ(s)M

), ϕ(s)

k =M∑

j=1ϕj e− i 2π ( j−1)(k−1)

M , 1 ≤ k ≤ M .

A comparison of the above relations shows that(ϕ(s)

1 , . . . ,ϕ(s)M

) = Mp2a

(ϕ0, . . . ,ϕ1

2 M−1,ϕ− 12 M , . . . ,ϕ−1

).

Altogether, with the help of the command fftshift which swaps the left and right halves of avector, we obtain approximations to the spectral coefficients through

p2a

M fftshift(fft

(ϕ1, . . . ,ϕM

)) = p2a

M fftshift(ϕ(s)

1 , . . . ,ϕ(s)M

)= fftshift

(ϕ0, . . . ,ϕ1

2 M−1,ϕ− 12 M , . . . ,ϕ−1

)= (ϕ− 1

2 M , . . . ,ϕ−1,ϕ0, . . . ,ϕ12 M−1

).

Spectral to real. On the other hand, starting with given spectral coefficients(ϕ− 1

2 M , . . . ,ϕ−1,ϕ0, . . . ,ϕ12 M−1

)= (ϕ1

2 M , . . . ,ϕM−1,ϕ0, . . . ,ϕ12 M−1

)= (

ϕ(s)12 M+1

, . . . ,ϕ(s)M ,ϕ(s)

1 , . . . ,ϕ(s)12 M

),

an application of the command ifft results in

ifft(ϕ(s)

1 , . . . ,ϕ(s)M

)= (ϕ1, . . . ,ϕM

), ϕk = 1

M

M∑j=1

ϕ(s)j e i 2π ( j−1)(k−1)

M , 1 ≤ k ≤ M .

Moreover, making use of the fact that

Mp2aϕk = 1p

2a

M∑j=1

ϕ(s)j e i 2π ( j−1)(k−1)

M = 1p2a

M∑j=1

ϕj−1 e i 2π ( j−1)(k−1)M

= 1p2a

∑j∈J

ϕj e i 2π j (k−1)M = 1p

2a

∑m∈MM

ϕm e i 2π m(k−1)M =ϕ(xk−1) , 1 ≤ k ≤ M ,

32

and that the command ifftshift swaps the left and right halves of a vector, approximationsto the function values are obtained through

Mp2a

ifft(ifftshift

(ϕ− 1

2 M , . . . ,ϕ−1,ϕ0, . . . ,ϕ12 M−1

)= Mp

2aifft

(ifftshift

(ϕ(s)

12 M+1

, . . . ,ϕ(s)M ,ϕ(s)

1 , . . . ,ϕ(s)12 M

)= Mp2a

ifft(ϕ(s)

1 , . . . ,ϕ(s)M

)= Mp

2a

(ϕ1, . . . ,ϕM

) = (ϕ(x 0), . . . ,ϕM−1(xM−1)

).

4.2. Approach in several space dimensions

Notation. For ai > 0, 1 ≤ i ≤ d , we setΩ= (−a1, a1)×·· ·× (−ad , ad ) ⊂Rd and further M =Zd .

Approach. The considerations for one space dimensions are extended to the general case.


Fourier basis functions. In d space dimensions, the Fourier basis functions (Fm)m∈M aregiven by

Fm(x ) =Fm1(x1) · · · Fmd(xd ) , x ∈Ω , m ∈M ,

Fmi(xi ) = 1p2ai

eiπmi

(1

aix i+1

), 1 ≤ i ≤ d .

In particular, the orthonormality relation (4.1) holds.

Fourier series expansion. The family (Fm)m∈M is complete in L2(Ω), i.e., for any functionϕ ∈ L2(Ω) the representation (4.2) holds with spectral coefficients (ϕm)m∈M .

Parseval’s identity. Relations (4.1) and (4.2) imply (4.3).

Eigenvalue relation. The Laplace operator A(x ) =−∆ fulfills the following eigenvalue rela-tion with eigenfunctions (Fm)m∈M and associated eigenvalues (λm)m∈M

−∆Fm =λm Fm , λm =d∑

i=1λmi =π2

d∑i=1

1a 2

im 2

i , m ∈M . (4.9)

Explanation. Due to the fact that

−∆Fm =−Fm2 · · · Fmd ∂2x1

Fm1 −·· ·−Fm1 · · · Fmd−1 ∂2xd

Fmd = (λm1 +·· ·+λmd

)Fm ,

the above relation follows by means of (4.4).

33


Notations. For M ∈Nd with Mi > 0 an even integer number for 1 ≤ i ≤ d , we set

MM = m ∈Zd : −1

2 Mi ≤ mi ≤ 12 Mi −1,1 ≤ i ≤ d

,

J = j ∈Zd : 0 ≤ ji ≤ Mi −1,1 ≤ i ≤ d

.

Further, we employ the short notation

ec j ·mM = e

c∑d

i=1ji miMi , c ∈C , j ∈J , m ∈MM .

Approach. We consider a complex-valued regular periodic function f : Ω→ Cd with con-tinuous extension to Ω, i.e. it holds f (−a) = f (a). For the quadrature approximation of themultiple integral ∫

Ωf (x ) dx

we apply the trapezoidal rule with equidistant nodes and corresponding weights (xj ,ωj )j∈J ,which are given by the quadrature nodes and weights of the one-dimensional trapezoidal rule

∑j∈J


f (x ) dx , xj = (xj1 , . . . , xjd ) , ωj =ωj1 · · ·ωjd , j ∈J ,

xji =−ai + 2aiMi

ji , ωji = 2aiMi

, 1 ≤ ji ≤ Mi , 1 ≤ i ≤ d .

(4.10)

Approximation of spectral coefficients. Note that

Fm(xj ) =d∏

i=1

1p2ai

e i 2π j ·mM , m ∈MM , j ∈J . (4.11)

An application of the trapezoidal rule (4.10) yields the following approximations to the Fourierspectral coefficients

ϕm ≈d∏

i=1

p2ai

Mi

∑j∈J

ϕ(xj ) e− i 2π j ·mM , m ∈MM , (4.12)

see also (4.2).

Approximation of function values. On the other hand, from the Fourier spectral coeffi-cients (ϕm)m∈MM approximations to the values of ϕ at the grid points (xj )j∈J are retainedthrough

ϕ(xj ) ≈d∏

i=1

1p2ai

∑m∈MM

ϕm e i 2π j ·mM , j ∈J , (4.13)

see also (4.2) and (4.11).

34


Notations. As before, for M ∈Nd with Mi > 0 an even integer number for 1 ≤ i ≤ d , we set

MM = m ∈Zd : −1

2 Mi ≤ mi ≤ 12 Mi −1,1 ≤ i ≤ d

,

J = j ∈Zd : 0 ≤ ji ≤ Mi −1,1 ≤ i ≤ d

.

Implementation. It is straightforward to extend the considerations of Section 4.1.3 to severalspace dimensions; again, the pseudo-spectral transformations (4.12) and (4.13) are realised byFast Fourier Techniques.

Real to spectral. In several space dimensions, approximations to the spectral coefficients areobtained through (

ϕm) = d∏

i=1

p2ai

Mifftshift

(fftn

(ϕ(xj )

)).

Spectral to real. In several space dimensions, approximations to the function values areobtained through (

ϕ(xj )) = d∏

i=1

Mip2ai

ifftn(ifftshift

(ϕm

)).

35

5. Hermite spectral method

Objective. In the following, we are concerned with the numerical solution of the linear partialdifferential equation (3.7) involving the second order differential operator

A(x ) =−∆+Uγ(x ) , Uγ(x ) =d∑

i=1γ4

i x 2i , γi > 0, 1 ≤ i ≤ d , (5.1)

on the unbounded domainΩ=Rd ; furthermore, we impose asymptotic boundary conditions.We first restrict ourselves to the case d = 1 and then extend our considerations to arbitraryspace dimensions.

5.1. Approach in one space dimension

Notation. We setΩ=R and M =N ; further, we denote by γ> 0 a positive weight.

Approach. As before, we also refer to [46] for the construction of the Hermite basis func-tions (H γ

m)m∈M and the derivation of basic relations. Combining the theories of Sobolevspaces and selfadjoint linear operators on Hilbert spaces, it is shown that the linear differentialoperator A(x ) =−∂ 2

x +γ4x 2, subject to asymptotic boundary conditions, is selfadjoint on asuitably chosen domain D(A ) ⊂ L2(Ω). Further, the corresponding eigenfunctions (H γ

m)m∈M ,which form a complete orthonormal system of the function space L2(Ω), and the eigenval-ues (λm)m∈M are determined.

5.1.1. Hermite basis functions

Objective. In the following, we are concerned with constructing the orthonormal Hermitebasis functions

(H

γm

)m∈M which fulfill the eigenvalue relation

A Hγ

m =λm Hγ

m , m ∈M , (5.2)

with associated eigenvalues (λm)m∈M .

Ladder operators. The construction of the Hermite basis functions is based on the approachof ladder operators The algebraic identity a 2 −b 2 = (a −b)(a +b) motivates the consideration

37

of the differential operators

A(x ) =−∂ 2x +γ4x 2 , P (x ) = ∂x +γ2x , Q(x ) =−∂x +γ2x .

Although the operators P and Q do not commute, that is, it holds A 6=QP and A 6=P Q, wemay take advantage of the fact that the product of P and Q is close to A ; more precisely, wehave

QP =A −γ2I , P Q =A +γ2I ,

and, as a consequence, we further obtain

PA = (A +2γ2I

)P , QA = (

A −2γ2I)Q .

Explanations. For a regular function y , it follows

Q(x )P (x ) y(x ) = (−∂x +γ2x)(∂x y(x )+γ2x y(x )

)=−∂ 2

x y(x )−γ2(y(x )±x ∂x y(x ))+γ4x 2 y(x ) =A(x ) y(x )−γ2 y(x ) ,

P (x )Q(x ) y(x ) = (∂x +γ2x

)(−∂x y(x )+γ2x y(x ))

=−∂ 2x y(x )+γ2(y(x )±x ∂x y(x )

)+γ4x 2 y(x ) =A(x ) y(x )+γ2 y(x ) .

Hence, using that A =QP +γ2I =P Q−γ2I we obtain

PA =P(QP +γ2I

)= (P Q+γ2I

)P = (

A +2γ2I)P ,

QA =Q(P Q−γ2I

)= (QP −γ2I

)Q = (

A −2γ2I)Q .

First Hermite basis function. The first Hermite basis function H γ0 is related to the weight

function w(x ) = e−12 γ

2x 2; namely, using that P w = 0 it follows A w = (

QP +γ2I)

w = γ2w .Due to the fact that ∥∥w

∥∥2L2 =

∫Ω

e− γ2x 2

dx =√

πγ2 ,

it is seen that the first normalised Hermite basis function Hγ

0 is given by

Hγ

0 (x ) = 4

√γ2

π e−12 γ

2x 2, x ∈Ω ,

with associated eigenvalue λ0 = γ2.

Preliminaries. We first note that by partial integration the relation∫Ω

(∂xH

γm(x )

)2 dx =−∫Ω

Hγ

m(x )∂ 2x H

γm(x ) dx

38

follows. Thus, the eigenvalue relation (5.2) and the normalisation condition∥∥H

γm

∥∥L2 = 1 imply∫

Ω

(∂x H

γm(x )

)2 +γ4x 2(H γm(x )

)2 dx =∫Ω

Hγ

m(x )(− ∂ 2

x Hγ

m(x )+γ4x 2Hγ

m(x ))

dx

=∫Ω

Hγ

m(x )A(x )H γm(x ) dx =λm

∫Ω

(H

γm(x )

)2 dx =λm ;

in a similar manner, by partial integration it follows∫Ω

x Hγ

m(x )∂xHγ

m(x ) dx =− 12 .

Altogether, we obtain the following identities

∥∥QHγ

m

∥∥2L2 =

∫Ω

(−∂xHγ

m(x )+γ2x Hγ

m(x ))2 dx

=∫Ω

(∂xH

γm(x )

)2 −2γ2x Hγ

m(x )∂xHγ

m(x )+γ4x 2(H γm(x )

)2 dx =λm +γ2 ,∥∥P Hγ

m

∥∥2L2 =

∫Ω

(∂xH

γm(x )+γ2x H

γm(x )

)2 dx

=∫Ω

(∂xH

γm(x )

)2 +2γ2x Hγ

m(x )∂xHγ

m(x )+γ4x 2(H γm(x )

)2 dx =λm −γ2 .

Up. We consider the eigenvalue relation (5.2) for the m-th Hermite basis function Hγ

m withcorresponding eigenvalue λm . Applying the operator Q and making use of the previouslyderived relation QA = (

A −2γ2I)Q, we obtain

AQHγ

m = (λm +2γ2) QH

γm ;

that is, QHγ

m is also an eigenfunction of A with associated eigenvalue λm+1 =λm +2γ2. Dueto the fact that λ0 = γ2, it follows

λm = γ2(1+2m) , m ∈M .

The above considerations imply∥∥QH

γm

∥∥2L2 = 2(m +1)γ2 and thus H

γm+1 = 1

‖QHγ

m‖L2QH

γm , i.e.

Hγ

m+1(x ) = 1p2(m+1) γ

(− ∂xHγ

m(x )+γ2x Hγ

m(x ))

, x ∈Ω . (5.3)

Down. We consider the eigenvalue relation (5.2). Applying the operator P and employingthe relation PA = (

A +2γ2I)P , we obtain

A P Hγ

m = (λm −2γ2) P H

γm ;

39

that is, P Hγ

m is also eigenfunction of A with associated eigenvalue λm −2γ2. Due to the factthat

∥∥P Hγ

m

∥∥2L2 = 2mγ2 and H

γm−1 = 1

‖P Hγ

m‖L2P H

γm , we thus have

Hγ

m−1(x ) = 1p2m γ

(∂xH

γm(x )+γ2x H

γm(x )

), x ∈Ω . (5.4)

Recall that P Hγ

0 = 0, i.e. Hγ

0 is indeed the first eigenfunction of A .

Recurrence relation. The identities (5.3) and (5.4) yield the recurrence relation

Hγ

0 (x ) = 4

√γ2

πe−

12 γ

2x 2, H

γ1 (x ) = 4

√4γ6

πx e−

12 γ

2x 2,

Hγ

m+1(x ) = 1pm+1

(p2 γx H

γm(x )−p

m Hγ

m−1(x ))

, m ≥ 1, x ∈Ω .(5.5)

Note that Hγ

m(x ) is of the form

Hγ

m(x ) = H γm(x ) e−

12 γ

2x 2, x ∈Ω , m ∈M ,

with H γm a polynomial of degree m. Clearly, the Hermite polynomials

(H γ

m)m∈M also fulfill the

recurrence relation in (5.5).


Hermite basis functions. The Hermite basis functions(H

γm

)m∈M are given by

Hγ

m(x ) = H γm(x ) e−

12 γ

2x 2, x ∈Ω , m ∈M ; (5.6a)

here, we denote by H γm the m-th Hermite polynomial which fulfills the recurrence relation

H γ0 (x ) = 4

√γ2

π , H γ1 (x ) = 4

√4γ6

π x ,

H γm+1(x ) = 1p

m+1

(p2 γx H γ

m(x )−pm H γ

m−1(x ))

, m ≥ 1, x ∈Ω ,(5.6b)

see also (5.5). The Hermite basis functions satisfy the orthonormality relation(H

γ

k

∣∣H γm

)L2 = δkm , k,m ∈M . (5.7)

Hermite series expansion. The family(H

γm

)m∈M is complete in L2(Ω), i.e., for any function

ϕ ∈ L2(Ω) the representation

ϕ= ∑m∈M

ϕm Hγ

m , ϕm = (ϕ

∣∣H γm

)L2 , (5.8)

40

holds with spectral coefficients (ϕm)m∈M ; the convergence of the infinite series versus ϕ isensured in L2(Ω).

Parseval’s identity. The above relations (5.7) and (5.8) imply (4.3).

Eigenvalue relation. The space dependent differential operator A(x ) =−∂ 2x +γ4x 2 fulfills

the following eigenvalue relation with eigenfunctions(H

γm

)m∈M and eigenvalues (λm)m∈M(− ∂ 2

x +γ4x 2) Hγ

m(x ) =λm Hγ

m(x ) , λm = γ2(1+2m) , x ∈Ω , m ∈M .


Notations. For a positive integer K > 0 we set MM = m ∈ N : 0 ≤ m ≤ M −1

and further

J = j ∈N : 0 ≤ j ≤ K −1

.

Approach. We first consider a real-valued regular function f :Ω→ R. For the quadratureapproximation of an integral of the form∫

Ωf (x ) e− γ

2x 2dx

we apply the Gauß–Hermite quadrature formula with nodes and weights (xj ,ωj )j∈J

∑j∈J


f (x ) e− γ2x 2

dx . (5.9)

The above quadrature approximation extends to complex-valued function f :Ω→C by consid-ering the real and imaginary part of f .

Approximation of spectral coefficients. An application of the Gauß–Hermite quadratureformula (5.9) yields

ϕm = (ϕ

∣∣H γm

)L2 =

∫Ωϕ(x )H γ

m(x ) dx =∫Ω

e12 γ

2x 2ϕ(x ) H γ

m(x ) e− γ2x 2

dx

≈ ∑j∈J

ωj e12 γ

2x2j ϕ(xj ) H γ

m(xj ) , m ∈MM ,

see also (5.8); we thus obtain the following approximations to the Hermite spectral coefficients

ϕm ≈ ∑j∈J

ωj e12 γ

2x2j ϕ(xj ) H γ

m(xj ) , m ∈MM . (5.10)

41

Approximation of function values. On the other hand, from the Hermite spectral coeffi-cients (ϕm)m∈MM approximations to the values of ϕ at the grid points (xj )j∈J are retainedthrough

ϕ(xj ) ≈ ∑m∈MM

ϕm Hγ

m(xj ) , j ∈J , (5.11)

see also (5.8).

5.1.4. Gauß–Hermite quadrature formula

Order. We recall that a quadrature formula is said to be of order p iff the quadrature ap-proximation yields the exact result for any polynomial f with deg f ≤ p −1. In particular, theGauß–Hermite quadrature formula (xj ,ωj )j∈J is said to be of order p iff

∑j∈J

ωj f (xj ) =∫Ω

f (x ) w(x)dx , deg f ≤ p −1, (5.12)

with weight function w(x ) = e−γ2x 2

, x ∈Ω.

Approach. The construction of the Gauß–Hermite quadrature formula (xj ,ωj )j∈J is in thelines of the construction of the Gauß quadrature formula. For the Gauß–Hermite quadratureformula, the associated orthogonal polynomials are the Hermite polynomials

(H γ

m)

m∈N, i.e., itholds (

H γ

k w∣∣H γ

m)L2 =

(H

γ

k

∣∣H γm

)L2 = δkm , k,m ∈N ,

see also (5.7). The Gauß–Hermite quadrature nodes (xj )j∈J are the roots of H γ

K ; the correspond-ing weights (ωj )j∈J are obtained through the order conditions for order K . By construction,the Gauß–Hermite quadrature formula is of order 2K .

Computation of quadrature nodes. The Gauß–Hermite quadrature nodes (xj )j∈J are com-puted numerically through the solution of an eigenvalue problem; this approach is closelyrelated to Sturm’s chains. Namely, we make use of the fact that the characteristic polynomialassociated with a symmetric tridiagonal matrix

Am =

a1 b1

b1 a 2 b 2

. . .

bm−1 a m

∈Rm×m (5.13a)

fulfills a three-term recurrence relation. More precisely, we consider the polynomial

χm :R→R : x 7−→χ(x) = cm det(Am −xI ) , m ≥ 1,

42

with leading coefficient cm . Expanding the determinant of Am − xI with respect to the last rowand with respect to the last column, respectively, yields

det(

Am+1 −xI)= det

a1 −x b1

b1 a 2 −x b 2. . .

. . .bm−2 a m−1 −x bm−1

bm−1 a m −x bm

bm a m+1 −x

= (a m+1 −x) det(

Am −xI)− bm det

a1 −x b1

. . .

bm−2 a m−1 −xbm−1 bm

= (a m+1 −x) det

(Am −xI

)− b2m det

(Am−1 −xI

),

and, as a consequence, the recurrence relation

χ0(x ) = c0 , χ1(x ) = c1 (a1 −x) ,

χm+1(x ) = dm+1((a m+1 −x)χm(x )−dm b 2

m χm−1(x ))

, m ≥ 1,

follows, where dm = cmcm−1

for m ≥ 1. Comparing the above relation with the recurrence rela-

tion (5.6b) for the Hermite polynomials, we conclude that χm = H γm , provided that

c0 = 4

√γ2

π , a m = 0, bm = 1γ

√m2 , m ≥ 1. (5.13b)

Thus, the Gauß–Hermite quadrature nodes (xj )j∈J , that is, the roots of the K -th Hermitepolynomial, coincide with the eigenvalues of the associated matrix AK , see (5.13a). Note that

for any xj , j ∈J , the function values(H γ

0 (xj ), . . . , H γ

K−1(xj ))T form an eigenvector of AK with

associated eigenvalue xj .

Computation of quadrature weights. Inserting the Hermite polynomials(H γ

m)m∈J into the

order conditions (5.12) and applying the orthonormality relation (5.7), yields the followingsystem of linear equations for the Gauß–Hermite quadrature weights (ωj )j∈J

Hω= 4√

πγ2 e1 ,

H =

H γ0 (x 0) . . . H γ

0 (xK−1)...

...H γ

K−1(x 0) . . . H γ

K−1(xK−1)

, ω= (ω0, . . . ,ωK−1

)T , e1 =(1,0, . . . ,0

)T ∈Rk .

43

Furthermore, due to the fact that

H TH = D = diag(d0, . . . ,dK−1

), dj =

∑m∈J

(H γ

m(xj ))2 , j ∈J ,

H Te1 =(H γ

0 (x 0), . . . , H γ0 (xK−1)

)T = 4

√γ2

π

(1, . . . ,1

)T , D−1(1, . . . ,1)T = ( 1

d0, . . . , 1

dK−1

)T ,

it follows ω= 4√

πγ2 D−1H Te1, that is

ω= ( 1d0

, . . . , 1dK−1

)T .

Interpolation. The above considerations further imply∑m∈J

H γm(xj ) H γ

m(xk ) = 1ωjδj k , j ,k ∈J .

As a consequence, for any (regular) function of the form

ϕ= ∑m∈M

ϕm Hγ

m

the following interpolatory relation at the quadrature nodes

ϕ(xk ) = ∑m∈J

ϕm Hγ

m(xk ) , ϕm = ∑j∈J

ωj e12 γ

2x2j ϕ(xj ) H γ

m(xj ) , m ∈J ,

follows. Namely, a short argument shows that∑m∈J

ϕm Hγ

m(xk ) = ∑m∈J

∑j∈J

ωj e12 γ

2(x2j −x2

k )ϕ(xj ) H γ

m(xj ) H γm(xk )

= ∑j∈J

ωj e12 γ

2(x2j −x2

k )ϕ(xj )

∑m∈J

H γm(xj ) H γ

m(xk ) =ϕ(xk ) .


Notations. As before, for integers M ,K > 0 we set MM = m ∈N : 0 ≤ m ≤ M −1

as well as

J = j ∈N : 0 ≤ j ≤ K −1

.

Approach. The implementation of the Hermite spectral transformations (5.10) and (5.11) inMATLAB relies on matrix–matrix multiplications. For notational simplicity, we do not employdifferent notations for the exact spectral coefficients or function values and the numericalapproximations.

44

Preliminaries. Clearly, the multiplication of a full matrix A by a diagonal matrix D yields

A D =a11 a12 a13

a21 a22 a23

a31 a32 a33

d1

d2

d3

=d1a11 d2a12 d3a13

d1a21 d2a22 d3a23

d1a31 d2a32 d3a33

,

D A =d1

d2

d3

a11 a12 a13

a21 a22 a23

a31 a32 a33

=d1a11 d1a12 d1a13

d2a21 d2a22 d2a23

d3a31 d3a32 d3a33

.

Real to spectral. For given function values ϕ= (ϕ(x 0), . . . ,ϕ(xK−1)

)T approximations to the

spectral coefficients ϕ(s) = (ϕ0, . . . ,ϕM−1

)T are computed through

ϕ(s) =Tr2sϕ , Tr2s = H Dr2s ,

H =

H γ0 (x 0) . . . H γ

0 (xK−1)...

...H γ

M−1(x 0) . . . H γ

M−1(xK−1)

, Dr2s =

ω0 e

12γ

2x 20

. . .

ωK−1 e12γ

2x 2K−1

,

see also (5.10).

Spectral to real. On the other hand, for given spectral coefficients ϕ(s) approximations to thefunction values ψ are computed through

ϕ=Ts2rϕ(s) , Ts2r = Ds2r H T ,

Ds2r =

e−

12γ

2x 20

. . .

e−12γ

2x 2K−1

,

see also (5.11).

5.2. Approach in several space dimensions

Notation. We setΩ=Rd and further M =Nd .

Approach. The considerations for one space dimensions are extended to the general case.


Hermite basis functions. In d space dimensions, the Hermite basis functions(H

γm

)m∈M are

given byH

γm(x ) =H

γ1m1

(x1) · · · Hγd

md(xd ) , x ∈Ω , m ∈M ,

45

see also (5.6). In particular, the orthonormality relation (5.7) holds.

Hermite series expansion. The family(H

γm

)m∈M is complete in L2(Ω), i.e., for any function

ϕ ∈ L2(Ω) the representation (5.8) holds with spectral coefficients (ϕm)m∈M .

Parseval’s identity. Relations (5.7) and (5.8) imply (4.3).

Eigenvalue relation. The differential operator A =−∆+Uγ, see also (5.1), fulfills the follow-ing eigenvalue relation with eigenfunctions

(H

γm

)m∈M and associated eigenvalues (λm)m∈M

(− ∆+Uγ

)H

γm =λm H

γm , λm =

d∑i=1

λmi =d∑

i=1γ2

i (1+2mi ) , m ∈M . (5.14)


Notations. For K ∈Nd with Ki > 0 for 1 ≤ i ≤ d we set

MM = m ∈Nd : 0 ≤ mi ≤ Mi −1,1 ≤ i ≤ d

, J =

j ∈Nd : 0 ≤ ji ≤ Ki −1,1 ≤ i ≤ d

.

Further, we employ the short notation

ec γ2·x 2 = ec∑d

i=1γ2

i x 2i , c ∈R , γ ∈Rd , x ∈Ω .

Approach. We consider a complex-valued regular function f :Ω→Cd . For the quadratureapproximation of a multiple integral of the form∫

Ωf (x ) e− γ

2·x 2dx

we apply the Gauß–Hermite quadrature formula with nodes and weights (xj ,ωj )j∈J given bythe quadrature nodes and weights of the one-dimensional Gauß–Hermite quadrature formula

∑j∈J


f (x ) e− γ2·x 2

dx , xj = (xj1 , . . . , xjd ) , ωj =ωj1 · · ·ωjd , j ∈J . (5.15)

Approximation of spectral coefficients. An application of the Gauß–Hermite quadratureformula (5.15) yields the following approximations to the Hermite spectral coefficients

ϕm ≈ ∑j∈J

ωj e12 γ

2·x 2ϕ(xj ) H γ

m(xj ) , m ∈MM , (5.16)

see also (5.8).

46

Approximation of function values. On the other hand, from the Hermite spectral coeffi-cients (ϕm)m∈MM approximations to the values of ϕ at the grid points (xj )j∈J are retainedthrough (5.11), see also (5.8).

5.2.3. Approximation result

Approximation result. A result on the accuracy of the Hermite spectral method is found inthe recent work [26]. For M ∈Nd with Mi > 0 for 1 ≤ i ≤ d , we set

M max = max

Mi : 1 ≤ i ≤ d

,

MM = m ∈Nd : 0 ≤ mi ≤ Mi −1,1 ≤ i ≤ d

, J =

j ∈Nd : 0 ≤ ji ≤ Mi −1,1 ≤ i ≤ d

.

Moreover, we employ the notations

ϕ= ∑m∈MM

ϕm Hγ

m , ϕm = ∑j∈J

ωj e12 γ

2·x 2ϕ(xj ) H γ

m(xj ) , m ∈MM .

Then, the following spatial error estimate is valid

∥∥A α(ϕ−ϕ)∥∥

L2 ≤C M−(β−α−d

3

)max

∥∥A βϕ∥∥

L2 .


Approach. Especially, for two space dimensions an efficient implementation of the Hermitespectral transformations (5.16) and (5.11) relies on matrix–matrix multiplications.

47

Part III.

Time integration ofGross–Pitaevskii systems

49

6. Gross–Pitaevskii systems

6.1. Original formulation

Gross–Pitaevskii systems. In certain respects, a multi-component Bose–Einstein condensateis well described by a system of J coupled Gross–Pitaevskii equations

iħ ∂tΨj (x, t ) =(− ħ2

2mj∆+Vj (x )+ħ2

J∑k=1

gj k∣∣Ψk (x, t )

∣∣2)Ψj (x, t ) ,∥∥Ψj (·,0)

∥∥2L2 = Nj , x ∈Rd , t ≥ 0, 1 ≤ j ≤ J .

(6.1)

Here, we denote byΨj :Rd×R≥0 →C : (x, t ) 7→Ψj (x, t ) the order parameters (wave functions),by ħ≈ 1.054571628 ·10−34 Planck’s constant, by mj the masses of the atomic species, and by Nj

their total particle numbers, 1 ≤ j ≤ J . In the most relevant case of three space dimensions, theintra-species coupling constants gj j and the inter-species coupling constants gj k are given

by gj k = 2πσj kmj+mk

mj mkwith scattering lengths σj k , where σj k = σk j , 1 ≤ j ,k ≤ J . The external

trapping potentials are descreibed by real-valued functions Vj : Rd → R, 1 ≤ j ≤ J . Further,the partial differential equations in (6.1) are subject to asymptotic boundary conditions andcertain initial conditions.

External potentials. In many cases, the external trapping potentials are modelled by scaledharmonic potentials

Vj (x ) = mj

2

d∑i=1

ω2ji (x i −ζj i )2 , x ∈Rd , 1 ≤ j ≤ J ,

with positive weights ωji > 0 and center displacements ζj i ∈R, 1 ≤ i ≤ d , 1 ≤ j ≤ J .

6.2. Normalised formulation

Normalisation. Employing a linear transformation of the spatial variable

ξ=pc x , c = 1

ħJp

m1 · · ·mJ ,

ψj (ξ, t ) = 1C Ψj (x, t ) , C = 4

√cd , Uj (ξ) = 1

ħ Vj (x ) , ϑj k =ħC 2gj k , 1 ≤ j ,k ≤ J ,

51

we obtain the following normalised formulation of the Gross–Pitaevskii system (6.1)

i ∂tψj (ξ, t ) =(− cj ∆+Uj (ξ)+

J∑k=1

ϑj k∣∣ψk (ξ, t )

∣∣2)ψj (ξ, t ) , cj =

Jpm1···mJ

2mj,∥∥ψj (·,0)

∥∥2L2 = Nj , ξ ∈Rd , t ≥ 0, 1 ≤ j ≤ J .

(6.2)

In accordance with (6.1), the partial differential equations in (6.2) are subject to asymptoticboundary conditions and certain initial conditions. The constant C is chosen such that thetotal particle numbers are conserved.

Explanation. Multiplying the partial differential equation in (6.1) with 1ħ yields

i ∂tΨj (x, t ) =(− ħ

2mj∆+ 1

ħ Vj (x)+ħJ∑

k=1gj k

∣∣Ψk (x, t )∣∣2

)Ψj (x, t ) .

Note that ∂xi =p

c ∂ξi , 1 ≤ i ≤ d , and thus ∆x = c∆ξ; therefore, substitutingΨj (x, t ) =C ψj (ξ, t )as well as Vj (x) =ħUj (ξ) gives

iC ∂tψj (ξ, t ) =C(− ħc

2mj∆+Uj (ξ)+ħC 2

J∑k=1

gj k∣∣ψk (ξ, t )

∣∣2)ψj (ξ, t ) .

Multiplying with 1C and using that ħc = J

pm1 · · ·mJ and ϑj k =ħC 2gj k , yields the partial differ-

ential equation in (6.2). Further, due to dξ=p

cd dx =C 2 dx, it follows

∥∥ψj (·,0)∥∥2

L2 =∫Rd

∣∣ψj (ξ,0)∣∣2 dξ=

∫Rd

∣∣Ψj (x,0)∣∣2 dx = ∥∥Ψj (·,0)

∥∥2L2 .

6.3. Special case

Special case. As an illustration, we specify a Gross–Pitaevskii system in three space dimen-sions involving two coupled equations, that is, we set J = 2 and d = 3 in (6.1) and obtain

iħ ∂tΨ1(x, t ) =(− ħ2

2m1∆+V1(x )+ħ2g11

∣∣Ψ1(x, t )∣∣2 +ħ2g12

∣∣Ψ2(x, t )∣∣2

)Ψ1(x, t ) ,

iħ ∂tΨ2(x, t ) =(− ħ2

2m2∆+V2(x )+ħ2g12

∣∣Ψ1(x, t )∣∣2 +ħ2g22

∣∣Ψ2(x, t )∣∣2

)Ψ2(x, t ) ,∥∥Ψ1(·,0)

∥∥2L2 = N 1 ,

∥∥Ψ2(·,0)∥∥2

L2 = N 2 , x ∈R3 , t ≥ 0;

the scaled harmonic potentials are of the form

V1(x ) = m12

(ω2

11(x1 −ζ11)2 +ω212(x2 −ζ12)2 +ω2

13(x3 −ζ13)2) , x ∈R3 ,

V2(x ) = m22

(ω2

21(x1 −ζ21)2 +ω222(x2 −ζ22)2 +ω2

23(x3 −ζ23)2) , x ∈R3 .

52

In compact vector and matrix notation, we have

m = (m1,m2) ∈R2 , N = (N 1, N 2) ∈R2 ,

g =(

g11 g12

g12 g22

)∈R2×2 , ω=

(ω11 ω12 ω13

ω21 ω22 ω23

)∈R2×3 , ζ=

(ζ11 ζ12 ζ13

ζ21 ζ22 ζ23

)∈R2×3 .

In the present situation, the linear transformation

ξ=pc x , c =

pm1m2ħ ,

ψj (ξ, t ) = 1C Ψj (x, t ) , C = 4

√c3 , Uj (ξ) = 1

ħ Vj (x ) , j = 1,2, ϑ=ħC 2g ,

yields the following normalised system

i ∂tψ1(ξ, t ) =(−

pm1m22m1

∆+U1(ξ)+ϑ11∣∣ψ1(ξ, t )

∣∣2 +ϑ12∣∣ψ2(ξ, t )

∣∣2)ψ1(ξ, t ) ,

i ∂tψ2(ξ, t ) =(−

pm1m22m2

∆+U2(ξ)+ϑ12∣∣ψ1(ξ, t )

∣∣2 +ϑ22∣∣ψ2(ξ, t )

∣∣2)ψ2(ξ, t ) ,∥∥ψ1(·,0)

∥∥2L2 = N 1 ,

∥∥ψ2(·,0)∥∥2

L2 = N 2 , ξ ∈R3 , t ≥ 0.

53

7. Ground state solution

7.1. Energy functional

Energy functional. For a nonlinear Schrödinger equation such as (6.2), the energy func-tional E is given by

E(ϕ) =J∑

j=1Ej (ϕ) ,

Ej (ϕ) =((− cj ∆+Uj

)ϕj

∣∣∣ϕj

)L2+ 1

2

J∑k=1

ϑj k

(|ϕk |2ϕj

∣∣∣ϕj

)L2

, 1 ≤ j ≤ J ,

(7.1)

where ϕ= (ϕ1, . . . ,ϕJ

)with ϕj :Rd →C for 1 ≤ j ≤ J .

Notations. In accordance with the eigenvalue relation (5.14) for the Hermite basis functions,we henceforth denote

Gj (ϕ) = (Aj +Bj (ϕ)

)ϕj , Gj (ϕ) = (

Aj +Bj (ϕ))ϕj ,

Bj (ϕ) =B(0)j +

J∑k=1

ϑj k B(k)j (ϕ) , Bj (ϕ) =B(0)

j + 12

J∑k=1

ϑj k B(k)j (ϕ) ,

Aj = cj(− ∆+Uγ

), B(0)

j =Uj − cj Uγ , B(k)j (ϕ) = |ϕk |2 , 1 ≤ j ≤ J .

(7.2)

With these abbrevitaions, the partial differential equations in (6.2) and the associated energiesare written in compact form

i ∂tψj (ξ, t ) =Gj(ψ(ξ, t )

), Ej (ϕ) =

(Gj (ϕ)

∣∣∣ϕj

)L2

, 1 ≤ j ≤ J ,

see also (7.1).

Energy conservation. For the solution of the normalised Gross–Pitaevskii system (6.2) thetotal energy (7.1) is a conserved quantity, that is, it holds

E(ψ(·, t )

)= E(ψ(·,0)

), t ≥ 0. (7.3)

55

7.2. Ground state solution

Ground state solution. The ground state solution of the normalised Gross–Pitaevskii sys-tem (6.2) is a solution of the special form

ψj (ξ, t ) = e− iµj t ϕj (ξ) , ξ ∈Rd , t ≥ 0, 1 ≤ j ≤ J ,

that minimises the energy functional E , see (7.1). Hereby, the chemical potentials µj ∈ R,1 ≤ j ≤ J , are given by

Nj µj =(Gj (ϕ)

∣∣∣ϕj

)L2

=((− cj ∆+Uj

)ϕj

∣∣∣ϕj

)L2+

J∑k=1

ϑj k

(|ϕk |2ϕj

∣∣∣ϕj

)L2

, 1 ≤ j ≤ J .

7.3. A single Gross–Pitaevskii equation

Gross–Pitaevskii equation. For simplicity, we meanwhile consider a single Gross–Pitaevskiiequation of the form

i ∂tψ(ξ, t ) =(− 1

2 ∆+U (ξ)+ϑ ∣∣ψ(ξ, t )∣∣2

)ψ(ξ, t ) ,∥∥ψ(·,0)

∥∥2L2 = N , ξ ∈Rd , t ≥ 0,

(7.4)

see also (6.2); in accordance with (7.2), we denote

G (ϕ) = (A +B(ϕ)

)ϕ , G (ϕ) = (

A +B(ϕ))ϕ ,

B(ϕ) =B(0) +ϑB(1)(ϕ) , B(ϕ) =B(0) + 12 ϑB(1)(ϕ) ,

A = 12

(− ∆+Uγ

), B(0) =U − 1

2 Uγ , B(1)(ϕ) = |ϕ|2 .

Particle number conservation. With the help of the eigenvalue relation (5.14) and Parseval’sidentity (4.3), it it seen that (

ψ(·, t )∣∣∣G (

ψ(·, t )))

L2∈R .

As a consequence, for the solution of (7.4) we further obtain

∂t∥∥ψ(ξ, t )

∥∥2L2 =

(∂tψ(ξ, t )

∣∣ψ(ξ, t ))L2 +

(ψ(ξ, t )

∣∣∂tψ(ξ, t ))L2 = 2ℜ(

ψ(ξ, t )∣∣∂tψ(ξ, t )

)L2

= 2ℜ(− i

(ψ(·, t )

∣∣∣G (ψ(·, t )

))L2

)= 0, t ≥ 0,

which shows that the total particle number is conserved.

Energy conservation. In the present situation, the energy functional is given by

E(ϕ) = (G (ϕ)

∣∣ϕ)L2 =

((− 12 ∆+U + 1

2 ϑ |ϕ|2)ϕ

∣∣∣ϕ)L2

.

56

We note that ∂ϕ|ϕ|2 =ϕ (·)+ϕ (·) and further

12 ϑ

(ϕϕ (·)+ϕ2 (·) ∣∣ϕ)

L2 = 12 ϑ

(|ϕ|2ϕ ∣∣ (·))L2 + 12 ϑ

((·) ∣∣ |ϕ|2ϕ)

L2 .

Thus, making use of the fact that the Laplacian is a selfadjoint operator and that the potential Uis real-valued, the Fréchet derivative of E equals

∂ϕE(ϕ) =((− 1

2 ∆+U +ϑ |ϕ|2)ϕ ∣∣∣ (·))L2+

((− 12 ∆+U +ϑ |ϕ|2) (·)

∣∣∣ϕ)L2

= 2ℜ(G (ϕ)

∣∣ (·))L2 .

As a consequence, the energy conservation (7.3) follows; namely, for the solution of (7.4), weobtain

∂t E(ψ(·, t )

)= ∂ϕE(ψ(·, t )

)∂tψ(·, t ) = 2ℜ

(G

(ψ(·, t )

)∣∣∣∂tψ(·, t ))L2

= 2ℜ(− i

∥∥G(ψ(·, t )

)∥∥2L2

)= 0, t ≥ 0.

Generalisation. Similar considerations apply to Gross–Pitaevskii systems (6.2) showing thatthe total particle numbers and the total energy is conserved; more precisely, it follows

∂ϕj E(ϕ) = 2ℜ(Gj (ϕ)

∣∣ (·))L2 , ∂t∥∥ψj (·, t )

∥∥L2 = 0,

∂t E(ψ(·, t )

)= ∂ϕE(ψ(·, t )

)∂tψ(·, t ) = 0, t ≥ 0, 1 ≤ j ≤ J .

7.3.1. Groundstate solution

We next determine the ground state solution of the Gross–Pitaevskii equation (7.4) for thelimiting cases ϑ= 0 and ϑ>> 1; in particular, we consider a scaled harmonic potential

U (ξ) = 12

d∑i=1

ω2i ξ

2i , ωi > 0, ξ ∈Rd .

These special solutions will serve as suitable initial values in the ground state computation bythe imaginary time method and a minimisation approach, respectively.

Linear Schrödinger equations

For ϑ= 0, problem (7.4) simplifies to a linear Schrödinger equation. In the above situation, theground state solution is given by the first Hermite basis function. More precisely, it holds

ψ(ξ, t ) =p

N e− iµtϕ(ξ) , ξ ∈Rd , t ≥ 0,

ϕ=Hγ

0 , γi =pωi , 1 ≤ i ≤ d , µ= 1

2

d∑i=1

ωi .

57

Namely, inserting into (7.4) yields (due to λ0 = γ21 +·· ·+γ2

d =ω1 +·· ·+ωd )

µψ(ξ, t ) = i ∂tψ(ξ, t ) = (− 12 ∆+U (ξ)

)ψ(ξ, t ) = 1

2

pN e− iµt (− ∆+Uγ(ξ)

)H

γ0 (ξ)

= 12

pN e− iµtλ0 H

γ0 (ξ) = 1

2

d∑i=1

ωi ψ(ξ, t ) ,

see also (5.14). Note that this is consistent with

µ= 1N

(12

(− ∆+Uγ

)ψ(·, t )

∣∣∣ψ(·, t ))L2

= 12

(λ0 H

γ0

∣∣H γ0

)L2 = 1

2 λ0 = 12

d∑i=1

ωi .

Thomas–Fermi approximation

For large values ofϑ, neglecting the Laplace operator in (7.4), the Thomas–Fermi approximationyields an approximate ground state through

ψ(ξ, t ) =p

N e− iµtϕ(ξ) ,

ϕ(ξ) =√

1N ϑ

(µ−U (ξ)

), if U (ξ) <µ ,

0 , otherwise,, ‖ϕ‖L2 = 1, ξ ∈Rd , t ≥ 0.

This is seen by inserting the above relation into the differential equation

µψ(ξ, t ) = i ∂tψ(ξ, t ) =(U (ξ)+ϑ ∣∣ψ(ξ, t )

∣∣2)ψ(ξ, t )

which yields µ=U (ξ)+ϑ ∣∣ψ(ξ, t)∣∣2. In particular, for a scaled harmonic potential Uγ, due to

the normalisation condition

1 = ‖ϕ‖2L2 = 1

N ϑ

∫U (ξ)<µ

(µ−U (ξ)

)dξ

the chemical potential µ is given by

µ=

14

3√

18(Nϑω1

)2 , d = 1,2√

1πNϑω1ω2 , d = 2,

14

5√

450( 1πNϑω1ω2ω3

)2 , d = 3.

58

8. Time-splitting pseudo-spectral methods

8.1. Abstract formulation

Abstract evolutionary problem. For the specification of the time integration method aswell as for theoretical considerations, it is convenient to formulate a nonlinear Schrödingerequation as an abstract ordinary differential equation on a function space by formally omittingthe spatial variable. In particular, for the normalised Gross–Pitaevskii system (6.2) this yieldsthe following abstract initial value problem for u(t ) =ψ(·, t ) = (

ψ1(·, t ), . . . ,ψJ (·, t ))

iu′(t ) = A u(t )+B(u(t )

)u(t ) , t ≥ 0, u(0) given. (8.1)

Approach. In the following, we apply time-splitting spectral methods for discretising Gross–Pitaevskii systems in space and time. More precisely, the time integration of (6.2) and (8.1),respectively, relies on high-order exponential operator splitting methods, see Part I. For thenumerical solution of the associated initial value problems

iu′(t ) = A u(t ) , t ≥ 0, u(0) given, (8.2a)

iu′(t ) = B(u(t )

)u(t ) , t ≥ 0, u(0) given, (8.2b)

we make use of Hermite and Fourier spectral methods, see Part II. In the subsequent Sec-tions 8.2 and 8.3, we specify the definition of the unbounded operators A : D(A) ⊂ X → Xand B(v) : D(B) ⊂ X → X , v ∈V , which is closely related to the choice of the spectral method.Moreover, we briefly discuss the numerical solution of the initial value problems (8.2).

8.2. Time-splitting Fourier pseudo-spectral method

For the Fourier spectral method, the numerical solution of the associated initial value prob-lem (8.2a) relies on techniques that were the content of Section 4.2. Due to the fact thatGross–Pitaevskii systems fulfill a certain invariance properties, the exact solution of (8.2b) isavailable.

8.2.1. First part

Initial value problem. Regarding (8.2a), we consider the following initial value problem

i ∂tψ(·, t ) =Aψ(·, t ) , t ≥ 0, ψ(·,0) given, (8.3a)

59

where the application of the Fourier spectral method suggests the choice

A =

A1. . .

AJ

, Aj =−cj ∆ , cj =Jpm1···mJ

2mj, 1 ≤ j ≤ J . (8.3b)

We are thus concerned with the componentwise numerical solution of the above problem, i.e.,we consider

i ∂tψj (·, t ) =Aj ψj (·, t ) , t ≥ 0, ψj (·,0) given, 1 ≤ j ≤ J . (8.3c)

Exact solution. It is straightforward to extend the approach of Section 4.2. The eigenvaluerelation (4.9) for the differential operator −∆ with corresponding eigenfunctions (Fm)m∈M

and eigenvalues (λm)m∈M implies

Aj Fm = cjλm Fm , m ∈M , 1 ≤ j ≤ J .

Employing a spectral decomposition of the initial value into Fourier basis functions, we thusobtain the following representation

ψj (·, t ) = ∑m∈M

e− icjλm t ψj m(0) Fm , ψj (·,0) = ∑m∈M

ψj m(0)Fm , 1 ≤ j ≤ J , (8.4)

see also (4.2) and (8.3).

Numerical solution. The numerical realisation of (8.4) relies on techniques that were dis-cussed in Section 4.2.2.

8.2.2. Second part

Initial value problem. Regarding (8.2b), we consider the initial value problem

i ∂tψ(·, t ) =B(ψ(·,0)

)ψ(·, t ) , t ≥ 0, ψ(·,0) given, (8.5a)

where the application of the Fourier spectral method for the first part (8.3) suggests the follow-ing choice

B(ψ(·, t )

)=B1

(ψ(·, t )

). . .

BJ(ψ(·, t )

) ,

Bj(ψ(·, t )

)=Uj +J∑

k=1ϑj k

∣∣ψk (·, t )∣∣2 , 1 ≤ j ≤ J .

(8.5b)

60

We are thus concerned with the numerical solution of the initial value problem

i ∂tψj (·, t ) =Bj(ψ(·,0)

)ψj (·, t ) , ψj (·,0) given, 1 ≤ j ≤ J . (8.5c)

Invariance and exact solution. Noting that the exact solution of (8.5) fulfills

∂t∣∣ψj (·, t )

∣∣2 = 2 ℜ(ψj (·, t ) ∂tψj (·, t )

)= 2 ℜ

(− i Bj

(ψ(·, t )

)∣∣ψj (·, t )∣∣2

)= 0,

we conclude that the following invariance property holds

Bj(ψ(·, t )

)=Bj(ψ(·,0)

), t ≥ 0, 1 ≤ j ≤ J . (8.6)

Consequently, the exact solution of the initial value problem (8.5) is obtained by a pointwisemultiplication

ψj (ξ, t ) = e− i t Bj (ψ(ξ,0))ψj (ξ,0) , ξ ∈Ω , t ≥ 0. (8.7)

Numerical solution. The numerical realisation of (8.7) relies on collocation at the trapezoidquadrature nodes, see also Section 4.2.2.

8.3. Time-splitting Hermite pseudo-spectral method

For the Hermite spectral method, the numerical solution of the associated initial value prob-lem (8.2a) relies on techniques that were the content of Section 5.2, see also Section 7.1. Due tothe fact that Gross–Pitaevskii systems fulfill a certain invariance properties, the exact solutionof (8.2b) is available. We recall the definition of the scaled harmonic potential

Uγ(ξ) =d∑

i=1γ4

i ξ2i , γi > 0, ξ ∈Rd , 1 ≤ i ≤ d ,

see also (5.1).

8.3.1. First part

Initial value problem. Regarding (8.2a), we consider the initial value problem (8.3a), wherethe application of the Hermite spectral method suggests the choice

A =

A1. . .

AJ

, Aj =−cj(∆−Uγ

), cj =

Jpm1···mJ

2mj, 1 ≤ j ≤ J .

In the same way as for the Fourier spectral method, we are thus concerned with the numericalsolution of (8.3c).

61

Exact solution. It is straightforward to extend the approach of Section 5.2. The eigenvaluerelation (5.14) for the differential operator −∆+Uγ with eigenfunctions (H γ

m)m∈M and eigen-values (λm)m∈M implies

Aj Hγ

m = cjλm Hγ

m , m ∈M , 1 ≤ j ≤ J .

Employing a spectral decomposition of the initial value into Hermite basis functions, we thusobtain the following representation

ψj (·, t ) = ∑m∈M

e− icjλm t ψj m(0) Hγ

m , ψj (·,0) = ∑m∈M

ψj m(0)H γm , 1 ≤ j ≤ J , (8.8)

see also (5.8) and (8.3c).

Numerical solution. The numerical realisation of (8.8) relies on techniques that were dis-cussed in Section 5.2.2.

8.3.2. Second part

Initial value problem. Regarding (8.2b), we consider the initial value problem (8.5a), wherethe application of the Hermite spectral method for the first part suggests the choice

B(ψ(·, t )

)=B1

(ψ(·, t )

). . .

BJ(ψ(·, t )

) ,

Bj(ψ(·, t )

)=Uj − cj Uγ+J∑

k=1ϑj k

∣∣ψk (·, t )∣∣2 , 1 ≤ j ≤ J .

Similarly to before, due to the validity of the invariance property (8.6), the numerical solu-tion of (8.5c) is realised by a pointwise multiplication and collocation at the Gauß–Hermitequadrature nodes, see also Section 5.2.2.

8.4. Numerical illustrations

In the following, we illustrate the favourable behaviour of time-splitting Fourier and Hermitepseudo-spectral methods for systems of coupled Gross–Pitaevskii equations (6.1). A detaileddescription of the numerical examples is found in [13], see also [36, 44].

8.4.1. Computation time

A comparison of the computation time of the Fourier and Hermite spectral method in one andtwo space dimensions is given in Figure 1.

62

101

102

10−5

10−4

10−3

10−2

degree of freedom

CP

U s

econ

ds

Hermite 1DFourier 1D

101

102

10−5

10−4

10−3

10−2

10−1

degree of freedom

CP

U s

econ

ds


Figure 1.: Computation time of the Fourier and Hermite spectral methods in one (left picture)and two (right picture) space dimensions.

8.4.2. Spatial error

The accuracy of the Fourier and Hermite spectral methods is illustrated in Figure 2.

8.4.3. Temporal convergence order

The numerical convergence orders of various exponential operator splitting methods appliedto a two-dimensional Gross–Pitaevskii equation with external harmonic potential and couplingconstant ϑ= 1 and ϑ= 100, respectively, are given in Figures 3 and 4.

10 25 50 100 250 50010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

102

degree of freedom

erro

r

ϑ = 1ϑ = 10ϑ = 100ϑ = 1000

10 25 50 100 250 50010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

102

degree of freedom

erro

r

ϑ = 1ϑ = 10ϑ = 100ϑ = 1000

Figure 2.: Spatial error of the Fourier (left picture) and Hermite (right picture) spectral method.

63

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

StrangMcLachlan

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

Y4S4M4BM4−1BM4−2

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

Y6KL6S6BM6−1BM6−2BM6−3

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

StrangMcLachlan

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r


10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r


Figure 3.: Temporal orders of various time-splitting Fourier (first row) and Hermite (secondrow) spectral methods when applied to a Gross–Pitaevskii equation with ϑ= 1.

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

StrangMcLachlan

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r


10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r


10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

StrangMcLachlan

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r


10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r


Figure 4.: Temporal orders of various time-splitting Fourier (first row) and Hermite (secondrow) spectral methods when applied to a Gross–Pitaevskii equation with ϑ= 100.

64

tol. method d.o.f. #transf. ∆pn ∆E

< 10−2 Hermite 2 32×32 16384 2.6 ·10−11 4.2 ·10−6

< 10−2 Fourier 2 64×64 32768 3.6 ·10−13 1.6 ·10−6

< 10−2 Hermite 4 32×32 6144 9.7 ·10−12 1.1 ·10−5

< 10−2 Fourier 4 64×64 12288 1.7 ·10−13 9.1 ·10−7

< 10−2 Hermite 6 32×32 14337 2.3 ·10−11 3.2 ·10−8

< 10−2 Fourier 6 64×64 7169 1.1 ·10−13 6.8 ·10−6

< 10−2 Hermite rk4 32×32 65532 2.1 ·10−5 1.2 ·10−4

< 10−2 Fourier rk4 64×64 524284 6.4 ·10−10 3.7 ·10−9

< 10−2 Hermite ode45 32×32 208376 2.6 ·10−8 1.5 ·10−7

< 10−2 Fourier ode45 64×64 1132436 5.6 ·10−12 3.1 ·10−11

< 10−6 Hermite 4 64×64 24576 1.0 ·10−10 1.1 ·10−10

< 10−6 Fourier 4 128×128 49152 6.7 ·10−12 1.2 ·10−11

< 10−6 Hermite 6 64×64 28673 1.2 ·10−8 2.1 ·10−10

< 10−6 Fourier 6 128×128 28673 4.2 ·10−12 8.7 ·10−12

< 10−6 Hermite rk4 64×64 524284 6.4 ·10−10 3.7 ·10−9

< 10−6 Fourier rk4 128×128 524284 6.4 ·10−10 3.7 ·10−9

< 10−6 Hermite ode45 64×64 509816 3.6 ·10−10 2.1 ·10−9

< 10−6 Fourier ode45 128×128 1411448 2.2 ·10−12 1.1 ·10−11

Table 1.: Time integration of a two-dimensional Gross–Pitaevskii equation up to T = 400. For atolerance (tol.), the degree of freedom (d.o.f.), the number of spectral transformations(#transf.), the particle number conservation error ∆pn, and the energy conservationerror ∆E are displayed.

8.4.4. Long-term integration

The long-term behaviour of time-splitting Fourier and Hermite spectral methods for a two-dimensional Gross–Pitaevskii equation with external harmonic potential and coupling con-stant ϑ= 1 is illustrated in Table 1. For the time integration, the second-order Strang splittingmethod, fourth- and sixth-order splitting methods proposed by BLANES & MOAN, and fourth-order explicit Runge–Kutta methods are applied, see also Table 1. For a certain prescribedtolerance, the required number of basis functions, the number of spectral transformations, theparticle number conservation error ∆pn = ∣∣∥∥ψ(·,0)

∥∥2L2 −

∥∥ψ(·,T )∥∥2

L2

∣∣, and the energy conserva-tion error ∆E = ∣∣E(

ψ(·,0))−E

(ψ(·,T )

)∣∣ are displayed.

65

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Appendix

71

High-order time-splitting Hermite and Fourier spectral methods q

Mechthild Thalhammer b,*, Marco Caliari a, Christof Neuhauser b

a Dipartimento di Informatica, Università degli Studi di Verona, Ca’ Vignal 2, Strada Le Grazie 15, I-37134 Verona, Italyb Institut für Mathematik, Leopold-Franzens Universität Innsbruck, Technikerstraße 13/7, A-6020 Innsbruck, Austria

a r t i c l e i n f o

Article history:Received 1 July 2008Received in revised form 24 September2008Accepted 5 October 2008Available online 17 October 2008

Keywords:Nonlinear Schrödinger equationsGross–Pitaevskii equationPseudospectral methodsExponential operator splitting methods

a b s t r a c t

In this paper, we are concerned with the numerical solution of the time-dependent Gross–Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider dis-cretisations that are based on spectral methods in space and higher-order exponentialoperator splitting methods in time. The resulting methods are favourable in view of accu-racy and efficiency; moreover, geometric properties of the equation such as particle num-ber and energy conservation are well captured.

Regarding the spatial discretisation of the GPE, we consider two approaches. In theunbounded domain, we employ a spectral decomposition of the solution into Hermite basisfunctions; on the other hand, restricting the equation to a sufficiently large boundeddomain, Fourier techniques are applicable. For the time integration of the GPE, we studyvarious exponential operator splitting methods of convergence orders two, four, and six.

Our main objective is to provide accuracy and efficiency comparisons of exponentialoperator splitting Fourier and Hermite pseudospectral methods for the time evolution ofthe GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting meth-ods compared to standard integrators in a long-term integration.

2008 Elsevier Inc. All rights reserved.

1. Introduction

In the present paper, we are concerned with the numerical solution of the time-dependent Gross–Pitaevskii Equation (GPE)[14,21]

ih@twðx; tÞ ¼ ð h2

2mDþ VðxÞ þ gjwðx; tÞj2Þwðx; tÞ; ð1Þ

describing the wave function w : Rd RP0 ! C of a Bose–Einstein condensate. Our main objective is to compare space andtime discretisations that are based on Hermite and Fourier spectral methods and exponential splitting methods oforders two, four, and six; moreover, we illustrate the effectiveness of higher-order splitting methods compared to stan-dard integrators in a long-term integration. In most cases, we use the ground state of the GPE as a reliable referencesolution, computed by employing the Hermite spectral decomposition and directly minimising the energy functional,see [3,8].

Over the past years, numerous works were devoted to the discretisation of nonlinear Schrödinger equations; we mention[1,2,4,10,11,20,24,25], where a particular emphasis is given to accuracy and the preservation of geometric properties. For thespatial discretisation of the GPE, Hermite pseudospectral methods are used in [2,11]; on the other hand, restricting the

0021-9991/$ - see front matter 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jcp.2008.10.008

q Version of 23 september 2008.* Corresponding author.

E-mail address: [email protected] (M. Thalhammer).

Journal of Computational Physics 228 (2009) 822–832

Contents lists available at ScienceDirect

Journal of Computational Physics

journal homepage: www.elsevier .com/locate / jcp

problem to a bounded domain, Fourier spectral methods are applicable, see Bao et al. [1]. The favourable behaviour of thesecond-order Strang splitting and a fourth-order time-splitting scheme regarding accuracy, efficiency, and the conservationof geometric properties is illustrated in [1,2]. For the cubic Schrödinger equation, numerical comparisons of different spaceand time discretisations are provided by Pérez–García and Liu [20].

The present work is organised as follows. In Section 2, we restate the time-dependent d-dimensional GPE in a norma-lised form. Further, we briefly discuss the special case of a harmonic potential and vanishing interaction that leads to thetime-dependent linear Schrödinger equation; in this situation, the ground and the excited states are given by the Hermitefunctions. The linear Schrödinger equation also motivates the consideration of a Hermite spectral decomposition for thenonlinear GPE. Sections 3 and 4 are devoted to numerical discretisations of the GPE based on Hermite and Fourier spec-tral methods in space and exponential operator splitting methods in time; we note that an extension to systems of cou-pled GPEs is straightforward, see also Caliari et al. [8]. In Section 5, we present several illustrations regarding accuracy,efficiency, and the preservation of geometric properties. The numerical experiments are carried out for problems in twospace dimensions; however, in view of the tensor product structure of the spatial discretisation, we expect our conclu-sions to be maintained in three space dimensions as well. In Section 6, we finally summarise our results and discuss openquestions.

The following notations are tacitly employed throughout. For a multi-index of integer numbersm ¼ ðm1;m2; . . . ;mdÞ 2 Zd, the relation 6 is understood componentwise. For an element x ¼ ðx1; x2; . . . ; xdÞ 2 Rd, we denoteby jxj its Euclidean norm. As usual, the d-dimensional Laplacian is defined through D ¼ @2

x1þ þ @2

xd. The Lebesgue space

L2ðXdÞ of square integrable complex-valued functions on Xd # Rd is endowed with scalar product ðjÞL2ðXdÞ and associatednorm k kL2ðXdÞ defined by

ðf jgÞL2ðXdÞ ¼Z

Xdf ðxÞgðxÞdx; kfkL2ðXdÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf jf ÞL2ðXdÞ

q; f ; g 2 L2ðXdÞ;

for notational simplicity, we omit the domain in the scalar product and norm.

2. Gross–Pitaevskii equation

In the present section, we state a normalisation of the d-dimensional GPE (1) that is obtained by a linear transformation ofthe spatial variable and a rescaling of the wave function, see also Caliari et al. [8]; moreover, we introduce the ground andexcited state solutions of the GPE by means of a nonlinear eigenvalue problem. Existence and uniqueness results for time-dependent Schrödinger equations are found in Cazenave [9, Chapters 4 and 6].

2.1. Time-dependent Gross–Pitaevskii equation

Henceforth, we consider the following normalisation of the time-dependent Gross–Pitaevskii equation (GPE):

i@twðx; tÞ ¼ ð 12 Dþ VðxÞ þ #jwðx; tÞj2Þwðx; tÞ; t P 0; ð2aÞ

the equation is subject to asymptotic boundary conditions on the unbounded spatial domain Rd, i.e., we require wðx; tÞ ! 0 asjxj ! 1. Without any loss of generality, we further suppose the initial value wð;0Þ 2 L2ðRdÞ to satisfy the normalisationcondition

kwð;0Þk2L2 ¼ 1: ð2bÞ

In the present paper, we restrict ourselves to the case of a scaled harmonic potential V ¼ 12 VH where

VHðxÞ ¼Xd

j¼1

c4j x2

j ; cj > 0; 1 6 j 6 d; ð2cÞ

however, as indicated in Section 4, our approach extends to more general real-valued potentials V. Also, we assume the cou-pling constant # to be non-negative, that is, we restrict ourselves to a defocusing condensate.

As proven in Cazenave [9, Thm 4.1.1], a fundamental property of (2) is the preservation of the particle number

kwð; tÞk2L2 ¼ kwð;0Þk2

L2 ¼ 1; t P 0; ð3aÞ

see (2b). Moreover, the energy functional

Eðwð; tÞÞ ¼ ðð12Dþ V þ 1

2#jwð; tÞj2Þwð; tÞjwð; tÞÞL2 ð3bÞ

is time-independent, that is, it holds Eðwð; tÞÞ ¼ Eðwð;0ÞÞ for t P 0; we further require Eðwð;0ÞÞ <1.

M. Thalhammer et al. / Journal of Computational Physics 228 (2009) 822–832 823

2.2. Ground and excited states

The ground state solution of the GPE (2) is a solution of the form

wðx; tÞ ¼ eilt/ðxÞ; x 2 Rd; t P 0; ð4aÞ

with l 2 R and / a real-valued (positive) function that minimises the energy functional Eðwð; tÞÞ ¼ Eð/Þ, see (3b). Inserting(4a) into (2a) and using (3a) yields

ð12Dþ V þ #/2Þ/ ¼ l/; l ¼ Eð/Þ þ 1

2#ð/3j/ÞL2 : ð4bÞ

Further solutions of the GPE (2) that allow a decomposition (4a) and thus solve the nonlinear eigenvalue problem (4b) arecalled excited state solutions.

We next consider the special case where the parameter # vanishes and the potential V is given by the scaled harmonicpotential VH, see (2c). In this situation, the GPE (2) reduces to the linear Schrödinger equation

i@twðx; tÞ ¼ 12 Dþ VHðxÞð Þwðx; tÞ; t P 0;

and, as well known, the Hermite functions with associated eigenvalues

HmðxÞ ¼Yd

j¼1

HmjðxjÞe

12ðcjxjÞ2

; km ¼

Xd

j¼1

c2j mj þ

12

; ð5aÞ

solve (4b) for # ¼ 0, that is, for any m P 0 it holds

12 Dþ VHð ÞHm ¼ kmHm; ð5bÞ

here, we let Hmjdenote the Hermite polynomial of degree mj, normalised with respect to the weight function wjðxjÞ ¼ eðcjxjÞ2 .

The Hermite functions ðHmÞ form an orthonormal basis of the function space L2ðRdÞ; in particular, it holds ðHkjHmÞL2 ¼ dkm

with Kronecker delta dkm.

3. Pseudospectral methods

In the following, we discuss two approaches for the spatial discretisation of the GPE that are based on Hermite and Fourierspectral decompositions of the solution, respectively.

3.1. Hermite pseudospectral method

In this section, we let m 2 Zd be a multi-index with non-negative components, i.e., we suppose m P 0; hence, for a family ðamÞwe writeX

m

am ¼XmP0

am

for short. Using that the Hermite functions ðHmÞ form an orthonormal basis of the function space L2ðRdÞ, the representation

wð; tÞ ¼X

m

wmðtÞHm; wmðtÞ ¼ wð; tÞjHmð ÞL2 ; ð6aÞ

follows, see also Section 2.2; besides, due to Parseval’s equality, the identity

kwð; tÞk2L2 ¼

Xm

jwmðtÞj2 ð6bÞ

is valid. Truncating the infinite sum in (6a) yields

wMð; tÞ ¼X

Mm

wmðtÞHm ¼X

m6M1

wmðtÞHm ð7aÞ

with coefficient vector wðtÞ ¼ ðwmðtÞÞm6M1 given by (6a); the above relation (6b) implies

kwMð; tÞk2L2 ¼ jwðtÞj2 ¼

XM

m

jwmðtÞj2: ð7bÞ

Results on the approximation error of the Hermite spectral method are found in Boyd [6, Chapter 17.4]. For computingnumerically the coefficients wmðtÞ given by (6a), we apply the following approximation:

wmðtÞ ¼Z

Rdwðx; tÞHmðxÞdx

XM

k

xkejnk j2 wðnk; tÞHmðnkÞ ð8Þ

824 M. Thalhammer et al. / Journal of Computational Physics 228 (2009) 822–832

with nk ¼ ðnk1; . . . ; nkd

Þ and xk ¼ xk1 . . . xkd; here, nkj

and xkjdenote the nodes and weights of the Gauss–Hermite quad-

rature formula relative to wjðxjÞ ¼ eðcjxjÞ2 .

3.2. Fourier pseudospectral method

In order to apply Fourier techniques for the spatial discretisation of (2), we restrict the unbounded domain to a boundedset Xd; for simplicity, we assume X ¼ ½a; a to be a symmetric interval with a > 0 chosen sufficiently large. For the remain-der of this section, we denote by m 2 Zd a multi-index of integer numbers; the Lebesgue space L2ðXdÞ is endowed with thescalar product

ðf jgÞL2 ¼Z

Xdf ðxÞgðxÞdx; f ; g 2 L2ðXdÞ;

and corresponding norm. In contrast to (5b), we now employ the eigenvalue decomposition

12DFm ¼ kmFm

involving the Fourier basis functions ðFmÞ and associated eigenvalues ðkmÞ that are given by

FmðxÞ ¼Yd

j¼1

FmjðxjÞ; Fmj

ðxÞ ¼ 1ffiffiffiffi2ap eimjp 1

axþ1ð Þ; km ¼ 12a2p2

Xd

j¼1

m2j ;

in particular, it holds (FkjFmÞL2 ¼ dkm. Therefore, similarly to (6a), the representation

wð; tÞ ¼X

m

wmðtÞFm; wmðtÞ ¼ ðwð; tÞjFmÞL2 ; ð9Þ

follows for elements in L2ðXdÞ; as before, the coefficients ðwmðtÞÞ satisfy (6b). For some integer M1 P 0 we henceforth setM ¼ 2M1. Truncating the infinite sum in (9) yields

wMð; tÞ ¼X

Mm

wmðtÞFm ¼X

M16m6M11

wmðtÞFm; ð10Þ

the coefficient vector wðtÞ ¼ ðwmðtÞÞM16m6M11 fulfills relation (7b). Results on the favourable approximation behaviour of theFourier spectral method are found in Boyd [6, Chapter 2]. For computing numerically the coefficients wmðtÞ given by (9), weapply the trapezoidal rule

wmðtÞ ¼Z

Xdwðx; tÞFmðxÞdx x

XM

k

wðnk; tÞFmðnkÞ; ð11Þ

here, we set nk ¼ ðnk1; . . . ; nkd

Þ with equidistant grid points nkj¼ 1

M1akj and further x ¼ ð 1

M1aÞd.

4. Time-splitting methods

In this section, we introduce exponential operator splitting methods for the time integration of evolutionary nonlinearSchrödinger equations such as (2). For a detailed treatment of splitting and composition methods for ordinary differentialequations, we refer to [15,19]. In particular for the GPE, the favourable behaviour of a second-order Strang type splittingand a fourth-order splitting scheme regarding accuracy, efficiency, and the preservation of geometric properties is confirmedby numerical experiments given in [1,2]; a convergence analysis for Strang type splitting methods is provided by Caliari et al. [7],see also Lubich [17].

In order to state the considered numerical method class, it is convenient to formulate the partial differential equation (2a)as an abstract ordinary differential equation; more precisely, suppressing the spatial variable in the equation and settinguðtÞ ¼ wð; tÞ, we obtain an initial value problem of the form

u0ðtÞ ¼

Aþ BðuðtÞÞ

uðtÞ; t P 0; uð0Þ given: ð12Þ

Exponential operator splitting methods rely on a decomposition of the right-hand side of the differential equation into twoparts in such a way that the resulting differential equations

v 0ðtÞ ¼ AvðtÞ; t P 0; vð0Þ given; ð13aÞw0ðtÞ ¼ B wðtÞð ÞwðtÞ; t P 0; wð0Þ given; ð13bÞ

are solvable in a favourable way. Regarding the Hermite and Fourier pseudospectral methods, we distinguish the followingapproaches:

Hermite : A ¼ 12iðD VHÞ; BðuðtÞÞ ¼ i V VH þ #juðtÞj2

;

Fourier : A ¼ 12iD; BðuðtÞÞ ¼ i V þ #juðtÞj2

;


see also (2).On the one hand, the solution of the initial value problem (13a) equals

vðtÞ ¼ etAvð0Þ; t P 0: ð14aÞ

The evaluation of v relies on the representation of the initial value with respect to the Hermite and Fourier basis functions,see Section 3; more precisely, provided that vð0Þ can be decomposed into the basis functions ðBmÞ, where Bm ¼Hm orBm ¼Fm, respectively, the exact solution value at time t P 0 is given by

vðtÞ ¼X

m

vmeitkmBm; t P 0; vð0Þ ¼X

m

vmBm: ð14bÞ

For the numerical evaluation of vðtÞ, we collocate (13a) at the nodes ðnkÞ and approximate the coefficients ðvmÞ by means ofthe Gauss–Hermite or the trapezoid quadrature formula, respectively, see (8) and (11). Clearly, the numerical approximationto vðtÞ can be evaluated at any x; however, much less computational effort is required when vðtÞ is evaluated numerically atthe quadrature nodes. In fact, for the Hermite pseudospectral method the values HmðnkÞ can be stored; for the Fourierpseudospectral method the Fast Fourier Transform is applicable. On the other hand, regarding the initial value problem(13b), the exact solution is available; namely, due to the fact that the differential equation for w leaves jwðtÞj invariant, itfollows BðwðtÞÞ ¼ Bðwð0ÞÞ and thus

wðtÞ ¼ etBðwð0ÞÞwð0Þ; t P 0; ð14cÞ

see also Caliari et al. [7]. In the numerical computation, we again collocate the equation at the quadrature nodes ðnkÞ; then,the approximate solution is obtained by a rapid componentwise multiplication.

The basic idea of exponential operator splitting methods is to compose the solutions of (13) in a suitable way. A widelyused scheme is based on the second-order Strang [22] or symmetric Trotter [27] splitting; for a step size h > 0 and an initialvalue u0 uð0Þ, approximations un to the exact solution values uðnhÞ, n P 0, are given by the recurrence formula

un ¼ e12hBðUnÞUn; Un ¼ ehAe

12hBðun1Þun1; or ð15aÞ

un ¼ e12hAe

12hBðUnÞUn; Un ¼ e

12hAun1; ð15bÞ

respectively. We note that for the Fourier spectral method the solution (14a) satisfies the periodic boundary conditions im-posed implicitly by the spectral approximation; therefore, this is also true for the auxiliary stage Un and the numerical solu-tion value un in (15b).

Higher-order exponential operator splitting methods for (12) can be cast into the following form:

un ¼ ebshBðUn;sÞUn;s;

Un;1 ¼ ea1hAun1; Un; j ¼ eajhAebj1hBðUn; j1ÞUn; j1; 2 6 j 6 s;ð16Þ

with real coefficients aj; bj 2 R, 1 6 j 6 s. Example methods were proposed in [5,16,18,23,28], e.g. see also [15,19]. InSection 5, we include numerical experiments for the splitting schemes of orders two, four, and six, respectively, that arecollected in Table 1.

As shown in Thalhammer [26], any splitting method retains its classical convergence order for time-dependent linearSchrödinger equations, provided that the initial value and the potential fulfill suitable regularity requirements. The numer-ical experiments presented in Section 5.2 and the theoretical analysis for the second-order Strang type splitting (15) given in[7,17] indicate that this result is also true for nonlinear Schrödinger equations with sufficiently regular solutions; however,extending the convergence analysis to general exponential operator splitting methods is out of the scope of the presentwork.

Table 1Exponential operator splitting methods of order p involving s compositions.

Method Order #Compositions

McLachlan McLachlan [15, V. 3.1, (3.3), pp. 138–139] p = 2 s = 3Strang Strang (15a) p = 2 s = 2BM4-1 Blanes and Moan [5, Table 2, PRKS6] p = 4 s = 7BM4-2 Blanes and Moan [5, Table 3, SRKNb

6] p = 4 s = 7M4 McLachlan [15, V. 3.1, (3.6), p. 140] p = 4 s = 6S4 Suzuki [15, II. 4, (4.5), p. 41] p = 4 s = 6Y4 Yoshida [15, II. 4, (4.4), p. 40] p = 4 s = 4BM6-1 Blanes and Moan [5, Table 2, PRKS10] p = 6 s = 11BM6-2 Blanes and Moan [5, Table 3, SRKNb

11] p = 6 s = 12BM6-3 Blanes and Moan [5, Table 3, SRKNa

14] p = 6 s = 15KL6 Kahan and Li [15, V. 3.2, (3.12), pp. 144] p = 6 s = 10S6 Suzuki [15, II. 4, (4.5), pp. 41] p = 6 s = 26Y6 Yoshida [15, V. 3.2, (3.11), pp. 144] p = 6 s = 8


We finally note that the total particle number (3a) is preserved by exponential operator splitting methods (16) applied tothe GPE (2); this follows from the conservation of the L2-norm when solving the differential equations in (13).

5. Numerical experiments

In this section, we present several numerical experiments comparing time-splitting spectral methods when applied to thetwo-dimensional GPE (2) involving a harmonic potential. Our experiments mainly rely on the computation of the groundstate and its propagation, see Caliari et al. [8]. Consequently, making use of the fact that the solutions are even functions,it would be possible to reduce the number of required basis functions for the Hermite and Fourier spectral methods; how-ever, in our presentation, we did not take into account this reduction. For the Fourier spectral method, we henceforth setX ¼ ½15;15.

The numerical experiments were implemented in MATLAB; the code is available from the authors on request. In theHermite case, we compute and store once and for all the values HmðnkÞ; in two space dimensions, it is then possible toevaluate each of the double sums in (7a) and (8) by two matrix–matrix multiplications at a cost of OðM3Þ. For the Fou-rier transforms (10) and (11) the MATLAB–functions ifft2 and fft2 of cost OðM2 log MÞ are used. The long-term com-putations were carried out on the Opteron cluster.1 of the High Performance Computing Consortium at the University ofInnsbruck.

5.1. Spatial error

In order to illustrate the accuracy of the Hermite and Fourier spectral methods, we use the ground state solution of thetwo-dimensional GPE (2) computed with 256 256 degrees of freedom as reference solution w, see (4a); we chooseVðxÞ ¼ VHðxÞ ¼ 1

2 ðx21 þ x2

2Þ. For the Hermite spectral method, we evaluate wðx;0Þ at the Gauss–Hermite quadrature pointscorresponding to M ¼ 2i, 4 6 i 6 8. We then compute an approximation ewMðx;0Þ to wMðx;0Þ by means of (7a), where thespectral coefficients are obtained from (8). Finally, the difference kwMð;0Þ wð;0ÞkL2 is computed through (7b). The sameapproach is employed for the Fourier spectral method. First, the reference solution wðx;0Þ is evaluated at 256 256equidistant grid points in the square ½15;15 ½15;15; then, an approximation corresponding to M ¼ 2i, 4 6 i 6 8, isdetermined by (10) and (11), and, finally, the error is computed through (7b).

The results displayed in Fig. 1 show that for M 6 128 and # 6 100 the Hermite spectral error is smaller than the Fourierspectral error. Further, for the Hermite spectral method it is possible to retain the original solution only up to a Hermite trans-form error of about 1014, even using the same degree of freedom 256 256, whereas the Fourier transform error is of themagnitude of the machine precision.

The Gauss–Hermite quadrature nodes and weights are obtained as solutions of an eigenvalue problem, see Gautschi[12,13] and references therein; furthermore, the Hermite functions are computed through a recurrence relation. Numericalexperiments showed that it would be possible to reduce the Hermite transform error by using variable precision arithmetic forthe computation of the quadrature nodes and weights; however, due to the additional computational effort required, we didnot further exploit this approach.

We finally note that the artificial boundary conditions introduced by the Fourier spectral method seem to have no effecton the approximation of the ground state.

1 See http://unix-docu.uibk.ac.at/zid/systeme/unix-hosts/hc-cluster/.

10 25 50 100 250 50010

10

10

10

10

10

10

10

100

102

degree of freedom

erro

r

ϑ = 1ϑ = 10ϑ = 100ϑ = 1000

10 25 50 100 250 50010

10

10

10

10

10

10

10

100

102

degree of freedom

erro

r

ϑ = 1ϑ = 10ϑ = 100ϑ = 1000

Fig. 1. Spatial error of the Hermite (left picture) and Fourier (right picture) spectral method.


5.2. Temporal order

We next determine the convergence orders of various exponential operator splitting methods listed in Table 1 when ap-plied to the two-dimensional GPE (2) with harmonic potential VðxÞ ¼ VHðxÞ.

To this purpose, we consider the time evolution of the ground state wðx;0Þ up to a final time T ¼ 1, see also (4a). First ofall, we verified the reliability of our code comparing the two numerical reference solutions obtained for 128 128 Hermiteand Fourier basis functions, respectively, and the time step size h ¼ 211 with the exact solution given bywðx; TÞ ¼ eilTwðx;0Þ. We then computed the temporal convergence orders in a standard way for different time step sizesranging from 29 to 1. An accuracy comparison of different time-splitting spectral methods with respect to a common ref-erence solution will be given in Section 5.3.

The results obtained for # ¼ 1 and # ¼ 100, respectively, are displayed in Figs. 2 and 3; the slope of the dashed-dotted linereflects the expected classical order. We refer to the fourth- and sixth-order splitting schemes by the initials of the authorsand their orders of convergence, see also Table 1. In particular, the partitioned Runge–Kutta methods PRKS6 and PRKS10 aswell as the Runge–Kutta–Nyström methods SRKNb

6, SRKNb11, and SRKNa

14 by Blanes and Moan [5, Tables 2 and 3] are denotedby BM4-1, BM6-1 and BM4-2, BM6-2, BM6-3, respectively; we note that the schemes SRKNb

6 (BM4-2) and SRKNa14 (BM6-3)

are claimed to be favourable in view of their small error constants.The Hermite and Fourier space discretisations show a similar behaviour. As expected, for # ¼ 1 the temporal convergence

order is clearly obtained for each splitting method. As soon as the nonlinear part increases, i.e. for # ¼ 100, the error in-creases; furthermore deflections in the temporal order might occur for larger time step sizes.

In order to illustrate the efficiency of the considered splitting methods, we further include the temporal error versus thetotal number of the spectral transformations reflecting the principal computational cost in the time integration, see Figs. 4and 5; the displayed results confirm the favourable behaviour of the schemes BM4-2 and BM6-3. Although the cost of theFast Fourier Transform in two space dimensions is OðM2 log MÞ compared with a cost of OðM3Þ for the Hermite transform,in the present situation, for values M 6 128, the latter turns out to be comparable or even faster; this behaviour is wellknown, see Boyd [6, Chapter 10] and also observed in Fig. 6 (right picture), where the mean computational cost of a singlespectral transform in two space dimensions is given.

5.3. Long-term integration

In order to illustrate the effectiveness of higher-order time-splitting Fourier and Hermite spectral methods in long-termintegrations, we consider the two-dimensional time-dependent GPE (2) with harmonic potential VðxÞ ¼ 2VHðxÞ and # ¼ 1 onthe time interval ½0; T where T ¼ 400; as initial value we choose the ground state of the GPE with harmonic potential

10 10 10 100

10

10

100

step size

erro

r

StrangMcLachlan

10 10 10 100

10

10

100

step size

erro

r

Y4S4M4

10 10 10 100

10

10

100

step size

erro

r

Y6KL6S6

10 10 10 100

10

10

100

step size

erro

r

StrangMcLachlan

10 10 10 100

10

10

100

step size

erro

r

Y4S4M4

10 10 10 100

10

10

100

step size

erro

r

Y6KL6S6

Fig. 2. Temporal orders of various time-splitting Hermite (first row) and Fourier (second row) spectral methods when applied to the two-dimensional GPE(2) with # ¼ 1.


VðxÞ ¼ VHðxÞ at t ¼ 0. Following Dion and Cancès [11] we call this experiment breathing; for 0 6 t 6 13 the solution is dis-played in Fig. 7.

Among the previously considered time-splitting methods, we select the widely used Strang splitting as well as the meth-ods SRKNb

6 (BM4-2) and SRKNa14 (BM6-3) by Blanes and Moan [5]. A reference solution is computed by means of the scheme

10 10 10 100

10

10

100

step size

erro

r

StrangMcLachlan

10 10 10 100

10

10

100

step size

erro

r

Y4S4M4

10 10 10 100

10

10

100

step size

erro

r

Y6KL6S6

10 10 10 100

10

10

100

step size

erro

r

StrangMcLachlan

10 10 10 100

10

10

100

step size

erro

r

Y4S4M4

10 10 10 100

10

10

100

step size

erro

r

Y6KL6S6

Fig. 3. Temporal orders of various time-splitting Hermite (first row) and Fourier (second row) spectral methods when applied to the two-dimensional GPE(2) with # ¼ 100.

100

101

102

103

10

10

100

number of transformations

erro

r

StrangMcLachlan

100

101

102

103

10

10

100


erro

r

Y4S4M4

100

101

102

103

10

10

100


erro

r

Y6KL6S6

100

101

102

103

10

10

100


erro

r

StrangMcLachlan

100

101

102

103

10

10

100


erro

r

Y4S4M4

100

101

102

103

10

10

100


erro

r

Y6KL6S6

Fig. 4. Efficiency of various time-splitting Hermite (first row) and Fourier (second row) spectral methods when applied to the two-dimensional GPE (2) with# ¼ 1.


SRKNa14 with 128 128 degrees of freedom and a temporal step size corresponding to N ¼ 217 time steps. For different com-

binations of degrees of freedom and temporal step sizes, constrained to be equal to powers of two, we compute the globalerror in the L2-norm and the total number of spectral transformations; further, we measure the particle number (3a) andenergy (3b) conservation with respect to the initial values. Prescribing certain tolerances for the global discretisation error,the optimal performances corresponding to the smallest values of the required degrees of freedom and the number of spec-tral transformations are displayed in Table 2.

In the present situation, for any time integration method, the number of basis functions required for the Hermite spectralmethod is always smaller than the number of basis functions required for the Fourier spectral method; moreover, in manycases, the number of Hermite spectral transformations is smaller than the number of Fourier spectral transformations. Thisobservation is in accordance with Figs. 2 and 4 showing that the Hermite spectral method is slightly more accurate. Com-paring the time-splitting methods, the fourth and sixth-order schemes, which behave in a similar manner, require less spec-tral transformations than the second-order Strang splitting and thus are more efficient; moreover, for the Strang splitting, itwas not possible to reach a tolerance smaller than 104 within the maximal number of 215 timesteps. For each time-splittingspectral method, the particle number and the energy are well preserved.

100

101

102

103

10

10

100


erro

r

StrangMcLachlan

100

101

102

103

10

10

100


erro

r

Y4S4M4

100

101

102

103

10

10

100


erro

r

Y6KL6S6

100

101

102

103

10

10

100


erro

r

StrangMcLachlan

100

101

102

103

10

10

100


erro

rY4S4M4

100

101

102

103

10

10

100


erro

r

Y6KL6S6

Fig. 5. Efficiency of various time-splitting Hermite (first row) and Fourier (second row) spectral methods when applied to the two-dimensional GPE (2) with# ¼ 100.

101

102

10

10

10

10

degree of freedom

CP

U s

econ

ds


101

102

10

10

10

10

10

degree of freedom

CP

U s

econ

ds


Fig. 6. Computation time of the Hermite and Fourier spectral methods in one (left picture) and two (right picture) space dimensions.


We also performed this long-term integration using two standard explicit methods; we chose a constant step size Runge–Kutta method of order four, see also Dion and Cancès [11], and further the adaptive Runge–Kutta method by Dormand andPrince implemented in the MATLAB-routine ode45. As the stiffness of the problem restricts the maximal temporal step size,the time-splitting methods outperform the explicit Runge–Kutta methods.

01

23

02

46

810

12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x1

t

Fig. 7. Numerical solution wðx1;0; tÞ of the GPE (‘‘breathing”) for # ¼ 1 up to a final time T ¼ 13. Slice along x2 ¼ 0.

Table 2Time integration of the GPE (‘‘breathing”) with # ¼ 1 up to T ¼ 400. For a tolerance (Tol.), the degree of freedom (D.o.f.), the number of transformations(#Transf.), the particle number conservation error (Dpn ¼ jkwð;0Þk2

L2 kwð; TÞk2L2 j), and the energy conservation error (DE ¼ jEðwð;0ÞÞ Eðwð; TÞÞj) are reported.

Tol. M D.o.f. #Transf. Dpn DE

< 102 Hermite 2 32 32 16384 2:6 1011 4:2 106

< 102 Fourier 2 64 64 32768 3:6 1013 1:6 106

< 102 Hermite 4 32 32 6144 9:7 1012 1:1 105

< 102 Fourier 4 64 64 12288 1:7 1013 9:1 107

< 102 Hermite 6 32 32 14337 2:3 1011 3:2 108

< 102 Fourier 6 64 64 7169 1:1 1013 6:8 106

< 102 Hermite rk4 32 32 65532 2:1 105 1:2 104

< 102 Fourier rk4 64 64 524284 6:4 1010 3:7 109

< 102 Hermite ode45 32 32 208376 2:6 108 1:5 107

< 102 Fourier ode45 64 64 1132436 5:6 1012 3:1 1011

< 104 Hermite 4 32 32 12288 1:9 1011 2:6 109

< 104 Fourier 4 128 128 12288 1:6 1012 1:8 109

< 104 Hermite 6 32 32 14337 2:3 1011 3:2 108

< 104 Fourier 6 128 128 14337 2:0 1012 2:5 108

< 104 Hermite rk4 32 32 131068 6:5 107 3:8 106

< 104 Fourier rk4 128 128 524284 6:4 1010 3:7 109

< 104 Hermite ode45 32 32 208376 2:6 108 1:5 107

< 104 Fourier ode45 128 128 1411226 1:3 109 9:4 107

< 106 Hermite 4 64 64 24576 1:0 1010 1:1 1010

< 106 Fourier 4 128 128 49152 6:7 1012 1:2 1011

< 106 Hermite 6 64 64 28673 1:2 108 2:1 1010

< 106 Fourier 6 128 128 28673 4:2 1012 8:7 1012

< 106 Hermite rk4 64 64 524284 6:4 1010 3:7 109

< 106 Fourier rk4 128 128 524284 6:4 1010 3:7 109

< 106 Hermite ode45 64 64 509816 3:6 1010 2:1 109

< 106 Fourier ode45 128 128 1411448 2:2 1012 1:1 1011


For larger values of #, additional experiments not reported here showed that the Fourier spectral method becomes favour-able; moreover, smaller time step sizes are required in order to reach the prescribed tolerances, see also Section 5.1. Theinfluence of # on the convergence behaviour of time-splitting spectral methods should be investigated further.

6. Conclusions and future work

We devoted the present paper to high-accuracy discretisations of the time-dependent GPE (2), based on Hermite and Fou-rier pseudospectral methods and exponential operator splitting methods. In particular, we presented numerical comparisonsregarding accuracy and efficiency. In most of our experiments, we used the ground state solution of (2) as a reliable referencesolution.

As expected, our numerical experiments showed that the spectral methods perform well regarding accuracy, efficiency,and the preservation of geometric properties. For the time integration we compared the second order Strang splitting withfourth and sixth-order splitting methods given in Blanes and Moan [5]; the higher-order schemes proved to be superiorwhen low tolerances are required or a long-term integration is carried out. Furthermore, each time-splitting method outper-formed the explicit Runge–Kutta methods in the ‘‘breathing” experiment.

Following [17,26], it remains open to provide a stability and convergence analysis for high-order exponential operatorsplitting methods when applied to the time-dependent GPE (2). Furthermore, it is of interest to investigate the accuracyof time-splitting methods when the nonlinear part increases.

References

[1] W. Bao, D. Jaksch, P. Markowich, Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation, J. Comp. Phys. 187 (2003) 318–342.

[2] W. Bao, J. Shen, A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose–Einstein condensates, SIAM J. Sci. Comput. 26/6(2005) 2010–2028.

[3] W. Bao, W. Tang, Ground-state solution of Bose–Einstein condensate by directly minimising the energy functional, J. Comp. Phys. 187 (2003) 230–254.[4] C. Besse, B. Bidégaray, S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 40/5

(2002) 26–40.[5] S. Blanes, P.C. Moan, Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods, J. Comput. Appl. Math. 142 (2002) 313–330.[6] J. Boyd, Chebyshev and Fourier Spectral Methods, second ed., Dover, New York, 2001.[7] M. Caliari, G. Kirchner, M. Thalhammer, Convergence and energy conservation of the Strang time-splitting Hermite spectral method for nonlinear

Schrödinger equations, Universität Innsbruck, 2007, Preprint.[8] M. Caliari, A. Ostermann, S. Rainer, M. Thalhammer, A minimisation approach for computing the ground state of Gross–Pitaevskii systems, J. Comp.

Phys., in press, doi:10.1016/j.jcp.2008.09.018.[9] Th. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos 26, I.M.U.F.R.J., Rio de Janeiro, 1989.

[10] E. Celledoni, D. Cohen, B. Owren, Symmetric exponential integrators for the cubic Schrödinger equation, Preprint Numerics 3/2006, University ofTrondheim, 2006.

[11] C.M. Dion, E. Cancès, Spectral method for the time-dependent Gross–Pitaevskii equation with a harmonic trap, Phys. Rev. E 67 (2003) 046706.[12] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, 2004.[13] W. Gautschi, Orthogonal polynomials (in Matlab), J. Comput. Appl. Math. 178 (2005) 215–234.[14] E.P. Gross, Structure of a quantized vortex in boson systems, Nuovo. Cimento. 20 (1961) 454–477.[15] E. Hairer, Ch. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer,

Berlin, 2002.[16] W. Kahan, R.-C. Li, Composition constants for raising the orders of unconventional schemes for ordinary differential equations, Math. Comput. 66

(1997) 1089–1099.[17] Ch. Lubich, On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations, Math. Comp. 77/264 (2008) 2141–2153.[18] R.I. McLachlan, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comput. 16 (1995)

151–168.[19] R.I. McLachlan, R. Quispel, Splitting methods, Acta Numer. 11 (2002) 341–434.[20] V.M. Pérez–García, X. Liu, Numerical methods for the simulation of trapped nonlinear Schrödinger systems, Appl. Math. Comp. 144 (2003) 215–235.[21] L.P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP 13 (1961) 451–454.[22] G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968) 506–517.[23] M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A 146

(1990) 319–323.[24] Y.-F. Tang, L. Vázquez, F. Zhang, V.M. Pérez–García, Symplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl. 32/5 (1996) 73–

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Comput. Phys. 55/2 (1984) 203–230.[26] M. Thalhammer, High-order exponential operator splitting methods for time-dependent Schrödinger equations, SIAM J. Numer. Anal. 46/4 (2008)

2022–2038.[27] H.F. Trotter, On the product of semi-groups of operators, Proc. Am. Math. Soc. 10 (1959) 545–551.[28] H. Yoshida, Construction of higher order symplectic integrators, Phys. Let. A 150 (1990) 262–268.


A minimisation approach for computing the groundstate of Gross–Pitaevskii systems

Marco Caliari a,*, Alexander Ostermann b, Stefan Rainer b, Mechthild Thalhammer b

a Dipartimento di Informatica, Università degli Studi di Verona, Ca’ Vignal 2, Strada Le Grazie 15, I–37134 Verona, Italyb Institut für Mathematik, Leopold–Franzens Universität Innsbruck, Technikerstraße 13/7, A–6020 Innsbruck, Austria

a r t i c l e i n f o

Article history:Received 20 March 2008Received in revised form 10 September2008Accepted 15 September 2008Available online 2 October 2008

Keywords:Nonlinear Schrödinger equationsGround stateSpectral methodsMinimisation

a b s t r a c t

In this paper, we present a minimisation method for computing the ground state of sys-tems of coupled Gross–Pitaevskii equations. Our approach relies on a spectral decomposi-tion of the solution into Hermite basis functions. Inserting the spectral representation intothe energy functional yields a constrained nonlinear minimisation problem for the coeffi-cients. For its numerical solution, we employ a Newton-like method with an approximateline-search strategy. We analyse this method and prove global convergence. Appropriatestarting values for the minimisation process are determined by a standard continuationstrategy. Numerical examples with two- and three-component two-dimensional conden-sates are included. These experiments demonstrate the reliability of our method and nicelyillustrate the effect of phase segregation.

2008 Elsevier Inc. All rights reserved.

1. Introduction

The field of low-temperature physics has been fascinating and inspiring many scientists, in particular in the last decade,see [16] and references given therein. Memorable achievements were the first experimental realisations [1,19,22] of a singleBose–Einstein Condensate (BEC) in 1995, and of BECs for a mixture of two and three different interacting atomic species,respectively. Mathematically, BECs are modelled by nonlinear time-dependent Schrödinger equations; more precisely, theorder parameters of the condensates are solutions of a system of coupled Gross–Pitaevskii Equations (GPEs, [17,20]).

In the present paper, we are concerned with computing the ground state of a system of GPEs, a special solution of minimalenergy. To this purpose, as suggested in [5], we directly minimise the energy functional. We mention that an alternative ap-proach for the ground state computation is provided by the imaginary time method which can also be considered as a steep-est descent method, see [2,3,25] e.g. resulting in a parabolic evolution equation. Besides, an optimal damping algorithmbased on the inverse power method is used in [11]; then, the conjugate gradient method is applied for the solution of thearising linear systems. We do not exploit these and other ([12,23]) approaches here. Our objectives are twofold. In a generalcontext, we present a Newton-like method for the numerical solution of a constrained minimisation problem and study itsconvergence. Moreover, we apply the minimisation approach specifically to a system of GPEs to simulate the ground statesolution. Numerical results for two- and three-component two-dimensional condensate illustrating the effect of phase seg-regation [6,7,24] are provided.

The structure of the paper is as follows. In Section 2, we first introduce the system of GPEs in a normalised form and thenstate the minimisation problem. To avoid technicalities, we give a detailed description for the case of a single equation and

0021-9991/$ - see front matter 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jcp.2008.09.018

* Corresponding author. Tel.: +39 0458027809.E-mail address: [email protected] (M. Caliari).

Journal of Computational Physics 228 (2009) 349–360

Contents lists available at ScienceDirect

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journal homepage: www.elsevier .com/locate / jcp

then sketch the extension to systems. Our approach relies on a spectral decomposition of the ground state solution into Her-mite basis functions (see also [4,10]); we point out that similar ideas could be used in combination with Fourier techniquesor other spectral methods. Inserting the resulting representation into the energy functional leads to a constrained nonlinearminimisation problem for the spectral coefficients. Section 3 is devoted to the description and analysis of a numerical meth-od for the minimisation problem. We use a Newton-like method involving an approximate line-search strategy and contin-uation techniques. Finally, in Section 4, we illustrate the capability of our method by three numerical examples for systemsof two or three coupled Gross–Pitaevskii equations in two space dimensions.

2. Ground state of Gross–Pitaevskii systems

In this section, we present a constrained minimisation approach for computing the ground state of systems of coupledGross–Pitaevskii equations. For our purposes, it is useful to employ a normalised form of the problem which we introducein Section 2.1. As the discussion of the general case would involve additional technicalities, we first restrict ourselves tothe case of a single Gross–Pitaevskii equation; the extension to systems of coupled Gross–Pitaevskii equations is thensketched in Section 2.4. For the convenience of the reader, we further recall basic results on Hermite functions in Section2.2.

We henceforth employ the vector notation x ¼ ðx1; . . . ; xdÞ 2 Rd and denote by D the d-dimensional Laplacian with respectto x. The Lebesgue space L2 ¼ L2ðRd;CÞ of square integrable complex-valued functions is endowed as usual with the scalarproduct ð j ÞL2 and the associated norm k kL2 given by

ðf jgÞL2 ¼Z

Rdf ðxÞgðxÞdx; kfkL2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiðf jf ÞL2

q; f ; g 2 L2:

2.1. Gross–Pitaevskii equation

We consider the d-dimensional nonlinear Schrödinger equation

ihotWðx; tÞ ¼ h2

2m Dþ VðxÞ þ h2gjWðx; tÞj2

Wðx; tÞ;

kWð;0Þk2L2 ¼ N

8<: ð1Þ

where W : Rd RP0 ! C, ðx; tÞ#Wðx; tÞ, V : Rd ! R, and h denotes the reduced Planck’s constant. In three space dimensionsthis equation is called Gross–Pitaevskii equation (see [17,20]) and describes the order parameter W of a Bose–Einstein con-densate of an atomic species of mass m and total number of particles N, trapped in an external potential V; moreover, thecoupling constant g equals 4pr

m , with r being the scattering length of the species. We restrict ourselves to the defocusing case,that is r P 0. Since, under certain assumptions on the potential V, the equation can be reduced to lower dimension [5], wewill use the same terminology and the name GPE also for the general d-dimensional case.

In the following, we employ a normalisation of (1)

iotwðn; tÞ ¼ 12 Dþ UðnÞ þ #jwðn; tÞj2

wðn; tÞ;

kwð;0Þk2L2 ¼ N

8<: ð2Þ

that is obtained by the linear transformation n ¼ffiffifficp

x with hc ¼ m. Then, setting Cwðn; tÞ ¼ Wðx; tÞ, where C ¼ffiffiffiffifficd4p

, we get theGPE (2), with real potential hUðnÞ ¼ VðxÞ and parameter # ¼ hgC2 P 0; in (2), D denotes the d-dimensional Laplacian withrespect to n. It is easy to see that the particle number N and the energy which, for the normalised GPE (2), is given by

Eðwð; tÞÞ ¼ 12

Dþ U þ 12#jwð; tÞj2

wð; tÞjwð; tÞ

L2

ð3Þ

are preserved quantities in time. We further assume that the energy is positive and finite.

2.2. Hermite spectral decomposition

For any integer j P 0 and real number c > 0, we denote by Hcj ðnÞ the uni-variate Hermite polynomial of degree j, norma-

lised with respect to the weight function wðnÞ ¼ ec2n2. Hermite polynomials satisfy the recurrence relation

Hc0ðnÞ ¼

ffiffiffiffiffic2

p4

r; Hc

1ðnÞ ¼ffiffiffiffiffiffiffiffi4c6

p4

rn;ffiffi

jp

Hcj ðnÞ ¼

ffiffiffi2p

cnHcj1ðnÞ

ffiffiffiffiffiffiffiffiffiffiffij 1

pHc

j2ðnÞ; j P 2:ð4Þ

We further recall that the derivative satisfies

onHcj ðnÞ ¼ c

ffiffiffiffiffi2j

pHc

j1ðnÞ: ð5Þ

350 M. Caliari et al. / Journal of Computational Physics 228 (2009) 349–360

The corresponding Hermite function Hcj ðnÞ is defined through

Hcj ðnÞ ¼ Hc

j ðnÞe1

2c2n2: ð6Þ

Hence, the Hermite functions ðHcj Þ form an orthonormal basis of the function space L2ðRÞ, i.e. it holds

ðHcj jH

ckÞL2ðRÞ ¼ djk ð7aÞ

with Kronecker djk. As a consequence, for every function u 2 L2ðRÞ, the representation

u ¼X

j

ujHcj ; uj ¼ ðujH

cj ÞL2ðRÞ ð7bÞ

is valid; moreover, Parseval’s equality follows:

kuk2L2ðRÞ ¼

Xj

jujj2: ð7cÞ

The above relations (5) and (6) imply

ðo2n þ c4n2ÞHc

j ðnÞ ¼ 2kjHcj ðnÞ; 2kj ¼ c2ð1þ 2jÞ; ð8Þ

that is, the Hermite functions ðHcj Þ are eigenfunctions of the differential operator o2

n þ c4n2, with corresponding eigenvalues2kj.

Using the tensor basis of the Hermite functions

Hcj ðnÞ ¼ Hc1

j1ðn1Þ . . . Hcd

jdðndÞe

12 c2

1n21þþc2

dn2

dð Þ

where, with abuse of notation, now we assume n ¼ ðn1; . . . ; ndÞ and j ¼ ðj1; . . . ; jdÞ, the extension to the d-variate case isstraightforward. We only notice that (8) rewrites

ðDþ UcðnÞÞHcj ðnÞ ¼ 2kjHc

j ðnÞ; 2kj ¼Xd

k¼1

c2kð1þ 2jkÞ; ð9Þ

with the standard harmonic potential UcðnÞ ¼Pd

k¼1c4kn

2k .

2.3. Minimisation approach

The ground state of the GPE (2) is a solution of the form

wðn; tÞ ¼ eiltuðnÞ; l 2 R; u 2 L2ðRd;RÞ ð10Þ

that minimises energy functional (3); in particular, it is required that u fulfills the relations

EðuÞ ¼ 12

Dþ U þ 12#u2

uju

L2!min;

GðuÞ ¼ kuk2L2 N ¼ 0:

ð11aÞ

In view of the computation of the chemical potential l, we consider the Lagrange function Eðu;gÞ ¼ EðuÞ þ gGðuÞ. Usingthe fact that the local minima of E are solutions of rEðu;gÞ ¼ 0, we obtain the nonlinear system

12

Dþ U þ #u2

u ¼ gu; kuk2L2 ¼ N; ð11bÞ

the first relation in (11b) implies EðuÞ þ 12#ku2k2

L2 ¼ Ng. We note that the constrained nonlinear eigenvalue problem (11b)also follows by inserting the representation (10) into (2); we conclude

Nl ¼ Ng ¼ EðuÞ þ 12#ku2k2

L2 ;

i.e., l coincides with the Lagrange multiplier g.For the numerical solution of (11a), we employ the spectral representation (7b) truncated to J 1 and property (8) to re-

write (11a) as follows:

EqðuÞ ¼XJ1

jjj¼0

kju2j þ q

ZRd

UðnÞ 12

UcðnÞ XJ1

jjj¼0

ujHcj ðnÞ

!2

dnþ 12#

ZRd

XJ1

jjj¼0

ujHcj ðnÞ

!4

dn!min; ð12aÞ

GðuÞ ¼XJ1

jjj¼0

u2j N ¼ 0; ð12bÞ

M. Caliari et al. / Journal of Computational Physics 228 (2009) 349–360 351

where j j j¼maxfj1; . . . ; jdg, and where q is an additional parameter. Its significance will become clear later in Section 3.3.Here, we only mention that the energy EqðuÞ to be minimised corresponds to the choice q ¼ 1. Moreover, we approximatethe integrals by means of the Gauss–Hermite quadrature formula with 2J 1 nodes; this allows the exact integration of thelast term of (12a).

The choice of c clearly depends on the potential UðnÞ and on #. For example, the classical and widely used (in [5,11,25],e.g.) harmonic potential

VðxÞ ¼ m2

Xd

k¼1

x2kx2

k

allows to have UðnÞ 12 UcðnÞ with c2

k ¼ xk, k ¼ 1; . . . ; d. On the other hand, the larger # is, the wider the region is where theparticles are mainly concentrated. Thus, smaller values of ck would help, with a better matching of the exponential decay ofthe Hermite functions (6).

Henceforth, we write the minimisation problem (12) in the abstract form

FðxÞ !min;GlðxÞ ¼ 0; 1 6 l 6 ‘:

ð13Þ

In the case of a single GPE, which we considered up to now, we have ‘ ¼ 1. The functions F and G1 ¼ G replace the(approximated) energy functional E and the constraint, respectively; further, the unknown x 2 Rn takes the role of finitelymany spectral coefficients uj.

2.4. Extension to Gross–Pitaevskii systems

In this section, we sketch how the above minimisation approach for the ground state computation extends to multicom-ponent systems of GPEs. Let us consider the system of ‘ GPEs.

ihotWðlÞ ¼ h2

2mlDþ Vl þ h2glljWðlÞj

2 þ h2 P‘k¼1k–l

glkjWðkÞj2

0B@

1CAWðlÞ;

kWðlÞk2L2 ¼ Nl; l ¼ 1; . . . ; ‘

8>>>><>>>>:

ð14Þ

describing the order parameters WðlÞ : Rd RP0 ! C, ðx; tÞ#WðlÞðx; tÞ of atomic species with masses ml, see [15] and also[11,26]. We call gll intra-species coupling constants and glk ¼ gkl, l–k inter-species coupling constants; for d ¼ 3, glk equals2prlk

mlþmkmlmk

, where rlk is the scattering length for the l-k species.Again, we restrict ourselves to the defocusing case rlk P 0. By a linear transformation, analogously to before, system (14)

takes the form

iotwðlÞ ¼ hc

2mlDþ Ul þ

P‘k¼1

#lkjwðkÞj2

wðlÞ;

kwðlÞk2L2 ¼ Nl; l ¼ 1; . . . ; ‘

8><>: ð15Þ

with

hc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim1 m‘

‘p

and #lk ¼ hglkC2; C ¼ffiffiffiffifficd4p

:

Let

w ¼ ðwð1Þ; . . . ;wð‘ÞÞ; u ¼ ðuð1Þ; . . . ;uð‘ÞÞ:

The ground state w of the GPEs system (15) is a special solution

wðlÞðn; tÞ ¼ eill tuðlÞðnÞ; l ¼ 1; . . . ; ‘

that minimises the energy functional

EðuÞ ¼X‘l¼1

hc2ml

Dþ Ul þ12

X‘k¼1

#lkjuðkÞj2 !

uðlÞjuðlÞ !

L2

:

The chemical potentials ll are given by

llNl ¼ hc2ml

Dþ Ul þX‘k¼1

#lkjuðkÞj2 !

uðlÞjuðlÞ !

L2

:

In order to compute the ground state of (15) we thus consider the constrained minimisation problem


EðuÞ !min;

GlðuÞ ¼ kuðlÞk2L2 Nl ¼ 0; 1 6 l 6 ‘:

As before, we employ a spectral decomposition of uðlÞ into a common basis of Hermite functions; truncating the infinite sumswe finally get

EqðuÞ ¼X‘l¼1

hcml

XJ1

jjj¼0

kjðuðlÞj Þ2 þ ql

ZRd

UlðnÞ hc

2mlUcðnÞ

XJ1

jjj¼0

uðlÞj Hcj ðnÞ

!2

dn

24

þ12

X‘k¼1

#lk

ZRd

XJ1

jjj¼0

uðkÞj Hcj ðnÞ

!2 XJ1

jjj¼0

uðlÞj Hcj ðnÞ

!2

dn

35!min; ð16aÞ

GlðuÞ ¼XJ1

jjj¼0

ðuðlÞj Þ2 Nl ¼ 0; 1 6 l 6 ‘: ð16bÞ

This results again in a minimisation problem of the form (13).

3. Constrained minimisation

For the numerical solution of the constrained minimisation problem (13), we apply a Newton-like method with line-search; the algorithm is described and analysed in the following sections. Note that in several space dimensions, a full New-ton iteration is computationally expensive due to the large number of unknowns. Our approach is based on a simplified iter-ation where the costs of the solution of the arising linear system grow only linearly with the number of unknowns. It turnsout that our new approach is more efficient than the standard Newton iteration even in one space dimension, if a higherspatial resolution is regarded, see Fig. 1.

3.1. A Newton-like method for minimisation

Let F : Rn ! R and G1; . . . ;G‘ : Rn ! R be sufficiently smooth functions. We aim at finding a (local) minimiser xI of (13).For this purpose we take up an idea presented in Han [18] and consider the exact penalty function

PðxÞ ¼ FðxÞ þ rX‘i¼1

jGiðxÞj ð17Þ

with an appropriate penalty parameter r > 0 to be chosen in Section 3.3. DenotingrG ¼ ½rG1; . . . ;rG‘, we recall that a solu-tion xI of (13) satisfies the first order conditions

rFðxIÞT þ yITrGðxIÞT ¼ 0 ð18aÞGðxIÞ ¼ 0 ð18bÞ

with a corresponding Lagrange multiplier yI. Any point xI satisfying these first order conditions is called a critical point of(13).

Starting from a given approximation xðkÞ to the minimiser xI, we consider the quadratic minimisation problem QðxðkÞ;HðkÞÞ

0 200 400 600 800 10000.2

0.4

0.6

0.8

1

1.2

Fig. 1. CPU time in seconds necessary to compute the ground state of the one-dimensional GPE (2) with the standard harmonic potential UðnÞ ¼ 12 n2, N ¼ 1,

with J ¼ 140 Hermite functions, as a function of #. The exact Newton method (dashed line) is more expensive than our new strategy (solid line), described inSection 3.


rFðxðkÞÞTsþ 12

sTHðkÞs!min ð19aÞ

subject to the linear constraints

GiðxðkÞÞ þ rGiðxðkÞÞTs ¼ 0; i ¼ 1; . . . ; ‘: ð19bÞ

Here, HðkÞ denotes a symmetric matrix that approximates the Hessian of F. We propose to take HðkÞ ¼ HðxðkÞÞ with an appro-priate function H. The precise choice of H will be discussed in Section 3.2 below. Han shows that a solution sðkÞ of (19) is adescent direction of (17), if HðkÞ is positive definite and the corresponding Lagrange multiplier is bounded by r, see [18, Thm.3.1]. For an appropriate step length kk satisfying

PðxðkÞ þ kksðkÞÞ < PðxðkÞÞ;

the new approximation xðkþ1Þ is defined by

xðkþ1Þ ¼ xðkÞ þ kksðkÞ: ð20Þ

In order to get a globally convergent method, the step length kk has to satisfy additional conditions. Among the different pos-sibilities, we consider the first Armijo–Goldstein condition

PðxðkÞ þ kksðkÞÞ 6 PðxðkÞÞ þ akkrPðxðkÞÞTsðkÞ ð21Þ

for some fixed 0 < a < 1, independently of k. The existence of the directional derivative rPðxðkÞÞTsðkÞ is verified in [18].We employ a backtracking line-search strategy as described in [8, Section 6.3.2]. This turns out to be an essential feature

of our method. The size of a in (21) is typically quite small. In literature, the value a ¼ 104 is recommended. Starting with aninitial guess k ¼ 1 for the step length, we reduce k step by a factor b 2 ½0:1;0:5 until (21) holds. In each step of this line-search, the size of b is determined anew from a quadratic or a cubic model of the restriction of P in search direction. For de-tails of this algorithm, see [8, Algorithm A6.3.1].

The convergence properties of our algorithm are collected in the following theorem which is obtained by a small mod-ification of Theorem 3.2 in [18]. In contrast to [18], we use a different (non-exact) line-search strategy.

Theorem 1. Let F and G ¼ ðG1; . . . ;G‘ÞT be continuously differentiable and let the following conditions hold, where fxðkÞg denotesthe sequence defined in (20):

(a) The function F is bounded from below and rG has full rank.(b) The matrices HðkÞ ¼ HðxðkÞÞ are positive definite, and for all critical points xI of (13) there exists a neighborhood where H is

continuous.(c) The solution of each quadratic minimisation problem QðxðkÞ;HðkÞÞ, given by (19), has a Lagrange multiplier that is bounded by

r in the maximum norm.

Then, the sequence fxðkÞg converges to a critical point xI of (13), or any of its accumulation points is a critical point.

Proof. Since F is bounded from below, the exact penalty function P has the same property. By construction, the sequencefPðxðkÞÞg is monotonically decreasing and thus converges to pI, say. Let xI be an accumulation point of the sequencefxðkÞg. Without loss of generality, we may assume that xðkÞ converges to xI. In particular, we have kksðkÞ ! 0 for k!1.

Since F and Gi; i ¼ 1; . . . ; ‘ are continuously differentiable, and HðkÞ ¼ HðxðkÞÞwith continuous H, for k sufficiently large, thecorresponding solutions sðkÞ of (19) converge to sI which solves QðxI;HðxIÞÞ.

If sI ¼ 0, then xI obviously satisfies the first order conditions (18).Otherwise, as sI–0, we conclude from [18, Thm. 3.1] that there exists k > 0 with PðxI þ ksIÞ < PðxIÞ. In accordance with

our backtracking strategy, let kI be the largest k 2 ð0;1 that satisfies the Armijo–Goldstein condition (21). Due to thecontinuity of the data, we have that kk P kI=2 for k sufficiently large which contradicts kksðkÞ ! 0. h

3.2. Application to the GPE

In order to apply the just described method to our constrained minimisation problem (13), we have to define the functionH.

For this, we choose a singularity level TOL and set

djðxÞ ¼o2Fox2

j

þX‘i¼1

yio2Gi

ox2j

: ð22Þ

If djðxÞ 6 TOL, we modify (22) to djðxÞ ¼ 1. A typical value for the singularity level is TOL ¼ 108. With

dðxÞ ¼max 1;TOLkrFðxÞk2

kDðxÞk2

; DðxÞ ¼ ðd1ðxÞ; . . . ; dnðxÞÞT ð23Þ


we finally define the function H as

HðxÞ ¼ dðxÞdiagðd1ðxÞ; . . . ;dnðxÞÞ: ð24Þ

This choice of H satisfies the conditions of Theorem 1.In some of our examples below, the generic choice a ¼ 104 in (21) required an unexpectedly high number of iterations.

This happens even in the one-dimensional single-equation case for particular values of #, see Fig. 2. To overcome this prob-lem, we vary the size of a in the iteration process. After accepting a step, say step k, we update a as follows.

If kk ¼ 1 and rPðxðkÞÞTsðkÞ < 0, we increase a according to

anew ¼minða;1:25a;0:99Þ; a ¼ Pðxðkþ1ÞÞ PðxðkÞÞrPðxðkÞÞTsðkÞ

; ð25aÞ

else, if kk < 1, we decrease a according to

anew ¼maxð104;minðk; 0:75aÞÞ: ð25bÞ

If anew does not admit an admissible step length in step kþ 1, we put anew ¼ 104 and restart the line-search with k ¼ 1.

3.3. Choice of the penalty parameter and the starting values

Appropriate starting values for the local optimisation procedure are determined with the help of a continuation method.For a general overview of such methods, see [9]. In our case, we choose #lk and ql as continuation parameters. For this pur-pose, we replace #lk by #lk in (16a) and continue, for appropriate initial values, the parameters #lk to #lk and ql to 1. As initialvalues for #lk, we take #lk ¼ ðNlNkÞ1=2. The choice of the initial values for ql is more tricky. It depends on the difference be-tween the given and the harmonic potential. If the potentials are very close to each other, the corresponding ql is chosenclose to 1, with ql ¼ 1 for the standard harmonic potential. Otherwise, ql is taken smaller.

In the first step of the continuation method, we take as starting value u(l) the ground state of the linear Schrödinger equa-tion with standard harmonic potential. In the subsequent steps, we then take the solution of the previous step as startingvalue for u(l).

For the actual values of #lk and ql, we compute the ground state by using the above procedure. Thereafter, we enlarge #lk

and ql according to the formulas

#newlk ¼ ð1þ jÞ#lk; qnew

l ¼ ð1þ jÞql: ð26Þ

The size of j in (26) has to be chosen appropriately. We take here into account the speed of convergence of the iteration. Ifthe residual is smaller than 106 after less than three iterations, we double the value j in the next continuation step; if theresidual does not meet the condition after ten iterations, we half j and redo the whole step; else, we keep the value of j inthe next step.

This procedure allows to determine appropriate starting values even for large values of #lk in a fast and reliable way.The penalty parameter r in (17) is chosen in accordance with the size of the chemical potentials l1; . . . ;l‘. Having com-

puted the ground state u of the current step, we calculate approximations to the chemical potential for #newlk by using the just

computed u. The maximal value of these approximations is taken for r. This choice guarantees that r is an upper bound forthe Lagrange multiplier in the step for #new

lk .

200 400 600 800 10000

100

200

300

400

500

600

200 400 600 800 10000

100

200

300

400

500

600

Fig. 2. Number of iterations necessary to compute the ground state of the one-dimensional GPE (2) with the standard harmonic potential UðnÞ ¼ 12 n2, N ¼ 1,

as a function of #. The left figure displays the results obtained with the traditional choice a ¼ 104, the right figure those obtained with the new strategy(25) for selecting a.


4. Numerical implementation and illustrations

We have implemented the above algorithm in MATLAB and it can be used without any restriction also in GNU OCTAVE, versiongreater or equal to 3.0.0. In the present version, our code can treat coupled systems of GPEs in one or two space dimensions.Gauss–Hermite quadrature points are computed using the stable and efficient routines provided in [13,14]. Hermite func-tions at quadrature points are computed once and for all using the stable recurrence relation (4). The program is freely avail-able from the authors on request.

The following examples illustrate the capability of our method. We note that the choice of the parameters is not strictlyrelated to physical experiments. In particular, in all the numerical examples we use the same masses for all the components,but vary the scattering lengths. We emphasise however that in our implementation there is no restriction at all in using dif-ferent masses. SI units are used throughout this section.

Example 1. As a first numerical example, we first consider the case of a two-component (‘ ¼ 2) two-dimensionalcondensate, modelled by (15), each component with the same atomic species 87Rb, mass ml ¼ m ¼ 1:44 1025, and the samenumber of particles Nl ¼ N ¼ 107. The intra-species coupling constants are #11 ¼ 1:3 106 and #22 ¼ 1:3 1011. Thepotentials are scaled and off-centered harmonic potentials (see, e.g., [2]), namely

Ulðn1; n2Þ ¼12½ðx1 lðn1 n1 lÞÞ

2 þ ðx2 lðn2 n2 lÞÞ2

with

x11 ¼ p; x21 ¼ p; n11 ¼ 0; n21 ¼ 0;x12 ¼ 3p; x22 ¼ 3p; n12 ¼ 0:19; n22 ¼ 0:

We perform four numerical experiments, with identical inter-species coupling constant #lk, l–k, assuming the values 0,2:5 107, 1:0 106 and 2:0 106, respectively. The number of Hermite functions was fixed to J ¼ 64 for each directionand component, resulting in a total of ‘J2 ¼ 8192 degrees of freedom for the spectral coefficients. The required CPU time(on a 2.2 GHz CPU) is about 2 s. The contour plots of the solution are given in Fig. 3.

Example 2. As a second example, we consider the case of a three-component (‘ ¼ 3) two-dimensional condensate, modelledby (15), each component with the same atomic species 87Rb, mass ml ¼ m ¼ 1:44 1025, and the same number of particlesNl ¼ N ¼ 107. The intra-species coupling constants are #11 ¼ #33 ¼ 1:3 105 and #22 ¼ 6:3 108. The potentials are scaled,off-centered and rotated harmonic potentials, namely

Ulðn1; n2Þ ¼12½ðx1 lððn1 n1 lÞ cos Xl þ ðn2 n2 lÞ sin XlÞÞ2 þ ðx2 lððn1 n1 lÞ sin Xl ðn2 n2 lÞ cos XlÞÞ2

with

x11 ¼32p; x21 ¼ p; X1 ¼ 0; n11 ¼ 0:37; n21 ¼ 0;

x12 ¼ p; x22 ¼ 2p; X2 ¼p4; n12 ¼ 0; n22 ¼ 0;

x13 ¼32p; x23 ¼ p; X3 ¼ 0; n13 ¼ 0:37; n23 ¼ 0:

We perform five numerical experiments, with identical inter-species coupling constant #lk, l–k, assuming the values 0,1:3 106, 2:5 106, 5:0 106 and 1:0 105, respectively. The number of Hermite functions was fixed to J ¼ 64 for eachdirection and component, resulting in a total of ‘J2 ¼ 12,288 degrees of freedom for the spectral coefficients. The requiredCPU time is about 2 s. The contour plots of the solution are given in Fig. 4.

Increasing the inter-species coupling constant #lk clearly shows the phase segregation phenomenon (see [21,24]) alreadydiscussed and proved in [6,7] for the two-component and the three-component condensate, respectively.

Example 3. Finally, we consider again the case of a three-component two-dimensional condensate, each component withthe same atomic species 87Rb, mass ml ¼ m ¼ 1:44 1025, and the same number of particles Nl ¼ N ¼ 107. The intra-speciescoupling constants are #ll ¼ 1:3 105, l ¼ 1;2;3 and #12 ¼ #21 ¼ 0. The potentials are scaled and strongly anisotropic har-monic potentials, namely

Ulðn1; n2Þ ¼12½ðx1 ln1Þ2 þ ðx2 ln2Þ2

with

x11 ¼ p; x21 ¼ 10p; x12 ¼ 10p; x22 ¼ p; x13 ¼ p; x23 ¼ p:

We perform five numerical experiments, with #13 ¼ #31 ¼ #23 ¼ #32 assuming the values 0, 5 106, 1:3 105, 2:5 105 and5 105, respectively. The number of Hermite functions was fixed to J ¼ 70 for each direction and component, resulting in a


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Fig. 3. Contour plots of the ground state ðuð1ÞðnÞ;uð2ÞðnÞÞ of Example 1 (left–right) for #lk , l–k, assuming the values 0, 2:5 107, 1:0 106 and 2:0 106

(top–bottom). The unit length for the function values is 103.


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1.5

2

2.5

3

3.5

4

4.5

ξ1

ξ2

0 1.5 3

0

1.5

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

ξ1

ξ2

0 1.5 3

0

1.5

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

ξ1

ξ2

0 1.5 3

0

1.5

3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

ξ1

ξ2

0 1.5 3

0

1.5

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 4. Contour plots of the ground state ðuð1ÞðnÞ;uð2ÞðnÞ;uð3ÞðnÞÞ of Example 2 (left–right) for #lk , l–k, assuming the values 0, 1:3 106, 2:5 106, 5:0 106

and 1:0 105 (top–bottom). The unit length for the function values is 103.


ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.2

0.4

0.6

0.8

1

1.2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.2

0.4

0.6

0.8

1

1.2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.2

0.4

0.6

0.8

1

1.2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.2

0.4

0.6

0.8

1

1.2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.5

1

1.5

2

ξ1

ξ2

0 2.5 5

0

2.5

5

0

0.2

0.4

0.6

0.8

1

1.2

Fig. 5. Contour plots of the ground state ðuð1ÞðnÞ;uð2ÞðnÞ;uð3ÞðnÞÞ of Example 3 (left–right) for #13 ¼ #31 ¼ #23 ¼ #32 assuming the values 0, 5 106,1:3 105, 2:5 105 and 5 105, (top–bottom). The unit length for the function values is 103.


total of ‘J2 ¼ 14,700 degrees of freedom for the spectral coefficients. The required CPU time is about 12 s. The contour plots ofthe solution are reported in Fig. 5.

5. Conclusions

In this paper, we were concerned with the numerical computation of the ground state of Gross–Pitaevskii systems. Bymeans of a spectral discretisation, we transformed the problem into a constrained minimisation problem and employed aNewton-like method with approximate line-search for its numerical solution. The algorithm was implemented in MATLAB

(and successfully tested in GNU OCTAVE); the code is available from the authors on request. The enclosed numerical examplesclearly demonstrate the reliability of the new method. We point out that the presented minimisation approach is neitherrestricted to Gross–Pitaevskii systems nor to Hermite basis functions.

Acknowledgment

The authors wish to thank Marco Squassina for providing the settings used for the experiment illustrated in Fig. 5.

References

[1] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Observation of Bose–Einstein condensation in a dilute atomic vapor, Science 269(5221) (1995) 198–201.

[2] W. Bao, Ground states and dynamics of multicomponent Bose–Einstein condensates, Multiscale Model. Simul. 2 (2) (2004) 210–236.[3] W. Bao, Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput. 25 (5) (2004)

1674–1697.[4] W. Bao, J. Shen, A fourth-order time-splitting Laguerre–Hermite pseudo-spectral method for Bose–Einstein condensates, SIAM J. Sci. Comput. 26 (6)

(2005) 2010–2028.[5] W. Bao, W. Tang, Ground-state solution of Bose–Einstein condensate by directly minimizing the energy functional, J. Comp. Phys. 187 (2003) 230–254.[6] M. Caliari, M. Squassina, Location and phase segregation of ground and excited states for 2D Gross–Pitaevskii systems, Dyn. Partial Differ. Equ. 5 (2)

(2008) 117–137.[7] M. Caliari, M. Squassina, Spatial patterns for the three species Gross–Pitaevskii system in the plane, Electron. J. Diff. Eqns. 79 (2008) 1–15.[8] J.E. Dennis, R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, 1998.[9] P. Deuflhard, Newton Methods for Nonlinear Problems, Springer, Berlin, 2004.

[10] C.M. Dion, E. Cancès, Spectral method for the time-dependent Gross–Pitaevskii equation with a harmonic trap, Phys. Rev. E 67 (2003) 046706.[11] C.M. Dion, E. Cancès, Ground state of the time-independent Gross–Pitaevskii equation, Comput. Phys. Commun. 177 (10) (2007) 787–798.[12] M. Edwards, R.J. Dodd, C.W. Clark, P.A. Ruprecht, K. Burnett, Properties of a Bose–Eistein condensate in an anisotropic harmonic potential, Phys. Rev. A

53 (4) (1996) 1950–1953.[13] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Clarendon Press, Oxford, 2004.[14] W. Gautschi, Orthogonal polynomials (in Matlab), J. Comput. Appl. Math. 178 (1–2) (2005) 215–234.[15] R. Graham, D. Walls, Collective excitations of trapped binary mixtures of Bose–Einstein condensed gases, Phys. Rev. A 57 (1) (1998) 484–487.[16] R. Grimm, Low-temperature physics: a quantum revolution, Nature 435 (2005) 1035–1036.[17] E.P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento 20 (1961) 454.[18] S.P. Han, A globally convergent method for nonlinear programming, J. Optim. Theory Appl. 22 (1977) 297–309.[19] C.J. Myatt, E.A. Burt, R.W. Ghrist, E.A. Cornell, C.E. Wieman, Production of two overlapping Bose–Einstein condensates by sympathetic cooling, Phys.

Rev. Lett. 78 (1997) 586–589.[20] L.P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP 13 (1961) 451.[21] F. Riboli, M. Modugno, Topology of the ground state of two interacting Bose–Einstein condensates, Phys. Rev. A 65 (2002) 063614.[22] C. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht, P. Vorderwisch, Bose–Einstein condensation of the triplet

states in the magnetic insulator TlCuCl3, Nature 423 (6935) (2003) 62–65.[23] B.I. Schneider, D.L. Feder, Numerical approach to the ground and excited states of a Bose–Einstein condensed gas confined in a completely anisotropic

trap, Phys. Rev. A 59 (3) (1999) 2232–2242.[24] E. Timmermans, Phase separation of Bose–Einstein condensates, Phys. Rev. Lett. 81 (1998) 5718–5721.[25] R.P. Tiwari, A. Shukla, A basis-set based Fortran program to solve the Gross–Pitaevskii equation for dilute Bose gases in harmonic and anharmonic

traps, Comput. Phys. Commun. 174 (12) (2006) 966–982.[26] H. Wang, A time-splitting spectral method for coupled Gross–Pitaevskii equations with applications to rotating Bose–Einstein condensates, J. Comput.

Appl. Math. 205 (2007) 88–104.


BIT Numer MathDOI 10.1007/s10543-009-0215-2

On the convergence of splitting methods for linearevolutionary Schrödinger equations involvingan unbounded potential

Christof Neuhauser · Mechthild Thalhammer

Received: 10 March 2008 / Accepted: 3 December 2008© Springer Science + Business Media B.V. 2009

Abstract In this paper, we study the convergence behaviour of high-order exponen-tial operator splitting methods for the time integration of linear Schrödinger equations

i ∂ tψ(x, t) = − 12 ψ(x, t) + V (x)ψ(x, t) , x ∈ Rd , t ≥ 0 ,

involving unbounded potentials; in particular, our analysis applies to potentials V

defined by polynomials. We deduce a global error estimate which implies that anytime-splitting method retains its classical convergence order for linear Schrödingerequations, provided that the exact solution of the considered problem fulfills suitableregularity requirements. Numerical examples illustrate the theoretical result.

Keywords Linear Schrödinger equations · Unbounded potential · Splittingmethods · Convergence

Mathematics Subject Classification (2000) 65L05 · 65M12 · 65J10

1 Introduction

In the present paper, our concern is to study the convergence behaviour of high-ordertime-splitting methods for linear Schrödinger equations

i ∂ tψ(x, t) = − 12 ψ(x, t) + V (x)ψ(x, t) , x ∈ Rd , t ≥ 0 , (1.1)

Communicated by Timo Eirola.

C. Neuhauser · M. Thalhammer ()Institut für Mathematik, Leopold-Franzens Universität Innsbruck, 6020 Innsbruck, Austriae-mail: [email protected]

C. Neuhausere-mail: [email protected]

C. Neuhauser, M. Thalhammer

involving unbounded real-valued potentials V : Rd → R; in particular, our analysisapplies to potentials that comprise a polynomial part and in addition a sufficientlyoften differentiable function with bounded derivatives. For the theoretical study oftime discretisations for (1.1), as standard, the partial differential equation (1.1) iswritten as a linear evolution equation for u(t) = ψ(·, t)

u′(t) = Au(t) + B u(t) , t ≥ 0 , u(0) given, (1.2)

involving unbounded linear operators A : D(A) → X and B : D(B) → X on theunderlying function space X. A second-order approximation to the value of the exactsolution at time h > 0 is obtained by the Strang splitting [20, 23]

u1 = e12 hB ehA e

12 hBu0 ≈ u(h) = eh(A+B) u(0) ; (1.3)

example methods of higher-order are found in [12, 17], see also Sect. 2.The main result of the present work, deduced in Sect. 3, is a convergence esti-

mate for exponential operator splitting methods of arbitrarily high order when appliedto linear evolutionary problems of the form (1.2). We employ an abstract analyti-cal framework that includes linear Schrödinger equations (1.1) and further evolutionequations of parabolic type. Extending techniques previously exploited in [13, 15,22], we show that any splitting method retains its classical convergence order, pro-vided that the exact solution of (1.2) satisfies suitable regularity requirements. Forsimplicity, we restrict ourselves to equidistant time grids; however, it is straightfor-ward to extend our convergence result to variable stepsizes, see also Remark 3.8.Applications to linear Schrödinger equations are the contents of Sect. 4. In particu-lar, we discuss polynomial potentials and illustrate our theoretical error estimate bynumerical examples.

The intention of the present work is to give insight in the convergence behaviourof high-order exponential operator splitting methods and makes a contribution to abetter understanding of efficient space and time discretisation methods for nonlinearSchrödinger equations; an application of particular interest which arises in quantumphysics is the phenomenon of Bose–Einstein condensation, modelled by a system ofcoupled Gross–Pitaevskii equations, see [2–5, 8–10, 18, 24], e.g.

Henceforth, we denote by C a generic constant with possibly different values atdifferent occurrences.

2 Splitting methods for linear evolution equations

In this section, we introduce exponential operator splitting methods for the time in-tegration of evolutionary equations (1.2) involving (unbounded) linear operators A

and B . We employ the following general form of a splitting method that includesthe example methods given in literature; for a detailed treatment of composition andsplitting methods, we refer to [12, 17].

Splitting methods rely on the fact that the initial value problems

v′(t) = Av(t) , t ≥ 0 , v(0) given, (2.1a)

Splitting methods for linear evolutionary Schrödinger equations

Table 1 Exponential operator splitting methods of order p involving s compositions

Method Order #com.

McLachlan McLachlan [12, V.3.1, (3.3), pp. 138–139] p = 2 s = 3

Strang Strang (1.3) p = 2 s = 2

BM4-1 Blanes & Moan [6, Table 2, PRKS6] p = 4 s = 7

BM4-2 Blanes & Moan [6, Table 3, SRKNb6] p = 4 s = 7

M4 McLachlan [12, V.3.1, (3.6), pp. 140] p = 4 s = 6

S4 Suzuki [12, II.4, (4.5), pp. 41] p = 4 s = 6

Y4 Yoshida [12, II.4, (4.4), pp. 40] p = 4 s = 4

BM6-1 Blanes & Moan [6, Table 2, PRKS10] p = 6 s = 11

BM6-2 Blanes & Moan [6, Table 3, SRKNb11] p = 6 s = 12

BM6-3 Blanes & Moan [6, Table 3, SRKNa14] p = 6 s = 15

KL6 Kahan & Li [12, V.3.2, (3.12), pp. 144] p = 6 s = 10

S6 Suzuki [12, II.4, (4.5), pp. 41] p = 6 s = 26

Y6 Yoshida [12, V.3.2, (3.11), pp. 144] p = 6 s = 8

w′(t) = B w(t) , t ≥ 0 , w(0) given, (2.1b)

can be solved numerically in an accurate and efficient way. For some initial valueu0 ≈ u(0) and a constant time step h > 0, approximations un to the exact solutionvalues u(tn) at time tn = nh are then determined through the recurrence relation

un = S un−1 , n ≥ 1 ,

S =s∏

j=1

ehbj B ehaj A = ehbsB ehasA · · · ehb1B eha1A ,(2.2)

involving certain (complex) coefficients (aj , bj )sj=1. We emphasise that for t = 0 the

operators etA and etB do not commute, in general; throughout, the product in (2.2) isdefined downwards.

A widely used symmetric second-order scheme, the so-called Strang [20] or sym-metric Lie–Trotter [23] splitting, can be cast into the form (2.2) for s = 2 and

a1 = 12 = a2 , b1 = 1 , b2 = 0 , or

a1 = 0 , a2 = 1 , b1 = 12 = b2 ,

(2.3a)

respectively; in both cases, the order conditions

a1 + a2 = 1 , b1 + b2 = 1 , b1a1 + b2 = 12 , (2.3b)

are fulfilled. Fourth- and sixth-order symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods were proposed in Blanes and Moan [6]; for further examplemethods, we refer to [12, 14, 16, 17, 21, 25], see also Table 1.


3 Convergence analysis

In the following, we deduce a global error estimate for exponential operator splittingmethods of the form (2.2) when applied to linear evolution equations (1.2). Hence-forth, we denote by (X,‖ · ‖X) a (complex) Banach space with corresponding op-erator norm ‖ · ‖X←X . Our hypotheses on the (closed and densely defined) linearoperators A : D(A) ⊂ X → X and B : D(B) ⊂ X → X are as follows, see Engel andNagel [11] for a detailed treatment of one-parameter (semi)groups.

Hypothesis 3.1 The linear operators A : D(A) → X and B : D(B) → X generateC0-groups (etA)t∈R and (etB)t∈R, respectively, such that

∥∥etA∥∥

X←X≤ eα|t | ,

∥∥etB∥∥

X←X≤ eβ|t | , t ∈ R ,

for constants α,β ≥ 0.

For evolution equations of parabolic type we employ a more general assumptioninstead.

Hypothesis 3.2 The linear operators A : D(A) → X and B : D(B) → X generateC0-semigroups (etA)t≥0 and (etB)t≥0, respectively, such that

∥∥etA∥∥

X←X≤ eαt ,

∥∥etB∥∥

X←X≤ eβt , t ≥ 0 ,

for constants α,β ≥ 0.

With the help of Hypothesis 3.1, it is straightforward to deduce the followingstability result for high-order splitting methods (2.2).

Lemma 3.3 Suppose that the method coefficients of the exponential operator split-ting method (2.2) are real and that the assumptions of Hypothesis 3.1 hold. Then, thesplitting operator S fulfills the bound

∥∥Sn∥∥

X←X≤ eC tn , n ≥ 0 ,

with constant C = α (|a1| + · · · + |as |) + β (|b1| + · · · + |bs |) .

Remark 3.4 The statement of Lemma 3.3 remains valid under Hypothesis 3.2 (replac-ing Hypothesis 3.1), provided that the method coefficients of the exponential operatorsplitting method (2.2) are non-negative (but possibly complex).

In order to derive a global error estimate for exponential operator splitting meth-ods (2.2) when applied to abstract evolution equations (1.2), we make use of a LadyWindermere argument; that is, we employ the identity

un − u(tn) = Sn(u0 − u(0)

) −n−1∑

j=0

Sn−j−1 dj+1 , n ≥ 0 , (3.1a)


where dj+1 = u(tj+1)−S u(tj ). Estimating (3.1a) with the help of Lemma 3.3 furtheryields

∥∥un − u(tn)∥∥

X≤ C

(∥∥u0 − u(0)∥∥

X+

n−1∑

j=0

∥∥dj+1∥∥

X

), n ≥ 0 . (3.1b)

It remains to deduce a bound for the defect; for this purpose, we employ the followinghypothesis. As standard, the iterated commutators are defined through

adj+1A (B) = [

A, ad jA(B)

] = A ad jA(B) − ad j

A(B)A , j ≥ 0 , (3.2)

where ad0A(B) = B , see Hairer et al. [12, Chap. III.4.1].

Hypothesis 3.5 Let p denote the (classical) order of the exponential operator split-ting method (2.2). Suppose that the estimate

p+1∑

k=1

∑

μ∈Nk

|μ|=p+1−k

∥∥∥∥∥

k∏

=1

(adμ

A (B) eζA)∥∥∥∥∥

X←Dp+1

≤ C , ζ = (ζ1, . . . , ζp+1) ∈ Rp+1 ,

remains valid with a (suitably chosen) normed space Dp+1 ⊂ X.

For the sake of brevity, we meanwhile consider the Strang splitting (2.3a) only. Inthis case, we require Hypothesis 3.5 to be fulfilled for p = 2; that is, we employ theassumption

∥∥B eζ3A B eζ2A B eζ1A∥∥

X←D3+ ∥∥B eζ2A adA(B) eζ1A

∥∥X←D3

+ ∥∥adA(B) eζ2A B eζ1A∥∥

X←D3+ ∥∥ad2

A(B) eζ1A∥∥

X←D3≤ C (3.3)

for ζ ∈ R3. Formally, the exact solution of the evolution equation (1.2) is givenby u(tn−1 + h) = eh(A+B) u(tn−1) for h ≥ 0; alternatively, the representation by thevariation-of-constants formula

u(tn−1 + h) = ehA u(tn−1) +∫ h

0e(h−τ)A B u(tn−1 + τ)dτ


is valid. A repeated application of the above formula yields the following relation forthe exact solution value

u(tn) = (ehA + I1 + I2 + R3

)u(tn−1) ,

I1 =∫ h

0g1(τ1)dτ1 , I2 =

∫ h

0

∫ τ1

0g2(τ1, τ2)dτ2dτ1 ,

R3 =∫ h

0

∫ τ1

0

∫ τ2

0f2(τ1, τ2) e(τ2− τ3)A B eτ3(A+B) dτ3dτ2dτ1 ,

f1(τ1) = e(h− τ1)A B , f2(τ1, τ2) = f1(τ1) e(τ1− τ2)A B ,

g1(τ1) = f1(τ1) eτ1A , g2(τ1, τ2) = f2(τ1, τ2) eτ2A .

(3.4a)

To obtain a similar expansion for the numerical approximation Su(tn−1), we applythe recurrence relation ϕj (z) = 1

j ! + zϕj+1(z) which holds true for the exponentialfunctions

ϕ0(z) = ez, ϕj (z) = 1(j−1)!

∫ 1

0τ j−1 e(1−τ)z dτ, j ≥ 1, z ∈ C ;

we note that ϕj (tB) : X → X is a bounded linear operator, see also Hypothesis 3.1and Hypothesis 3.2. Furthermore, we employ the algebraic identity

J∏

j=1

(Kj + Lj

) =J∏

=1

K +J∑

j=1

J∏

=j+1

K Lj

j−1∏

=1

(K + L

),

valid for arbitrary linear operators Kj , Lj , 1 ≤ j ≤ J . A stepwise expansion of thesplitting operator S with the help of the above relations gives

S u(tn−1) = (ehA + Q1 + Q2 + R3

)u(tn−1) ,

Q1 = h

2∑

j1=1

bj1 g1(cj1h) , Q2 = 12 h2

2∑

j1=1

j1∑

j2=1

ηj1j2bj1bj2 g2(cj1h, cj2h) ,

R3 = h3(b3

1ea2hAB3ϕ3(b1hB

)ea1hA + b2

1b2Bea2hAB2ϕ2(b1hB

)ea1hA

+b1b22B

2ϕ2(b2hB

)ea2hABϕ1

(b1hB

)ea1hA + b3

2B3ϕ3

(b2hB

)ehA

),

(3.4b)where c1 = a1 and c2 = a1 + a2 = 1, see (2.3b); moreover, we set η11 = η22 = 1 andη21 = 2. The above expansions (3.4) imply

dn = (S1 + S2 + R3 − R3

)u(tn−1) , S1 = I1 − Q1 , S2 = I2 − Q2 . (3.4c)

In order to expand S1 and S2 further, we employ Taylor series expansions of thefunctions g1 and g2 about zero. Due to the fact that the order conditions (2.3b) are


fulfilled, we obtain

S1 =2∑

j1=1

bj1

∫ 1

0

∫ h

0(1 − σ)

(G1(σ, τ1) − G1(σ, cj1h)

)dτ1 dσ ,

S2 =2∑

j1=1

j1∑

j2=1

ηj1j2 bj1bj2

×∫ 1

0

∫ h

0

∫ τ1

0

(G2(σ, τ1, τ2) − G2(σ, cj1h, cj2h)

)dτ2dτ1dσ ,

G1(σ, τ1) =τ 21 ∂τ 2

1g1(στ1) , G2(σ, τ1, τ2) =

2∑

=1

τ ∂τg2(στ1, σ τ2) ;

(3.4d)

the involved partial derivatives of g1 and g2 are equal to

∂τ 21g1(τ1) = e(h−τ1)A ad2

A(B) eτ1A ,

∂τ1 g2(τ1, τ2) = − e(h−τ1)A adA(B) e(τ1−τ2)AB eτ2A ,

∂τ2 g2(τ1, τ2) = − e(h−τ1)AB e(τ1−τ2)A adA(B) eτ2A .

(3.4e)

Estimating (3.4c) by means of (3.3) finally yields the local error estimate∥∥dn

∥∥X

≤ C hp+1 maxtn−1≤t≤tn

∥∥u(t)∥∥

Dp+1(3.5)

with p = 2 for the Strang splitting scheme (2.3a).Following Thalhammer [22], the above considerations generalise to arbitrary split-

ting methods of the form (2.2); estimating (3.1b) with the help of (3.5) we thus obtainthe convergence result stated below. We note that the derivation of Theorem 3.6 is noteffected by exchanging the roles of A and B in (2.2). Concerning the (classical) orderconditions for higher-order splitting methods, we refer to Hairer et al. [12], e.g., seealso Thalhammer [22].

Theorem 3.6 Suppose that the coefficients aj , bj ∈ R, 1 ≤ j ≤ s, of the exponentialoperator splitting method (2.2) fulfill the classical order conditions for p ≥ 1. Assumefurther that the unbounded linear operators A and B satisfy the requirements ofHypothesis 3.1 and 3.5 for a suitably chosen normed space Dp+1 ⊂ X. Then, theerror estimate

∥∥un − u(tn)∥∥

X≤ C

(∥∥u0 − u(0)∥∥

X+ hp max

0≤t≤tn

∥∥u(t)∥∥

Dp+1

), 0 ≤ tn ≤ T ,

is valid, provided that the exact solution of the linear evolutionary equation (1.2)remains bounded in Dp+1.

Remark 3.7 The statement of Theorem 3.6 remains valid under the assumptions ofHypothesis 3.2 (replacing Hypothesis 3.1), provided that the method coefficients of


the exponential operator splitting method (2.2) are non-negative (but possibly com-plex).

Remark 3.8 It is straightforward to extend the convergence estimate of Theorem 3.6to non-equidistant time grids 0 = t0 < t1 < · · · < tN = T with associated stepsizesdefined by hj = tj+1 − tj for 0 ≤ j ≤ N − 1; in this case, for 0 ≤ tn ≤ T the errorbound

∥∥un − u(tn)∥∥

X≤ C

(∥∥u0 − u(0)∥∥

X+

n−1∑

j=0

hp+1j max

tj ≤t≤tj+1

∥∥u(t)∥∥

Dp+1

)

follows.

4 Applications to linear Schrödinger equations

We next apply the convergence analysis given in Sect. 3 to linear Schrödinger equa-tions (1.1), subject to asymptotic boundary conditions on the unbounded domainand a certain initial condition. We are primarily interested in problems involving un-bounded real potentials V : Rd → R; in particular, in Sect. 4.3, we discuss the casewhere V comprises a polynomial and in addition a sufficiently often differentiablefunction with bounded derivatives. For the spatial discretisation of (1.1), we employthe Hermite spectral method. Basic properties of the Hermite basis functions are col-lected in Sect. 4.1; we refer to Boyd [7] for a detailed description of spectral methods.We note that similar arguments apply to linear Schrödinger equations (1.1) that aresubject to periodic boundary conditions and involve a large potential; in this case,Fourier techniques are applicable.

4.1 Hermite basis functions

Henceforth, we let μ = (μ1, . . . ,μd) ∈ Nd denote a multi-index of non-negative inte-gers and define |μ| = μ1 +· · ·+μd . Further, we employ the compact vector notationx = (x1, . . . , xd) ∈ Rd and set

xμ = xμ11 · · ·xμd

d , ∂μ = ∂x

μ11

· · · ∂x

μdd

, ∂x

μjj

= ∂μj

∂xμj

j

; (4.1)

as standard, = ∂2x1

+ · · · + ∂2xd

denotes the d-dimensional Laplace operator. Wedenote by C∞

0 (Rd) the space of infinitely many times differentiable functions withcompact support. The Lebesgue space L2(Rd) = L2(Rd ,C) of square integrablecomplex-valued functions is endowed with scalar product and associated norm givenby

(f

∣∣g)L2 =

∫

Rd

f (x)g(x)dx ,∥∥f

∥∥L2 =

√(f

∣∣f)L2 , f, g ∈ L2(Rd) .


Moreover, the Banach space L∞(Rd) is endowed with the norm

∥∥f∥∥

L∞ = ess supx∈Rd

∣∣f (x)∣∣.

The Sobolev space Hm(Rd) comprises all functions with partial derivatives up toorder m ≥ 0 contained in L2(Rd); the associated norm ‖·‖Hm is defined through

∥∥f∥∥2

Hm =∑

μ∈Nd

|μ|≤m

∥∥∂ μf∥∥2

L2 , f ∈ Hm(Rd) .

Detailed information on Sobolev spaces is found in the monograph Adams [1].We let Hμj

: R → R : xj → Hμj(xj ) denote the Hermite polynomial of degree

μj ≥ 0 that is normalised with respect to the weights wj(xj ) = exp(−(γj xj )2).

Then, for μ ∈ Nd the Hermite function Hμ : Rd → R : x → Hμ(x) is defined by

Hμ(x) =d∏

j=1

Hμj(xj ) , Hμj

(xj ) = Hμj(xj ) e− 1

2 (γj xj )2. (4.2a)

The functions (Hμ) form an orthonormal basis of the function space L2(Rd); thus,for any v ∈ L2(Rd) the representation

v =∑

μ∈Nd

vμ Hμ , vμ = (v

∣∣Hμ

)L2 , (4.2b)

follows. Furthermore, the eigenvalue relation

12

(− + VH

)Hμ = λμ Hμ , λμ =

d∑

j=1

γ 2j

(μj + 1

2

), (4.2c)

holds; here, we denote by VH : Rd → R a scaled harmonic potential

VH (x) =d∑

j=1

γ 4j x2

j (4.2d)

involving positive weights γj > 0, 1 ≤ j ≤ d . We note that the values of Hμjare

computed through the recurrence relation

H0(xj ) = 4

√γ 2j

π, H1(xj ) = √

2γj xj H0(xj ) ,

Hμj(xj ) = 1√

μj

(√2γj xjHμj −1(xj ) − √

μj − 1Hμj −2(xj )), μj ≥ 2 ;

(4.3a)


as a consequence, for the derivative of Hμjthe identity

∂xjHμj

= − γj√2

(√μj + 1Hμj +1 − √

μj Hμj −1), μj ≥ 1 , (4.3b)

is valid.

4.2 Splitting methods for Schrödinger equations

In this section, we relate linear Schrödinger equations of the form (1.1) to abstractevolution equations (1.2) and discuss the validity of Hypothesis 3.1; as underlyingfunction space, we consider the Hilbert space L2(Rd). Besides, we comment on thenumerical solution of (1.1) by exponential operator splitting Hermite spectral meth-ods.

Regarding the spatial discretisation, the unbounded linear operators A and B areconstructed as follows. Let A : C∞

0 (Rd) → L2(Rd) be defined by

(Av

)(x) = 1

2

(−v(x) + VH (x) v(x))

for v ∈ C∞0 (Rd). The Friedrich’s extension AF of A is a selfadjoint operator,

see Reed and Simon [19], e.g.; consequently, by Stone’s Theorem, see Engel andNagel [11], e.g., the operator A = − i AF : D(A) = D(AF ) → L2(Rd) generates aunitary group (etA)t∈R on L2(Rd). We note that the domain of A can be charac-terised as

D(A) =v ∈ L2(Rd) :

∑

μ∈Nd

λ2μ |vμ|2 < ∞

,

see (4.2). Moreover, we define the multiplication operator B through(Bv

)(x) = W(x)v(x) = (

V (x) − 12 VH (x)

)v(x) ;

under suitable assumptions on W = V − 12 VH , the operator B : C∞

0 (Rd) → L2(Rd)

is well-defined, positive, and symmetric. Thus, similarly to before, its Friedrich’sextension BF : D(BF ) → L2(Rd) is a selfadjoint operator; again, by Stone’s Theo-rem, it is ensured that B = − i BF : D(B) = D(BF ) → L2(Rd) generates a unitarygroup (etA)t∈R on L2(Rd). Altogether, it follows

∥∥etA∥∥

L2←L2 = 1 ,∥∥etB

∥∥L2←L2 = 1 , t ≥ 0 ; (4.4)

that is, Hypothesis 3.1 is fulfilled with α = 0 and β = 0. Furthermore, we concludethat the splitting operator S, defined in (2.2), is a unitary operator on L2(Rd); moreprecisely, the relations in (4.4) imply the stability result

∥∥Sn u0∥∥

L2 = ∥∥u0∥∥

L2 , n ≥ 0 , (4.5)

for any initial value u0 ∈ L2(Rd), see also Lemma 3.3.The practical realisation of exponential operator splitting methods (2.2) for linear

Schrödinger equations (1.1) relies on the following approach. On the one hand, the


numerical solution of (2.1a) is based on a spectral decomposition of the initial valueinto Hermite basis functions

v(t) = etA v(0) =∑

μ∈Nd

vμ e− i tλμHμ , t ≥ 0 , v(0) =∑

μ∈Nd

vμ Hμ , (4.6a)

see (4.2); on the other hand, a rapid componentwise multiplication yields an approx-imation to the solution w(t) = ψ(·, t) of (2.1b)

ψ(x, t) = e− i t W(x) ψ(x,0) , t ≥ 0 . (4.6b)

For details on the implementation, we refer to Caliari et al. [8] and references giventherein.

4.3 Potentials defined by polynomials

In this section, we study the special case where the unbounded linear operator

B = B1 + B2 (4.7)

comprises a polynomial part B1 and in addition an operator B2 that is defined througha sufficiently often differentiable function with bounded derivatives. As in Sect. 3,in the derivation, we restrict ourselves to the consideration of the Strang splittingmethod (2.3a); that is, we characterise the normed space D3 in assumption (3.3).Similar though more tedious calculations then yield the stated result for a generalsplitting method.

Regarding (3.3), we remark that the iterated commutator adkA(B) defined by (3.2)

is a differential operator of order k. More precisely, it holds

adA(B)v =d∑

j=1

(∂xj

W ∂xj+ 1

2 ∂x2jW

)v , (4.8a)

ad2A(B)v = i

d∑

j,=1

(∂xj x

W ∂xj x+ ∂xj x2

W ∂xj

+ 14 ∂x2

j x2W + 1

d∂xj

VH ∂xjW

)v ,

(4.8b)

where ∂xjVH (x) = 2γ 4

j xj , see (4.2d).The following considerations rely on the fact that the Hermite functions (Hμ)

form an orthonormal basis of the function space L2(Rd); thus, for any v ∈ L2(Rd)

the representation

v =∑

μ∈Nd

vμHμ (4.9)

is valid, see also (4.2).As a first step, we study the unbounded linear operators Pκ and Qκ that are defined

through(Pκ v

)(x) = xκ v(x) ,

(Qκ v

)(x) = ∂κv(x) , κ ∈ Nd , (4.10)


see (4.1); the following auxiliary result implies that Pκ and Qκ are well-defined onthe normed space

Dκ =v ∈ L2(Rd) : ‖v‖Dκ < ∞

, ‖v‖2

Dκ=

∑

μ∈Nd

(μ + κ)κ |vμ|2 ; (4.11)

here, we set (μ + κ)κ = (μ1 + κ1)κ1 · · · (μd + κd)κd .

Lemma 4.1 (i) The functions x → xκHμ(x) and ∂κHμ can be represented by afinite linear combination of Hermite basis functions; more precisely, the identities

(PκHμ

)(x) =

∑

ν∈Zd

|νj |≤κj

cμ+ν Hμ+ν(x) ,(QκHμ

)(x) =

∑

ν∈Zd

|νj |≤κj

dμ+ν Hμ+ν(x) ,

hold with real coefficients (cμ+ν) and (dμ+ν) that fulfill the bound

c 2μ+ν + d 2

μ+ν ≤ C (μ + κ)κ

for a constant C > 0 that depends on κ .(ii) The linear operator etA : Dκ → Dκ is unitary, that is, the relation

∥∥ etA∥∥

Dκ←Dκ= 1

is valid for all t ≥ 0.(iii) For any η ∈ Nk the following estimate

∥∥Pκ

∥∥Dη←Dη+κ

+ ∥∥Qκ

∥∥Dη←Dη+κ

≤ C (4.12)

remains valid with a constant C that depends on κ .

Proof (i) On the one hand, we repeatedly apply the relation

xj Hμj(xj ) = 1

γj

(√μj + 1

2Hμj +1(xj ) +

√μj

2Hμj −1(xj )

)

that is a direct consequence of (4.2a) and the recurrence formula (4.3); this yields

xκj

j Hμj(xj ) =

∑

∈Z||≤κj

cμj Hμj +(xj ) , c 2μj ≤ C (μj + κj )

κj ,

with constant C depending on κj . Thus, we finally obtain

xκHμ(x) =d∏

j=1

(x

κj

j Hμj(xj )

)=

∑

ν∈Zd

|νj |≤κj

cμ+ν Hμ+ν ,


with coefficients c 2μ+ν that satisfy the bound c 2

μ+ν ≤ C (μ+ κ)κ for some constant C

depending on κ . Using that the relation

∂xjHμj

(xj ) = γj

(−

√μj + 1

2Hμj +1(xj ) +

√μj

2Hμj −1(xj )

)

holds, see also (4.3b), similar considerations yield the statement for ∂κHμ.(ii) Due to the fact that A possesses the purely imaginary eigenvalues (−iλμ), see

also (4.2c), the identity

∥∥ etA v∥∥2

Dκ=

∥∥∥∑

μ∈Nd

vμ e−i tλμHμ

∥∥∥2

Dκ

=∑

μ∈Nd

(μ + κ)κ∣∣vμ

∣∣2 = ‖v‖2Dκ

remains true for all t ≥ 0.(iii) In the following, we derive the bound for ‖Pκ‖Dη←Dη+κ; similar arguments

then show the estimate involving Qκ . Suppose v to be given by a finite linear combi-nation of the form (4.9); by means of statement (i) we obtain

(Pκ v

)(x) = xκ v(x) =

∑

μ∈Nd

vμ xκHμ(x) =∑

ν∈Zd

|νj |≤κj

∑

μ∈Nd

cμ+ν vμ Hμ+ν(x) .

Furthermore, due to the fact that the relation

∥∥∥∑

μ∈Nd

cμ+ν vμ Hμ+ν

∥∥∥2

Dη

≤ C∑

μ∈Nd

(μ + η)η (μ + κ)κ |vμ|2 ≤ C ‖v‖2Dη+κ

is valid with a constant C depending on κ , the estimate

∥∥Pκ v∥∥

Dη≤

∑

μ∈Nd

|νj |≤κj

∥∥∥∑

μ∈Nd

cμ+ν vμ Hμ+ν

∥∥∥L2

≤ C ‖v‖Dκ (4.13)

follows. A standard limiting process finally yields the desired result for any v of theform (4.9).

We are now ready to characterise the normed space D3 in (3.3) for linear op-erators that are defined by a polynomial; namely, for a finite set K ⊂ Nd and realcoefficients (ωκ)κ∈K we set

B1 = − iW1 , W1 =∑

κ∈K

ωκPκ . (4.14a)

With the help of Lemma 4.1, we obtain the following estimate

∥∥B1v∥∥

Dη≤ C ‖v‖Dη+κ , κ ∈ Nd , κj = max

κ∈Kκj , (4.14b)


involving a constant C > 0 that depends on (ωκ)κ∈K and κ ∈ K . Moreover, the rela-tions in (4.8) and the bound (4.12) for the differential operator Qκ imply

∥∥B1 eζ3A B1 eζ2A B1 eζ1A v∥∥

L2 ≤ C ‖v‖D3κ,

∥∥B1 eζ2A adA(B1) eζ1A v∥∥

L2 + ∥∥adA(B1) eζ2A B1 eζ1A v∥∥

L2 ≤ C ‖v‖D2κ,

∥∥ad2A(B) eζ1A v

∥∥L2 ≤ C ‖v‖Dκ ,

for any ζ ∈ R3; in the present situation, the above estimates allow to choose D3 =D3κ . In case that B involves an additional bounded function, the following resultholds true; we omit the details.

Theorem 4.2 Suppose that the exponential operator splitting method (2.2) has (clas-sical) order p. Assume further that the unbounded linear operator (4.7) comprises apolynomial part of the form (4.14a) and that B2 = iW2 is defined by a real-valuedfunction W2 such that max

‖∂μW2‖L∞ : |μ| ≤ 2p

remains bounded. Then, Hypoth-esis 3.5 is satisfied for Dp+1 = D(p+1) κ , see also (4.11) and (4.14b).

4.4 Illustrations

In this section, we illustrate our theoretical convergence result by numerical examplesfor the two-dimensional linear Schrödinger equation

i ∂ tψ(x, t) = (− 12 + 1

2 x21 + x2

2 + W(x))ψ(x, t) , x ∈ R2 , t ≥ 0 ; (4.15)

we consider in particular the following unbounded potentials

W(x) = x21 + 2x2

2 , (4.16a)

W(x) = x21 + 2x2

2 + 32 cos(3x1) + 3

2 cos(3x2) , (4.16b)

W(x) = x1 − 32 x2 , (4.16c)

W(x) = 120

(x4

1 + 2x42

). (4.16d)

For the first (somewhat artificial) problem (4.15)–(4.16a) the exact solution is com-putable in an easy manner by a suitable choice of the weights in the Hermite func-tions; we thus have at hand a reliable reference solution. The solution of the modifiedequation (4.15)–(4.16b) comprising an additional bounded part is illustrated in Fig. 1.

Obviously, it holds κ = (2,2) for (4.16a) and (4.16b), κ = (1,1) for (4.16c), andκ = (4,4) for (4.16d), see (4.14b). To ensure that the exact solution of the problemfulfills the requirements of Theorem 3.6, see also Hypothesis 3.5 and Theorem 4.2,we choose ψ(x,0) = 1√

πexp(− 1

2 (x21 + x2

2)) as initial value for (4.16a)–(4.16c). Onthe other hand, for (4.16d), we evolve the corresponding ground state solution, com-puted by minimisation techniques, see also [8, 9].

The linear Schrödinger equation (4.15) is discretised in space by the Hermitepseudospectral method with 128 basis functions in each coordinate direction; in ourimplementation, we use Hermite polynomials, normalised with respect to the weight


Fig. 1 Numerical solution ofthe linear Schrödingerequation (4.15) involving theunbounded potential (4.16b).Value of |ψ(x1, x2, T )|2 at timeT = 8.98 versus (x1, x2)

Fig. 2 Temporal orders of various time-splitting Hermite spectral methods when applied to the linearSchrödinger equation (4.15) involving the unbounded potential (4.16a). Error versus stepsize

wj(xj ) = exp(−c2j x2

j ) for some cj ∈ R, so that the corresponding Hermite functions

are eigenfunctions of the operator − 12 + x2

1 + 2x22 .

For the time integration, we apply several time-splitting methods of orders two,four, and six, proposed in [6, 14, 16, 21, 25], and further the Strang splitting (1.3),see also (2.3a) and Table 1. In the present situation, due to the fact that the addi-tional condition [B, [B, [B,A]]] = 0 holds, also the Runge–Kutta–Nyström methodsdeveloped by Blanes and Moan [6] are consistent of orders four and six, respectively;however, in this case, exchanging the roles of A and B in (2.2) could lead to an or-der reduction. For time stepsizes h = 2−j , 0 ≤ j ≤ 9, we determine the numericalconvergence orders of the integration methods at time T = 1; reference solutions arecomputed with stepsize 2−11.

The numerical results, obtained for the potential (4.16a), are displayed in Fig. 2.We refer to the splitting schemes by the initials of the authors and their orders ofconvergence, see Table 1.

As expected, all splitting schemes retain their classical temporal convergence or-ders reflected by the slopes of the dashed-dotted lines. For the potentials (4.16b)–(4.16d), we included the results obtained for the second-order Strang splitting (1.3)and the fourth- and sixth-order Runge–Kutta–Nyström methods BM4-2 and BM6-3,which are favourable in view of their small error constants, see Fig. 3.


Fig. 3 Temporal orders of second, fourth and sixth-order time-splitting Hermite spectral methods whenapplied to the linear Schrödinger equation (4.15) involving the unbounded potential (4.16b) (left), (4.16c)(middle), and (4.16d) (right), respectively. Error versus stepsize

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. NUMER. ANAL. c© 2008 Society for Industrial and Applied MathematicsVol. 46, No. 4, pp. 2022–2038

HIGH-ORDER EXPONENTIAL OPERATOR SPLITTING METHODSFOR TIME-DEPENDENT SCHRODINGER EQUATIONS∗

MECHTHILD THALHAMMER†

Abstract. In this paper, we deduce high-order error bounds for exponential operator splittingmethods. The employed techniques are specific to linear differential equations of the form u′(t) =A u(t) + B u(t), t ≥ 0, involving an unbounded operator A. In particular, evolutionary Schrodingerequations with sufficiently regular initial values are included in the analysis.

Key words. exponential operator splitting, Schrodinger equations, high-order methods, con-vergence, stability

AMS subject classifications. 65L05, 65M12, 65J10

DOI. 10.1137/060674636

1. Introduction. In this paper, we study exponential operator splitting meth-ods for initial value problems of the form

(1.1) u′(t) = A u(t) + B u(t), t ≥ 0, u(0) given.

We are primarily interested in evolution equations that are related to time-dependentlinear Schrodinger equations or spatial discretizations thereof. That is, we allow theoperator norm of A to be of arbitrary size and suppose B to be a bounded linearoperator.

For the time integration of (1.1), we consider exponential operator splitting meth-ods composed by several exponentials

(1.2) un =s∏

j=1

e bjhB eajhA un−1, n ≥ 1, u0 given,

with coefficients aj , bj ∈ R for 1 ≤ j ≤ s. In particular, the symmetric second-ordersplitting scheme

(1.3) un = e12 hB ehA e

12 hB un−1, n ≥ 1, u0 given,

referred to as Strang [20] or symmetric Trotter [21] splitting, is contained in themethod class (1.2).

In Jahnke and Lubich [11] error bounds for (1.3) when applied to pseudospec-tral discretizations of time-dependent linear Schrodinger equations are given. In thepresent paper, we extend this approach to splitting methods of the general form (1.2).

So far, despite the fact that exponential operator splitting methods are widelyused in the time integration of partial differential equations, it remains open to providea convergence analysis for the numerical method class (1.2) when applied to stiffproblems. In this work, we deduce a theoretical result on the convergence and stabilitybehavior of exponential operator splitting methods that contributes to filling the blank

∗Received by the editors November 10, 2006; accepted for publication (in revised form) June 27,2007; published electronically April 23, 2008.

http://www.siam.org/journals/sinum/46-4/67463.html†Institut fur Mathematik, Leopold-Franzens-Universitat Innsbruck, Technikerstrasse 13/VII, A–

6020 Innsbruck, Austria ([email protected]).

2022


HIGH-ORDER SPLITTING FOR SCHRODINGER EQUATIONS 2023

for evolutionary linear Schrodinger equations and to justifying the practical use ofexample methods proposed in the literature for this type of problem; for instance, ourerror bound is applicable to symmetric symplectic fourth- and sixth-order methods,given in Blanes and Moan [3], and it shows that the schemes retain their convergenceorders for time-dependent linear Schrodinger equations with sufficiently regular data.

This paper is organized as follows. In section 2, we introduce the precise assump-tions on the linear evolution equation (1.1). Our abstract framework is based onthe theory of C0-(semi)groups on Banach spaces and includes evolutionary linearSchrodinger equations; a thorough treatment of one-parameter (semi)groups and ap-plications to partial differential equations is found in [7, 10, 17]. In section 3, we derivean expansion for the local error of the exponential operator splitting method (1.2) thatremains valid within the analytic framework of section 2; for that purpose, extend-ing Jahnke and Lubich [11, Proof of Thm. 2.1], we associate the order conditionswith quadrature order conditions for multiple integrals. In section 4, we then provea global error bound under reasonable regularity requirements on the initial value.Essential tools for our convergence analysis are estimates for repeated commutators;for instance, such bounds are valid for time-dependent linear Schrodinger equationssubject to periodic boundary conditions, provided that the potential is sufficientlyregular. The error estimate is finally illustrated by a numerical example.

A variety of works is concerned with exponential splitting methods for differentialequations; as a small excerpt from the literature we mention [1, 2, 12, 18, 19] and therecent contributions [4, 5, 6, 15]; see also the references therein. In particular, we referthe reader to Lubich [14], where a rigorous convergence analysis of the Strang splittingfor the cubic Schrodinger equation is given; Magnus integrators for linear Schrodingerequations involving a time-dependent potential are analyzed in Hochbruck and Lu-bich [9]. Basic information on splitting methods is also found in the survey article ofMcLachlan and Quispel [16] and the monograph of Hairer, Lubich, and Wanner [8].

2. Splitting methods for evolutionary Schrodinger equations. In thissection, we give the basic hypotheses on the abstract differential equation (1.1). Asillustration, we consider time-dependent linear Schrodinger equations and formulatethem as evolution equations; for notational simplicity, we restrict ourselves to onespace dimension. Moreover, we restate the exponential operator splitting method (1.2)and introduce several auxiliary abbreviations used throughout.

2.1. Evolution equations. We employ the following assumptions on the ab-stract initial value problem (1.1); see also Engel and Nagel [7, Thm. II.3.8]. To simplifymatters, following Jahnke and Lubich [11], we require etA to be bounded by one.

Hypothesis 1. Let (X, ‖ · ‖X) denote the underlying Banach space and ‖ · ‖X←X

the induced operator norm. We assume that the densely defined and closed linearoperator A : D ⊂ X → X is the infinitesimal generator of a C0-semigroup

(e tA

)t≥0

on X satisfying the bound

(2.1)∥∥e tA

∥∥X←X

≤ 1, 0 ≤ t ≤ T.

Further, we suppose B : X → X to be a bounded linear operator.The above assumptions ensure that for u(0) ∈ D the uniquely determined (clas-

sical) solution u ∈ C 1([0,∞), X) of the initial value problem (1.1) is given by

(2.2a) u(t) = et(A+B) u(0), t ≥ 0.


2024 MECHTHILD THALHAMMER

Moreover, the following representation by the variation-of-constants formula

(2.2b) u(t) = etA u(0) +∫ t

0

e(t−τ)A B u(τ) dτ, t ≥ 0,

is valid; see [7, Chap. II.6/III.1].For some ω ∈ R and ϑ > 0 the fractional powers Aϑ

ω of the shifted operatorAω = A − ωI are well defined on a subspace of X (see [7, Chap. II.5c]); we denoteby

(Xϑ, ‖·‖Xϑ

)the domain of Aϑ

ω, endowed with the graph norm

‖v‖Xϑ= ‖v‖X +

∥∥Aϑω v

∥∥X

, v ∈ Xϑ;

in particular, we set A0 = I and X0 = X.Henceforth, we assume that the linear operator B remains bounded on Xϑ for

some ϑ ≥ 0, that is, there exists a constant C > 0 such that

(2.3a)∥∥B

∥∥Xϑ←Xϑ

≤ C;

consequently, the bound

(2.3b)∥∥etB

∥∥Xϑ←Xϑ

≤ C, 0 ≤ t ≤ T,

holds with some constant C > 0. Hypothesis 1 implies that etA : X → X is uniformlybounded on finite time intervals; furthermore, for any ϑ ≥ 0 it follows that

(2.3c)∥∥etA

∥∥Xϑ←Xϑ

≤ 1,∥∥et(A+B)

∥∥Xϑ←Xϑ

≤ C, 0 ≤ t ≤ T,

with some constant C > 0.For integers j ≥ 0, the linear operators ϕj(tB) are defined through

(2.4a) ϕ0(z) = ez, ϕj(z) =1

(j − 1)!

∫ 1

0

τ j−1 e(1−τ)z dτ, j ≥ 1, z ∈ C.

We conclude from the above relation (2.3b) that the estimate

(2.4b)∥∥ϕj(tB)

∥∥Xϑ←Xϑ

≤ 1j!

C, 0 ≤ t ≤ T,

is valid for j ≥ 0.In the situation of Hypothesis 1, the exponential operator splitting method (1.2) is

well defined provided that aj ≥ 0 for 1 ≤ j ≤ s. For schemes with negative coefficients,we employ a stronger framework; in particular, relation (2.3c) then remains valid for0 ≤ |t| ≤ T .

Hypothesis 2. The densely defined closed linear operator A : D ⊂ X → Xgenerates a C0-group

(e tA

)t∈R on X such that the bound

(2.5)∥∥e tA

∥∥X←X

≤ 1, 0 ≤ |t| ≤ T,

is satisfied.In order to obtain high-order error bounds for (1.2), we require the solution of the

initial value problem (1.1) to fulfill certain commutator bounds on fractional powerspaces of A. We henceforth employ the standard notation

(2.6) ad 0A(B) = B, ad j

A(B) =[A, adj−1

A (B)], j ≥ 1,



where [A, L] = AL− LA for some linear operator L; e.g., see [8, Chap. III.4.1].Hypothesis 3. We suppose that for certain ϑ, ϑ ≥ 0 the estimate

∥∥ad jA(B)

∥∥Xϑ←X

ϑ

≤ C

is valid with constant C > 0.As indicated in section 2.2, the above hypothesis is reasonable in connection

with evolutionary linear Schrodinger equations that are subject to periodic boundaryconditions and involve a smooth potential.

2.2. Time-dependent Schrodinger equations. Let V : Ω = [−π, π] → R bea periodic map that fulfills suitable regularity requirements. We consider the time-dependent linear Schrodinger equation

(2.7) i ∂ tU(x, t) = −Δ U(x, t) + V (x) U(x, t), x ∈ Ω, t ≥ 0,

subject to periodic boundary conditions on Ω and an initial condition U(x, 0) = U0(x),x ∈ Ω.

The above initial-boundary value problem is interpreted as an initial value prob-lem of the form (1.1) by setting u(t) = U(·, t) and

(A v

)(x) = i Δ v(x),

(B v

)(x) = − iV (x) v(x)

for v : Ω → C a sufficiently regular function. Similarly as in Pazy [17, Chap. 7.5], it isshown that the linear operators A : D → X and B : X → X satisfy the assumptionsof Hypothesis 1 with X = L2(Ω, C) and D =

v ∈ H2(Ω, C) : v periodic on Ω.

Furthermore, it holds that Xk/2 =v ∈ Hk(Ω, C) : v periodic on Ω for integers

k ≥ 1.We note that Hypothesis 3 is fulfilled with ϑ = k/2 and ϑ = ϑ + j/2 for integers

j, k ≥ 0, provided that V is sufficiently often differentiable. In fact, the commutatorof the one-dimensional Laplace operator A = ∂ 2

x and a jth-order differential operator

B =j∑

=0

β(x) ∂ x

with smooth space-dependent coefficients β, 0 ≤ ≤ j, yields a differential operatorof order j + 1. More precisely, it follows that

[A ,B

]= AB −BA =

j∑

=0

(2 β′ ∂ +1

x + β′′ ∂ x

).

Proceeding by induction, we conclude that the iterated commutator adjA (V ) is a

differential operator of order j and therefore obtain∥∥ad j

A (V ) v∥∥

Hk(Ω)≤ C‖v‖Hk+j(Ω), v ∈ Hk+j(Ω),

for k, j ≥ 0.It is straightforward to generalize the above considerations to higher space di-

mensions.



2.3. Exponential operator splitting methods. Throughout, the gridpointsassociated with a constant stepsize h > 0 are denoted by tn = nh for n ≥ 0. Foran exponential operator splitting method with real coefficients aj , bj ∈ R, 1 ≤ j ≤ s,numerical approximation values un ≈ u(tn) to the true solution of (1.1) are given bythe recurrence formula

(2.8a) un = e bshB eashA · · · e b2hB ea2hA e b1hB ea1hA un−1, n ≥ 1, u0 given;

see Hypotheses 1 and 2. We note that the linear operators arising in the above productdo not commute in general.

In order to write (2.8a) in short notation, it is useful to employ the followingabbreviations. For linear operators Lj : X → X, ≤ j ≤ k, we define

k∏

j=

Lj = Lk · · ·L+1 L, k ≥ ,k∏

j=

Lj = I, k < .

Here, I : X → X denotes the identity operator on X. Furthermore, for k ≤ s we set

(2.8b) Pk =k∏

j=1

e bjhB eajhA,

and thus the exponential operator splitting method (2.8a) takes the compact form

(2.8c) un = Ps un−1, n ≥ 1, u0 given;

see (1.2).Inserting the exact solution values into the numerical scheme (2.8c) defines the

defect at time tn

(2.9) u(tn) = Ps u(tn−1) + dn, n ≥ 1.

As a consequence, a recurrence relation for the error of the splitting method

en = un − u(tn) = Ps en−1 − dn, n ≥ 1,

follows. Resolving this recursion, we finally obtain

(2.10) en = Pns e0 −

n−1∑

j=0

Pn−j−1s dj+1, n ≥ 0.

Our main result in section 4 is a convergence estimate for (2.8). Its proof is based onthe above representation; that is, suitable estimations of the splitting operator and thedefects are required; in this context, several auxiliary results provided in section 3.1are of use.

3. Local error. In the present section, our concern is to deduce a suitable ex-pansion of the local error

(3.1) dn = u(tn)− Ps u(tn−1), n ≥ 1,

that remains well defined within the analytical framework of section 2; see (2.9). Fur-thermore, from this representation, the order conditions for the exponential operatorsplitting method (2.8) follow.

We start with expanding the exact solution and the splitting operator.



3.1. Auxiliary expansions. (i) Exact solution. An expansion of the exactsolution value u(tn) that is specific to evolutionary Schrodinger equations is obtainedby means of a reapplication of the variation-of-constants formula (2.2b). Namely,replacing in

u(tn) = u(tn−1 + h) = ehA u(tn−1) +∫ h

0

e(h−τ1)A B u(tn−1 + τ1) dτ1

the solution value u(tn−1 + τ1) by

(3.2) u(tn−1 + τj) = eτjA u(tn−1) +∫ τj

0

e(τj−τj+1)A B u(tn−1 + τj+1) dτj+1,

where j = 1, yields the following relation:

u(tn) = ehA u(tn−1) +∫ h

0

e(h−τ1)A B eτ1A u(tn−1) dτ1

+∫ h

0

∫ τ1

0

e(h−τ1)A B e(τ1−τ2)A B u(tn−1 + τ2) dτ2 dτ1.

(3.3)

Henceforth, we employ the compact vector notation τ = (τ1, τ2, . . . , τk) ∈ Rk and set

(3.4a) fk(τ) =k∏

=1

(e(τk−− τk−+1)A B

), gk(τ) = fk(τ) eτkA,

with τ0 = h; as before, the product is defined downwards. Applying repeatedly thesubstitution (3.2) to (3.3), we therefore obtain

(3.4b)u(tn) = ehA u(tn−1) +

p∑

k=1

Ik u(tn−1) + R(1)p+1,

Ik =∫

Δk

gk(τ) dτ, R(1)p+1 =

∫

Δp+1

fp+1(τ) u(tn−1 + τp+1) dτ,

where Δk =τ ∈ Rk : 0 ≤ τ ≤ τ−1, 1 ≤ ≤ k

. We note that the above expansion

remains well defined in the situation of Hypothesis 1; see also section 4.(ii) Splitting operator. We next specify a representation for the splitting opera-

tor Ps that allows, in an easy manner, a comparison with (3.4b). Our main tool fordeducing such an expansion is a recurrence formula for the ϕ-functions

(3.5a) ϕj(z) =1j!

+ z ϕj+1(z), j ≥ 0, z ∈ C,

obtained from (2.4a) by a partial integration. Moreover, we make use of the identity

(3.5b)k∏

j=1

(Kj + Lj

)=

k∏

=1

K +k∑

j=1

k∏

=j+1

K Lj

j−1∏

=1

(K + L

)

that is valid for (noncommuting) linear operators Kj , Lj , 1 ≤ j ≤ k. By applying therelations (3.5) to Ps, we obtain

(3.6) Ps =s∏

j=1

e bjhB eajhA =s∏

j=1

(eajhA + h bj B ϕ1

(bjhB

)eajhA

),



and further the representation

(3.7) Ps = ecshA + hs∑

j=1

bj e(cs− cj) hA B ϕ1(bjhB) eajhA Pj−1.

Here, we employ the abbreviation

(3.8a) ck =k∑

j=1

aj , 1 ≤ k ≤ s.

Regarding the h-expansions (3.4b) and (3.7), it is seen that in (3.1) the O(1) termvanishes, provided that

(3.8b) cs = 1;

thus, we henceforth suppose the above order condition to be fulfilled. For the followingconsiderations, it is useful to introduce the abbreviations

(3.9)

Φj(λ) = F (λ) ϕj(bλkhB) eaλk

hA Pλk−1,

F (λ) =k∏

=1

bλfk(cλh), G(λ) = F (λ) ecλk

hA =k∏

=1

bλgk(cλh),

where we set cλ = (cλ1 , cλ2 , . . . , cλk) for multi-indices λ = (λ1, λ2, . . . , λk) ∈ Nk and

λ0 = s; see (3.4a). With the help of this notation, relation (3.7) can be written as

(3.10) Ps = ehA + hs∑

λ1=1

Φ1(λ1).

In order to expand the splitting operator Ps further, we proceed by induction; thatis, we repeatedly apply a recurrence relation for Φj to (3.10) which we derive next.Formula (3.5a) implies

Φj(λ) =1j!

F (λ) eaλkhA Pλk−1 + h Φj+1(λ, λk);

moreover, from the analogue of (3.7),

eaλkhA Pλk−1 = ecλk

hA

+ h

λk−1∑

λk+1=1

bλk+1 e(cλk− cλk+1 ) hA B ϕ1(bλk+1hB) eaλk+1hA Pλk+1−1,

we obtain the recurrence formula

(3.11) Φj(λ) =1j!

G(λ) + h

(Φj+1(λ, λk) +

λk−1∑

λk+1=1

1j!

Φ1(λ, λk+1))

.

As an illustration, we apply the above relation twice to (3.10). In a first step, we get

Ps = ehA + Q1 + r(2)2 ,

Q1 = hs∑

λ1=1

G(λ1), r(2)2 = h2

( s∑

λ1=1

Φ2(λ1, λ1) +s∑

λ1=1

λ1−1∑

λ2=1

Φ1(λ1, λ2))

.



A further expansion of r(2)2 by means of (3.11) yields

(3.12a)

Ps = ehA +2∑

k=1

Qk + r(2)3 ,

Q1 = hs∑

λ1=1

G(λ1), Q2 = h2

( s∑

λ1=1

12 G(λ1, λ1) +

s∑

λ1=1

λ1−1∑

λ2=1

G(λ1, λ2))

,

with remainder r(2)3 given by

r(2)3 = h3

( s∑

λ1=1

Φ3(λ1, λ1, λ1) +s∑

λ1=1

λ1−1∑

λ2=1

12 Φ1(λ1, λ1, λ2)

+s∑

λ1=1

λ1−1∑

λ2=1

Φ2(λ1, λ2, λ2) +s∑

λ1=1

λ1−1∑

λ2=1

λ2−1∑

λ3=1

Φ1(λ1, λ2, λ3))

.

(3.12b)

By reason of brevity, we write Q2 as

(3.12c) Q2 = h2s∑

λ1=1

λ1∑

λ2=1

αλ1λ2 G(λ1, λ2), αλ1λ2 =

12 if λ2 = λ1,

1 if λ2 ≤ λ1 − 1.

Applying (3.11) several times to (3.12), we finally end up with the expansion

(3.13a) Ps = ehA +p∑

k=1

Qk + r(2)p+1.

Here, the multiple sum Qk comprises terms of the form hk αλ G(λ) for certain αλ ∈ R;more precisely, we have

(3.13b) Qk = hk∑

λ∈Λk

αλ G(λ) = hks∑

λ1=1

λ1∑

λ2=1

· · ·λk−1∑

λk=1

αλ1λ2···λkG(λ1, λ2, . . . , λk),

where we denote Λk =λ ∈ Nk : 1 ≤ λ ≤ λ−1, 1 ≤ ≤ k

with λ0 = s, according

to (3.4). The coefficients αλ are displayed in Table A.1 for 1 ≤ k ≤ 4. The remain-der r

(2)p+1 is given by

(3.13c) r(2)p+1 = hp+1

p+1∑

j=1

∑

λ∈Λp+1

αjλ Φj(λ)

for certain αjλ ∈ R; we omit further details and refer the reader to the MATLAB coderecurrence, available at http://techmath.uibk.ac.at/mecht/research/research.html,that determines the coefficients αλ and αjλ.

(iii) Quadrature formulas. From the above considerations, we obtain the follow-ing expansion of the defect:

(3.14) dn =p∑

k=1

(Ik −Qk

)u(tn−1) + R

(1)p+1 −R

(2)p+1, R

(2)p+1 = r

(2)p+1 u(tn−1);



see also (3.1), (3.4), (3.8), and (3.13). We next relate the difference Ik − Qk to theerror of quadrature formulas for multiple integrals. Regarding (3.9) and (3.13b), wewrite Qk as

(3.15) Qk = hk∑

λ∈Λk

βλ gk(cλh), βλ = βλ1λ2···λk= αλ1λ2···λk

k∏

=1

bλ;

see (3.4a) for the definition of gk. As usual, in order to determine the defects of thequadrature formula (3.15), we employ a Taylor series expansion of order M = p− k:

(3.16a)gk(τ) =

p−k∑

m=0

1m!

g(m)k (0) τm + k,p−k+1(τ),

k,M+1(τ) =1

M !

∫ 1

0

(1− z)M g(M+1)k (z τ) τM+1 dz.

In view of section 4, we next specify the mth derivative of gk. It holds that

(3.16b) g(m)k (σ) τm =

∑

μ∈Nk

|μ|=m

m!μ!

∂μτ gk(σ) τμ;

here, for μ = (μ1, μ2, . . . , μk) ∈ Nk and τ = (τ1, τ2, . . . , τk) ∈ Rk, we let

(3.16c) |μ| =k∑

=1

μ, μ! =k∏

=1

μ!, τμ =k∏

=1

τμ

, ∂μτ =

k∏

=1

∂μτ

.

Proceeding by induction, it follows that

(3.16d) ∂μτ gk(τ) = (−1)|μ|

k∏

=1

(e(τk−− τk−+1)A adμk−+1

A (B))eτkA;

we recall definition (2.6) of the repeated commutators. In particular, we have

(3.16e) ∂μτ gk(0) = (−1)|μ| ehA

k∏

=1

adμk−+1A (B).

Inserting the Taylor series expansion (3.16) into

Ik −Qk =∫

Δk

gk(τ) dτ − hk∑

λ∈Λk

βλ gk(cλh)

(see (3.4b) and (3.15)) and making use of the fact that

(3.17)∫

Δk

τμ dτ = hk+|μ|k∏

=1

1μ + · · ·+ μk + k − + 1

,

we obtain the representation

Ik −Qk =p−k∑

m=0

hk+m∑

μ∈Nk

|μ|=m

1μ!

( k∏

=1

1μ + · · ·+ μk + k − + 1

−∑

λ∈Λk

βλ cμλ

)∂μ

τ gk(0)

+∫

Δk

k,p−k+1(τ) dτ − hk∑

λ∈Λk

βλ k,p−k+1(cλh).



Altogether, inserting the above relation into (3.14) implies

dn =p∑

k=1

p−k∑

m=0

hk+m∑

μ∈Nk

|μ|=m

1μ!

( k∏

=1

1μ + · · ·+ μk + k − + 1

−∑

λ∈Λk

βλ cμλ

)

× ∂μτ gk(0)u(tn−1) + R

(1)p+1 −R

(2)p+1 + R

(3)p+1,

R(3)p+1 =

p∑

k=1

( ∫

Δk


λ∈Λk

βλ k,p−k+1(cλh))

u(tn−1);

see also (3.4b) and (3.13c) for the definition of R(1)p+1 and R

(2)p+1.

3.2. Local error expansion and order conditions. In Lemma 1, we restatethe expansion for the defect (3.1) deduced in section 3.1. In view of section 4, weresume the employed abbreviations; see also (2.2a), (2.4a), (2.6), and (3.16c).

Lemma 1. We set Λk =λ = (λ1, λ2, . . . , λk) ∈ Nk : 1 ≤ λ ≤ λ−1, 1 ≤ ≤ k

and Δk =τ = (τ1, τ2, . . . , τk) ∈ Rk : 0 ≤ τ ≤ τ−1, 1 ≤ ≤ k

, where λ0 = s

and τ0 = h. Further, we denote ck = a1 + a2 + · · · + ak for 1 ≤ k ≤ s. Providedthat the condition cs = 1 is fulfilled, the defect of the exponential operator splittingmethod (2.8) equals

dn =p∑

k=1

∑

μ∈Nk

|μ|≤p−k

(−1)|μ|

μ!hk+|μ|

( k∏

=1

1μ + · · ·+ μk + k − + 1

−∑

λ∈Λk

αλ

k∏

=1

bλcμ

λ

)

× ehAk∏

=1

adμk−+1A (B) etn−1(A+B) u(0) + Rp+1.

(3.18)

The remainder Rp+1 = R(1)p+1 −R

(2)p+1 + R

(3)p+1 is given by

R(1)p+1 =

∫

Δp+1

p+1∏

j=1

(e(τp+1−j−τp+1−j+1)A B

)e(tn−1+τp+1)(A+B) u(0) dτ,

R(2)p+1 = hp+1

p+1∑

j=1

∑

λ∈Λp+1

αjλ

p+1∏

=1

(bλ

e(cλp−+1− cλp−+2 ) hA B)

×ϕj(bλp+1hB) eaλp+1hA Pλp+1−1 etn−1(A+B) u(0),

k,p−k+1(τ) = (p− k + 1)∫ 1

0

(1− z)p−k∑

μ∈Nk

|μ|=p−k+1

(−1)|μ|

μ!

×k∏

=1

(e(τk−− τk−+1) zA adμk−+1

A (B))

eτkzA τμ dz,

R(3)p+1 =

p∑

k=1

( ∫

Δk


λ∈Λk

αλ

k∏

=1

bλk,p−k+1(cλh)

)etn−1(A+B) u(0).



The coefficients αλ and αmλ are obtained by recurrence.1

Obviously, the first term in the local error expansion (3.18) vanishes whenever(combinations of) the coefficients

(3.19) Cμ =k∏

=1

1μ + · · ·+ μk + k − + 1

−∑

λ∈Λk

αλ

k∏

=1

bλcμ

λ

vanish for all 1 ≤ k ≤ p and |μ| ≤ p − k. In order to show that the classical orderconditions for a general exponential operator splitting method (2.8) coincide with thestiff order conditions, we meanwhile assume (1.1) to be a nonstiff problem; that is,we require the linear operator A to be bounded on X. In this situation, the expansion

ehA =∞∑

j=0

1j!

hjAj

is well defined on X and further yields

dμ =∑

k≥1, j≥0, μ∈Nk

k+j+|μ|≤p

(−1)|μ|

j!μ!hk+j+|μ| Cμ Ejμ(A, B) etμ−1(A+B) μ(0) + O(μp+1),

Ejμ(A, B) = Ajk∏

=1

adμk−+1A (B).

The splitting method is consistent of order p iff for any 1 ≤ q ≤ p the term involving hq

vanishes. Thus, taking into account all combinations of k ≥ 1, j ≥ 0, and μ ∈ Nk

such that k + |j|+ |μ| = q for some 1 ≤ q ≤ p fixed, we conclude that

∑

k≥1, j≥0, μ∈Nk

k+j+|μ|=q

(−1)|μ|

j!μ!Cμ Ejμ(A, B) = 0.

For j > 0, due to lower order conditions, the corresponding terms vanish; if j = 0,the same conditions as in the stiff case arise.

Alternatively, in order to show that for evolutionary Schrodinger equations thestiff and nonstiff order conditions coincide, one could also derive the classical orderconditions by means of the Campbell–Baker–Hausdorff formula (see [8]); however, asfurther tedious calculations are involved, we did not follow this approach here.

The proof of our convergence result for general exponential operator splittingmethods (2.8) relies on a suitable estimation of the local error expansion given inLemma 1. Provided that the conditions(3.20)

s∑

=1

a = 1,

∑

λ∈Λk

αλ

k∏

=1

bλcμ

λ=

k∏

=1

1μ + · · ·+ μk + k − + 1

, 1 ≤ k ≤ p, |μ| ≤ p− k,

1A MATLAB code recurrence is available at http://techmath.uibk.ac.at/mecht/research/research.html.



are fulfilled (see also (3.19)), the first term in the local error expansion vanishesand it remains to estimate Rp+1. Clearly, for nonstiff problems (1.1), the iteratedcommutators are bounded; however, whenever the differential equation involves anunbounded linear operator A, this is not true in general. For evolutionary Schrodingerequations, reasonable regularity assumptions on the initial value allow us to estimatethe decisive term R

(3)p+1 by means of Hypothesis 3.

3.3. Example methods. As an illustration, we next specify various exponentialoperator splitting methods and verify that the conditions (3.20) are satisfied; see alsoAppendix A. Our main result in section 4 ensures that the example methods retaintheir convergence orders for evolutionary Schrodinger equations.

Order 1. The conditions for p = 1 are given in (A.1a). In particular, for s = 1it follows that a1 = 1 = b1, and we retain the Lie–Trotter splitting

un = ehB ehA un−1, n ≥ 1, u0 given.

Order 2. For p = 2 the additional conditions are given in (A.1b). Choosing s = 2and simplifying (A.1a) and (A.1b) yields a1 + a2 = 1, b1 + b2 = 1, and b1a1 + b2 = 1

2 .For the choice a1 = 0 we thus retain the Strang splitting (1.3); alternatively, by settinga1 = 1

2 it follows that

un = e12 hA ehB e

12 hA un−1, n ≥ 1, u0 given.

Order 4/6. Symmetric symplectic methods that can be cast in the form (2.8)are proposed in Blanes and Moan [3, Tables 2–3]. The coefficients of a fourth- andsixth-order scheme are collected in Table 3.1; the conditions (3.20) are fulfilled (up tomachine precision) for p = 4 and p = 6, respectively (see also Appendix A).

Table 3.1Coefficients of a fourth-order method given in Blanes and Moan [3].

j aj

1,7 0.07920369643119572,6 0.35317290604977403,5 − 0.04206508035771954 1− 2 (a1 + a2 + a3)

j bj

1,6 0.2095151066133622,5 − 0.1438517731798183,4 1/2− (b1 + b2)7 0

4. Global error. In this section, we derive a convergence result for the ex-ponential operator splitting method (2.8) when applied to the abstract initial valueproblem (1.1) and further illustrate the error bound by a numerical example.

4.1. Global error bound in terms of the initial value. In the formulationof Theorem 1, we focus on evolutionary linear Schrodinger equations; that is, theunbounded linear operator A is related to the Laplacian. Provided that the methodcoefficients aj are nonnegative for 1 ≤ j ≤ s, it suffices to require that A generates aC0-semigroup; see Hypothesis 1.

Theorem 1. Assume that the coefficients of the exponential operator splittingmethod (2.8) fulfill the classical order conditions for some integer p ≥ 1. Supposefurther that the linear operators A and B satisfy the requirements of Hypotheses 2and 3 with ϑ = k/2 and ϑ = (j +k)/2 for integers j, k ≥ 0 such that j +k ≤ p. Then,provided that u(0) ∈ Xp/2, the error estimate

∥∥un − u(tn)∥∥

X≤ C

∥∥u(0)− u0

∥∥X

+ C hp∥∥u(0)

∥∥Xp/2

, 0 ≤ nh ≤ T,



is valid with some constant C > 0 depending in particular on T but not on n and h.Proof. Our main tools for deriving the above convergence result are a stability

bound for the splitting operator and an estimate for the defect of the exponentialoperator splitting method (2.8). For notational simplicity, we do not distinguish thearising constants.

We first verify the boundedness of Pns . To that purpose, we employ the relation

Ps =s∏

j=1

(I + h bj B ϕ1(bjhB)

)eajhA

obtained by means of (3.5a); see also (3.6). Applying (2.3) and (2.4) and using that1 + z ≤ ez for z > 0, it follows that

∥∥Ps

∥∥X←X

≤s∏

j=1

∥∥I + h bj B ϕ1

(bjhB

)∥∥X←X

∥∥eajhA∥∥

X←X

≤s∏

j=1

(1 + C h |bj |

)≤ eCh.

Furthermore, we conclude that for any h > 0 the estimate

(4.1)∥∥Pn

s

∥∥X←X

≤ C, 0 ≤ nh ≤ T,

holds true on finite time intervals with constant C > 0 depending on T but not on nand h.

We next deduce a suitable bound for the defects. Regarding the expansion givenin Lemma 1 and using that the order conditions (3.20) are fulfilled, it remains toestimate the remainder. By means of (2.3), we obtain

∥∥R(1)p+1

∥∥X≤

∫

Δp+1

p+1∏

j=1

(∥∥e(τk−j−τk−j+1)A∥∥

X←X‖B‖X←X

)

×∥∥e(tn−1+τp+1)(A+B)

∥∥X←X

∥∥u(0)∥∥

Xdτ

≤ C hp+1∥∥u(0)

∥∥X

;

set |μ| = 0 in (3.17). In a similar manner, it follows that∥∥R

(2)p+1

∥∥X≤ C hp+1

∥∥u(0)∥∥

X.

Hypothesis 3, together with (2.3), implies

(4.2)∥∥k,p−k+1(τ)

∥∥X←Xϑk

≤ C∑

|μ|=p−k+1

τμ

μ!, ϑk =

p− k + 12

.

Thus, by means of (3.17), we obtain

( ∫

Δk

∥∥k,p−k+1(τ)∥∥

X←Xϑk

dτ + hk∑

λ∈Λk

αλ

k∏

=1

|bλ|∥∥k,p−k+1(cλh)

∥∥X←Xϑk

)

×∥∥etn−1(A+B)

∥∥Xϑk

←Xϑk

∥∥u(0)∥∥

Xϑk

≤ C hp+1∥∥u(0)

∥∥Xϑk

.



Employing the reasonable assumption

(4.3) ‖v‖Xϑ≤ ‖v‖X

ϑ, ϑ ≤ ϑ, v ∈ Xϑ,

this finally gives the bound∥∥R

(3)p+1

∥∥X≤ C hp+1

∥∥u(0)∥∥

Xp/2.

From the above considerations, we obtain the following estimate for the defect:∥∥dn

∥∥X

=∥∥Rp+1

∥∥X≤ C hp+1

∥∥u(0)∥∥

Xp/2;

see also Lemma 1, (3.20), and (4.3). With the help of (4.1), it is now straightforwardto estimate the global error; see (2.10). Altogether, we obtain

∥∥en

∥∥X≤

∥∥Pns

∥∥X←X

∥∥e0

∥∥X

+n−1∑

j=0

∥∥Pn−j−1s

∥∥X←X

∥∥dj+1

∥∥X

≤ C∥∥e0

∥∥X

+ C hp∥∥u(0)

∥∥Xp/2

, 0 ≤ nh ≤ T,

with constant C > 0 depending in particular on T .In the present situation, as seen in the proof of Theorem 1, general exponential

splitting methods remain stable for any choice of the time stepsize 0 < h ≤ T ; however,for evolutionary Schrodinger equations (1.1) involving a time-dependent or solution-dependent operator B, in general, a stepsize restriction is expected; see [9, 14].

The above error bound implies that the example methods given in section 3.2retain their convergence orders when applied to time-dependent Schrodinger equationssubject to a periodic boundary condition, provided that the data are sufficientlydifferentiable. For less regular initial values, the following convergence result holdstrue; order reduction phenomena for the Strang splitting are also studied in [11, 13].

Corollary 1. Under the assumptions of Theorem 1, whenever u(0) ∈ Xk/2 butu(0) ∈ X(k+1)/2 for some 1 ≤ k ≤ p− 1, the error bound

∥∥un − u(tn)∥∥

X≤ C

∥∥u(0)− u0

∥∥X

+ C hk∥∥u(0)

∥∥Xk/2

, 0 ≤ nh ≤ T,

is valid with constant C > 0 depending on T but not on n and h.We note that the above convergence analysis also applies to equations involving a

differential operator of higher order; in this case, the relation between the quantitiesϑ, ϑ arising in Hypothesis 3 has to be adapted.

4.2. Numerical example. We next illustrate the error bound of Theorem 1by a numerical example for a time-dependent linear Schrodinger equation. In thepresent paper, we do not recapitulate how the obtained results for abstract differen-tial equations are applicable to pseudospectral discretizations of time-dependent linearSchrodinger equations; in this respect and for a more detailed description of the real-ization of the numerical example, we refer the reader to Jahnke and Lubich [11, sect. 3].

As test problem, we consider (2.7) with V (x) = 1 − cos x; in order to studynumerically the influence of the initial value on the temporal convergence order of anexponential operator splitting method, we choose various initial data that correspondto u(0) ∈ Xk/2 for 2 ≤ k ≤ 6. The initial-boundary value problem is discretized inspace by the Fourier pseudospectral method with M = 213 gridpoints.



10−3

10−2

10−1

100

10−12

10−10

10−8

10−6

10−4

10−2

100

Stepsize

rorrE

p = 2p = 4p = 6

10−3

10−2

10−1

100

10−12

10−10

10−8

10−6

10−4

10−2

Stepsize

rorrE

k=5k=4k=3k=2

Fig. 4.1. Numerically observed convergence orders of exponential operator splitting methods.Schemes of various orders (left picture). Initial values of various regularity (right picture).

For a sufficiently regular initial value, i.e., k = 6, we apply the second-orderStrang splitting and the fourth- and sixth-order schemes proposed by Blanes andMoan [3] for various time stepsizes (see section 3.3); a reference solution at final timeT = 1 is computed by the sixth-order splitting method with stepsize h = 2−11. Theobtained temporal errors, displayed in Figure 4.1 (left picture), confirm the assertion ofTheorem 1; the splitting methods retain their convergence orders for time-dependentSchrodinger equations.

For less regular initial values, the order reduction predicted by Corollary 1 is alsoobserved numerically; for the sixth-order scheme the numerical convergence ordersare displayed in Figure 4.1 (right picture).

Appendix A.. In this appendix, we state the coefficients αλ, λ ∈ Λk (see TableA.1), and further specify the conditions (3.20) for 1 ≤ k ≤ p = 4:

s∑

=1

a = 1,s∑

λ1=1

bλ1 = 1,(A.1a)

s∑

λ1=1

bλ1cλ1 =12,

s∑

λ1=1

12

b2λ1

+s∑

λ1=1

λ1−1∑

λ2=1

bλ1bλ2 =12,(A.1b)

Table A.1Coefficients αλκ = αλκ1λκ2 ···λκk

for κ = (κ1, κ2, . . . , κk).

k κ αλκ

1 1 12 11 1/2

12 13 111 1/6

112,122 1/2123 1

4 1111 1/241112,1222 1/6

1122 1/41123,1223,1233 1/2

1234 1



s∑

λ1=1

bλ1c2λ1

=13,

s∑

λ1=1

12

b2λ1

cλ1 +s∑

λ1=1

λ1−1∑

λ2=1

bλ1bλ2cλ1 =13,

s∑

λ1=1

12

b2λ1

cλ1 +s∑

λ1=1

λ1−1∑

λ2=1

bλ1bλ2cλ2 =16,

s∑

λ1=1

16

b3λ1

+s∑

λ1=1

λ1−1∑

λ2=1

12

bλ1bλ2

(bλ1 + bλ2

)+

s∑

λ1=1

λ1−1∑

λ2=1

λ2−1∑

λ3=1

bλ1bλ2bλ3 =16,

s∑

λ1=1

bλ1c3λ1

=14,

s∑

λ1=1

12

b2λ1

c2λ1

+s∑

λ1=1

λ1−1∑

λ2=1

bλ1bλ2c2λ1

=14,

s∑

λ1=1

12

b2λ1

c2λ1

+s∑

λ1=1

λ1−1∑

λ2=1

bλ1bλ2cλ1cλ2 =18,

s∑

λ1=1

12

b2λ1

c2λ1

+s∑

λ1=1

λ1−1∑

λ2=1

bλ1bλ2c2λ2

=112

,

s∑

λ1=1

16

b3λ1

cλ1 +s∑

λ1=1

λ1−1∑

λ2=1

12

bλ1bλ2cλ1

(bλ1 + bλ2

)

+s∑

λ1=1

λ1−1∑

λ2=1

λ2−1∑

λ3=1

bλ1bλ2bλ3cλ1 =18,

s∑

λ1=1

16

b3λ1

cλ1 +s∑

λ1=1

λ1−1∑

λ2=1

12

bλ1bλ2

(bλ1cλ1 + bλ2cλ2

)

+s∑

λ1=1

λ1−1∑

λ2=1

λ2−1∑

λ3=1

bλ1bλ2bλ3cλ2 =112

,

s∑

λ1=1

16

b3λ1

cλ1 +s∑

λ1=1

λ1−1∑

λ2=1

12

bλ1bλ2cλ2

(bλ1 + bλ2

)+

s∑

λ1=1

λ1−1∑

λ2=1

λ2−1∑

λ3=1

bλ1bλ2bλ3cλ3 =124

,

s∑

λ1=1

124

b4λ1

+s∑

λ1=1

λ1−1∑

λ2=1

bλ1bλ2

(16

b2λ1

+14

bλ1bλ2 +16

b2λ2

)

+s∑

λ1=1

λ1−1∑

λ2=1

λ2−1∑

λ3=1

12

bλ1bλ2bλ3

(bλ1 + bλ2 + bλ3

)

+s∑

λ1=1

λ1−1∑

λ2=1

λ2−1∑

λ3=1

λ3−1∑

λ4=1

bλ1bλ2bλ3bλ4 =124

.



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