modified magnus expansion in application to highly oscillatory differential equations ·...

MODIFIED MAGNUS EXPANSION IN APPLICATION TO HIGHLYOSCILLATORY DIFFERENTIAL EQUATIONS

MARIANNA KHANAMIRYAN

Abstract. In this work we introduce a novel approach, incorporating modified Magnus method with Filonquadrature, in application to the solution of systems of highly oscillatory linear and non-linear differential equations.Both methods are extremely efficiently in the setting of highoscillation compared to conventional ideas. This paperpresents and highlights the advantage and superior performance of numerical methods which are designed from theoutset for intrinsic dynamical systems of highly oscillatory nature.

Key words. High oscillation, dynamical systems, Filon quadrature, Magnus method, modified Magnus method,waveform relaxation methods.

AMS subject classifications.65L05, 65P, 34B, 34E05, 34C15, 34C60

1. Introduction. Stiff dynamical systems are active scientific research area, with nu-merous publications in the field. Classical and even advanced scientific literature providesus with effective methods for smooth solutions, however so far theory and computationaltools for highly oscillatory dynamical systems have been scarce. Even the most advancedtechniques using exponential integrators deliver very poor numerical solution in presence ofhigh oscillation, unless the step-sizeh of numerical integration is chosen to be infinitesimalcompared to the increasing frequencyω of oscillation, [7], introducing strong requirementon the producthω . Adversely, numerical techniques empowered by Filon quadrature andrelated techniques, based on the asymptotic expansion versus Taylor expansion, represent arather efficient and competitive approach compared to conventional methods, [13, 14, 16, 17].Not only large step-size becomes affordable, but also highly oscillatory nature of the prob-lem works in our advantage, and our approximation improves as the frequency of oscillationgrows.

We first proceed with introduction of the numerical methods and main results, and pro-vide a detailed computational framework for our methods in upcoming sections.

1.1. Matrix transformation. Given,

I [ f ] =∫ b

aXω(t) f (t)dt, X′

ω = AωXω , Xω = eΩ, (1.1)

with a matrix valued kernelXω depending on a real parameterω ≫ 1 and satisfying matrixlinear differential equation (1.1). We assume thatAω is a non-singular matrix with variablecoefficients,σ(Aω) ⊂ iR, ‖A−1

ω ‖ ≪ 1 and f ∈ Rd is a smooth vector-valued function. Nu-

merical solution of this type of integral, and in particularof the the matrix linear differentialequation in (1.1) is of practical importance in applicationto the solution of linear and non-linear systems of highly oscillatory dynamical systems,

y= Aωy+ f , y(0) = y0, (1.2)

with analytic time dependant solution,

y= Xωy0+

∫ t

0Xω f dτ = Xωy0+ I [ f ]. (1.3)

We apply matrix transformation, introduced in [17] , to obtain the asymptotic expansionof the integral in (1.1).

2 MARIANNA KHANAMIRYAN

The first step is to vectorize and transpose the matrixXω ,

Xω =

x11 x12 . . . x1d

x21 x22 . . . x2d...

. . .. . .

...xd1 xd2 . . . xdd

resulting in a vectorXω ,

Xω = [x1, x2, ..., xd]⊺,

whereby each entry element ¯xi is a column vector from the matrixXω above,

xi =

x1i

x2i...

xdi

.

DEFINITION 1.1. The matrix direct sum of n matrices constructs a block diagonal matrixfrom a set of square matrices,

d⊕

i=1

Ai = diag(A1,A2, ...,An) =

A1 0 . . . 00 A2 . . . 0...

.. .. . .

...0 0 . . . An

.

DEFINITION 1.2. Kronecker product, or direct product, denoted by⊗

, is an operationon two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensorproduct with respect to a standard choice of basis. If A is an m×n matrix and B is a p×qmatrix, then the Kronecker product is the mp×nq block matrix,

A⊗

B=

a11B . . . a1nB...

......

am1B . . . amnB

with elements defined by

cαβ = ai j bkl ,

where

α = p(i −1)+ k,

β = q( j −1)+ l .

Vector Xω satisfies a linear differential equationX′ω = Bω Xω , with Bω =

d⊕

i=1Aω repre-

senting a directd-tuple sum of the original matrixAω .The second step is to scale an identity matrixI by taking its direct product with the

vector-valued functionf := f ⊺ = [ f1, f2, . . . , fd], F = f 1×d⊗

Id×d.

MODIFIED MAGNUS METHOD USING FILON QUADRATURE 3

Finally, we arrive at equality,

∫ b

aXω f dt =

∫ b

aFXωdt.

The latter appears to be suitable for integration by parts and representation of the asymptoticexpansion of the integral (1.1), providing us with theoretical background to proceed with theFilon quadrature.

1.2. The asymptotic method.Hereafter we apply the bigO notation, defined element-wise for matrix- and vector-valued functions, [17].

THEOREM 1.3. [17] Postulate that

I [ f ] =∫ b

aXω(t) f (t)dt, and X′

ω = Aω(t)Xω ,

where Aω is an arbitrary non-singular matrix,σ(Aω)⊂ iR, ω is a real parameter and f∈Rd

is a smooth vector-valued function.If F ω = O(F), B−1

ω = O(B) andXω = O(Xω), then

I [ f ]−QAs [ f ] = (−1)s

∫ b

aσs(t)Xω (t)dt

= O(FBs+1X), as ω → ∞.

1.3. The Filon method. For the Filon quadrature, we construct anr-degree polynomialinterpolationv for the vector-valued functionf in (1.1),

v(t) =ν

∑l=1

θl−1

∑j=0

αl , j(t) f ( j)(tl ),

which agrees with function values and derivativesv( j)(tl ) = f ( j)(tl ) at node pointsa= t1 <t2 < ... < tν = b, with associatedθ1,θ2, ...,θν multiplicities, j = 0,1, ...,θl − 1, and l =1,2, ...,ν.

By definition,Filon-type method is

QFs [ f ] =

∫ b

aXω(t)v(t)dt =

ν

∑l=1

θl−1

∑j=0

βl , j f ( j)(tl ),

whereβl , j =∫ b

a Xω(t)αl , j(t)dt.

THEOREM 1.4. [17] Postulate that

I [ f ] =∫ b

aXω(t) f (t)dt, and X′

ω = Aω(t)Xω ,

Aω is an arbitrary non-singular matrix,σ(Aω )⊂ iR, ω is a real parameter and f∈ Rd

is a smooth vector-valued function.If F ω = O(F), B−1

ω = O(B) andXω = O(Xω ), then


I [ f ]−QFs [ f ] = (−1)s

∫ b

aσs(t)Xω(t)dt

= O(FBs+1X), as ω → ∞.

THEOREM 1.5. [17] Let θ1,θν ≥ s. Then the numerical order of the Filon-type methodis equal to r= ∑ν

l=1 θl −1.

Needless to mention thatθ1,θν ≥ s define the smoothness of the functionf at the endpoints, and hence contribute to the numerical order of the Filon method.

1.4. Magnus expansion.At this stage we find it suitable to draw our attention to thenumerical solution of the matrix ordinary differential equation

X′ω = AωXω , Xω(0) = I , Xω = eΩ. (1.4)

Here,Ω represents the Magnus expansion, an infinite recursive series,

Ω(t) =∫ t

0Aω(t)dt

+12

∫ t

0[Aω(τ),

∫ τ

0Aω(ξ )dξ ]dτ

+14

∫ t

0[Aω(τ),

∫ τ

0[Aω (ξ ),

∫ ξ

0Aω(ζ )dζ ]dξ ]dτ

+112

∫ t

0[[Aω(τ),

∫ τ

0Aω(ξ )dξ ],

∫ τ

0Aω(ζ )dζ ]dτ

+ · · · . (1.5)

From [5], we know thatΩ(t) satisfies the following non-linear differential equation,

Ω′ =∞

∑j=0

B j

j!adj

Ω(Aω ) = dexp−1Ω (Aω (t)), Ω0 = 0, (1.6)

with B j being Bernoulli numbers and the adjoint operator is defined as

ad0Ω(Aω) = Aω

adΩ(Aω) = [Ω,Aω ] = ΩAω −AωΩ

adj+1Ω (Aω) = [Ω,adj

ΩAω ]. (1.7)

Solving this equation forΩ with Picard iteration results in an infinite recursive series andwe use truncated expansionΩs in approximation ofΩ, [18].

Magnus methods preserve peculiar geometric properties of the solution, ensuring that ifXω is in matricial Lie groupG andAω is in associated Lie algebrag of G, then the numericalsolution after discretization stays on the manifold. Moreover, Magnus methods preserve time-symmetric properties of the solution after discretization, for anya> b, Xω(a,b)−1 =Xω(b,a)yieldsΩ(a,b) =−Ω(b,a). These properties appear to be valuable in applications.

1.5. Main results. In our previous works, [16, 17], we developed numerical methodsfor dynamical systems with constant and variable coefficients, where the representation ofthe matrixXω (1.1), an infinite Magnus series, was truncated and evaluated numerically. Inpresent work we expand our view. First, we employ Filon quadrature with Magnus expansion,to solve integral commutators in (1.6). Second, we introduce a combination of a modifiedMagnus method and Filon quadrature in evaluation of (1.6). Third, we apply developedsolutions for matrix exponential in the numerical approximation of (1.3).


2. The Magnus method. In this section we focus on Magnus methods for approxima-tion of a matrix-valued functionXω in (1.4). There is a large list of publications available ontheLie-group methods, here we refer to some of them: [1, 2, 4, 8, 9, 12, 15, 20, 21].

Given the representation forΩ(t) in (1.6), numerical evaluation of the commutator brack-ets with conventional quadrature rules is well presented inthe literature, [2], [12], [8]. Forexample, by choosing a symmetric gridc1,c2, ...,cν , and taking Gaussian points with respectto 1

2. Consider setA1,A2, ...,Aν, with Ak = hA(t0+ckh),k= 1,2, ...,ν. Linear combinationsof this basis form an adjoint basisB1,B2, ...,Bν, with

Ak =ν

∑l=1

(ck−12)l−1Bl , k= 1,2, ...,ν.

In this basis the six-order method, with Gaussian pointsc1 =12 −

√15

10 ,c2 =12,c3 =

12 +√

1510 , Ak = hA(t0+ ckh), is

Ω(t0+h)≈ B1+112

B3−112

[B1,B2]+1

240[B2,B3] (2.1)

+1

360[B1, [B1,B3]]−

1240

[B2, [B1,B2]]+1

720[B1[B1, [B1,B2]]], (2.2)

where

B1 = A2, B2 =

√153

(A3−A1), B3 =103(A3−2A2+A1).

This can be reduced further and written in a more compact manner [8], [12],

Ω(t0+h)≈ B1+112

B3+P1+P2+P3,

where

P1 = [B2,112

B1+1

240B3], (2.3)

P2 = [B1, [B1,1

360B3−

160

P1]], (2.4)

P3 =120

[B2,P1]. (2.5)

A more profound approach taking Taylor expansion ofA(t) around the pointt1/2 = t0+h2

was introduced in [2],

A(t) =∞

∑i=0

ai(t − t1/2)i , with ai =

1i!

dnA(t)dt i |t=t1/2

. (2.6)

This can be substituted in the univariate integrals of the form

B(i) =1

hi+1

∫ h/2

−h/2t iA

(

t +h2

)

dt, i = 0,1,2, ...

to obtain a new basis

B(0) = a0+112

h2a2+180

h4a4++1

448h6a6... (2.7)

B(1) =112

ha1+180

h3a3+1

448h5a5+ ... (2.8)

B(2) =112

a0+180

h2a2+1

448h4a4+

12304

h6a6+ ... (2.9)


In these terms a second order method will look as follows, eΩ = ehB(0) +O(h3). Whereas fora six-order method,Ω = ∑4

i=1 Ωi +O(h7), one needs to evaluate only four commutators,

Ω1 = hB(0) (2.10)

Ω2 = h2[B(1),32

B(0)−6B(2)] (2.11)

Ω3+ Ω4 = h2[B(0), [B(0),12

hB(2)− 160

Ω2]]+35

h[B(1),Ω2]. (2.12)

Numerical behaviour of the fourth and six order classical Magnus method is illustratedin Figures 2.1, 2.2, 2.3, 2.4, 2.5, and 2.6. The method is applied to solve Airy equationy′′(t) =−ty(t) with [1,0]⊺ initial conditions,t ∈ [0,1000], for varies step-sizes,h= 1

4, h= 110

andh = 125. Comparison shows that for a bigger interval stepsh both fourth and six order

methods give similar results, as illustrated in Figures 2.1, 2.2, 2.4 and 2.5. However,for smaller steps six order Magnus method has a more rapid improvement in approximationcompared to a fourth order method, Figures 2.3 and 2.6.

0 200 400 600 800 1000−4

−2

0

2

4x 10

−3

FIG. 2.1. Global error of the fourth order Magnus method for the Airy equation y′′(t) = −ty(t) with [1,0]⊺

initial conditions,0≤ t ≤ 1000and step-size h= 14 .

The following Theorem provides sufficient condition for convergence of the Magnusseries for a bounded linear operatorA(t) in Hilbert space, [3], extending Theorem 3 from[19], where the same results are stated for real matrices.

THEOREM 2.1. (F. Casas, [3]) Consider the differential equation X′ = A(t)X defined ina Hilbert spaceH with X(0) = I, and let A(t) be a bounded linear operator onH . Then,the Magnus seriesΩ(t) in (1.6) converges in the interval t∈ [0,T) such that

∫ T

0‖A(τ)‖dτ < π (2.13)

and the sumΩ(t) satisfiesexpΩ(t) = X(t).This result is applicable to bounded operators, whereas we consider matrices with a

propertyσ(Aω )⊂ iR, andω → ∞. In this respect we refer to work on Magnus integrators forSchrodinger equation, [6],

idψdt

= H(t)ψ , ψ(t0) = ψ0.


0 200 400 600 800 1000−8

−4

0

4

8x 10

−4


initial conditions,0≤ t ≤ 1000and step-size h= 110.

0 200 400 600 800 1000−1

−0.5

0

0.5

1x 10

−7


initial conditions,0≤ t ≤ 1000and step-size h= 125.

The time-dependant HamiltonianH(t) is a finite dimensional Hermitian operator. As a resultof discretization of an unbounded operator, typically it isa sum of discretized negative Lapla-cian and a time-dependent potential, and hence,H(t) my have an arbitrary large norm. In [6],authors provide asymptotically sharp error bounds for Magnus integrators in the frameworkapplied to time-dependent Schrodinger equation, where there are no restrictions on smallnessnor bounds onh‖H(t)‖, andh stands for a step-size of a numerical integrator.

In present work we introduce an alternative method to solve equations of the kind (1.4).We show that applying Filon quadrature in combination with Magnus method or modifiedMagnus method to evaluate integrals in Magnus expansion results in higher accuracy in thesolution of (1.4), and we extend the solution of the matrix exponential to the solution ofXωin (1.3), satisfying (1.4).

Initialy, we test our methods for the followingAiry-type equation,

y′′(t)+g(t)y(t) = 0, g(t)> 0 for t > 0, and limt→∞

g(t) = +∞. (2.14)


0 200 400 600 800 1000−4

−2

0

2

4x 10

−3

FIG. 2.4.Global error of the six order Magnus method for the Airy equation y′′(t) =−ty(t) with [1,0]⊺ initialconditions,0≤ t ≤ 1000and step-size h= 1

4 .

0 200 400 600 800 1000−8

−4

0

4

8x 10

−4


10.

We can rewrite this ordinary differential equation in a vector form,

y′ = Aω(t)y, where Aω(t) =

(

0 1−g(t) 0

)

. (2.15)

The equation is highly oscillatory due to the large imaginary spectrum of the matrixAω .We can now apply Filon quadrature to evaluate theΩ corresponding matrix exponential.

3. The modified Magnus method. In this section we continue our discussion on thesolution of the matrix differential equation, and introduce modified Magnus method in com-bination with Filon quadrature, to solve linear oscillator

y′ = Aω(t)y, y(0) = y0, σ(Aω)⊂ iR, ω → ∞. (3.1)

Introducing local change of variables, we write the solution in the form,

y(t) = e(t−tn)Aω (tn+ 12h)x(t − tn), t ≥ tn. (3.2)


0 200 400 600 800 1000−5

0

5x 10

−9


25.

Observe that,

y′ = Aω(

tn+12

h)

e(t−tn)Aω (tn+ 12h)x(t − tn)

+e(t−tn)Aω (tn+ 12h)x′(t − tn)

= Aω(t)y(t)

= Aω(t)e(t−tn)Aω (tn+ 1

2h)x(t − tn). (3.3)

Therefore,

Aω(

tn+12

h)

e(t−tn)Aω(tn+ 12h)x(t − tn)

−Aω(t)e(t−tn)Aω (tn+ 12h)x(t − tn)

=−e(t−tn)Aω(tn+ 12h)x′(t − tn).

(3.4)

In summary,x(t) satisfies the following differential equation,

x′ = B(t)x, x(0) = yn, (3.5)

with

B(t) = e−tAω (tn+ 12h)[Aω(tn+ t)−Aω(tn+ 1

2)]etAω (tn+ 1

2h), where tn+ 12= tn+

h2.

Recalling our original equation fory(t), we develop its numerical solution in terms ofx(t),

yn+1 = ehAω (tn+h/2)xn, (3.6)

xn = eΩnyn. (3.7)

It is evident that in the representation forΩn, the commutator brackets are now formedby the matrixB(t). It is possible to reduce cost of evaluation of the matrixB(t) by simplifying


it, [11]. Denoteq=√

g(tn+ 12h) andv(t) = g(tn+ t)−g(tn+ 1

2h). Then,

B(t) = v(t)

[

(2q)−1sin2qt q−2sin2qt−cos2qt −(2q)−1sin2qt

]

(3.8)

= v(t)

[

q−1sinqt−cosqt

]

[

cosqt q−1sinqt]

, (3.9)

and for the product

B(t)B(s) = v(t)v(s)sinq(s− t)

q

[

q−1sinqt−cosqt

]

[

cosqs q−1sinqs]

. (3.10)

It was shown in [11] that

‖B(t)‖= cos2wt+sin2wt

w2 and B(t) = O(t − tn+ 12).

Given the compact representation of the matrixB(t) with oscillatory entries, we solveΩ(t) in (1.6) with a Filon-type method, approximating functionsin B(t) by a polynomialv(t), for example Hermite interpolation, as in classical Filon-type method. Furthermore, inour approximation we use end points only, although the method is general and more nodes ofapproximation can be required.

Performance of the modified Magnus method is better than thatof the classical Mag-nus method due to a number of reasons. Firstly, the fact that the matrixB is small,B(t) =O(t − tn+ 1

2), contributes to higher order correction to the solution. Secondly, the order of the

modified Magnus method increases fromp= 2s to p= 3s+1, [10].In Figure 3.1 we present the global error of the fourth order modified Magnus method

with exact integrals for the Airy equationy′′(t) = −ty(t) with [1,0]⊺ initial conditions, 0≤t ≤ 2000 time interval and step-sizeh = 1

5. This can be compared with the global error oftheFM method applied to the same equation with exactly the same conditions and step-size,Figure 3.2. In Figures 3.3 and 3.4 we compare the fourth orderclassical Magnus method witha remarkable performance of the fourth orderFM method, applied to theAiry equation witha large step-size equal toh= 1

2. While in Figures 3.5 and 3.6 we solve theAiry equation withthe fourth orderFM method with step-sizesh= 1

4 andh= 110 respectively.

3.1. The modified WRFM. Applications of the modified Magnus method include sys-tems of highly oscillatory non-linear equations

y= Aω(t)y+ f (t,y), y(t0) = y0, (3.11)

with analytic solution

y(t) = Xω(t)y0+

∫ t

0Xω(t − τ) f (τ,y(τ))dτ = Xωy0+ I [ f ]. (3.12)

We apply modified Magnus techniques to the system of non-linear ODEs by introducing thefollowing change of variables

y(t) = e(t−tn)Aω(tn)x(t − tn), t ≥ t0, (3.13)


0 500 1000 1500 2000−3

−2

−1

0

1

2

x 10−10

FIG. 3.1.Global error of the fourth order modified Magnus method with exact evaluation of integral commu-tators for the Airy equation y′′(t) =−ty(t) with [1,0]⊺ initial conditions,0≤ t ≤ 2000and step-size h= 1

5 .

0 500 1000 1500 2000−3

−2

−1

0

1

2

x 10−10

FIG. 3.2.Global error of the fourth order FM method with end points only and multiplicities all2 for the Airyequation y′′(t) =−ty(t) with [1,0]⊺ initial conditions,0≤ t ≤ 2000and step-size h= 1

5 .

and solving the system locally with a constant matrixAω , while x(t) satisfies a non-linearhighly oscillatory equation

x′(t) = Bω(t)Xω(t)+e−tAω(tn) f (t,etAω (tn)x(t)), t ≥ 0, (3.14)

with exactly the same matrixBω as for linear equations,

Bω(t) = e−tAω (tn)[Aω(t)−Aω(tn)]etAω (tn). (3.15)

The analytic solution to the new system is of the form

x(t) = Xωt)y0+

∫ t

0Xω(t − τ)e−τAω(tn) f (τ,eτAω (tn)x(τ))dτ. (3.16)

We solve the system applyingWRFmethod, [16].

x[0](t) = y0, (3.17)

x[m+1] = X(t)y0+

∫ t

0X(t − τ)e−τA(tn) f (τ,eτA(tn)x[m](τ))dτ, (3.18)


0 500 1000 1500 2000−4

−2

0

2

4

x 10−7

FIG. 3.3.Global error of the fourth order FM method with end points only and multiplicities all2 for the Airyequation y′′(t) =−ty(t) with [1,0]⊺ initial conditions,0≤ t ≤ 2000and step-size h= 1

2 .

0 500 1000 1500 2000−0.015

−0.005

0.005

0.015


initial conditions,0≤ t ≤ 2000and step-size h= 12 .

REFERENCES

[1] A. M ARTHINSEN B. OWREN, Integration methods based on rigid frames, Norwegian University of Science& Technology Tech. Report, (1997).

[2] S. BLANES, F. CASAS, AND J. ROS, Improved high order integrators based on the Magnus expansion, BIT,40 (2000), pp. 434–450.

[3] F. CASAS, Sufficient conditions for the convergence of the Magnus expansion, J. Phys. A, 40 (2007),pp. 15001–15017.

[4] P. E. CROUCH AND R. GROSSMAN, Numerical integration of ordinary differential equationson manifolds,J. Nonlinear Sci., 3 (1993), pp. 1–33.

[5] F. HAUSDORFF, Die symbolische exponentialformel in der gruppentheorie, Berichte der SachsischenAkademie der Wissenschaften (Math. Phys. Klasse), 58 (1906), pp. 19–48.

[6] M ARLIS HOCHBRUCK AND CHRISTIAN LUBICH, On magnus integrators for time-dependent Schrodingerequations, SIAM J. Numer. Anal., 41 (2003), pp. 945–963.

[7] M ARLIS HOCHBRUCK AND ALEXANDER OSTERMANN, Exponential integrators, Acta Numerica, 19(2010), pp. 209–286.

[8] A. I SERLES, Brief introduction to Lie-group methods, in Collected lectures on the preservation of stabilityunder discretization (Fort Collins, CO, 2001), SIAM, Philadelphia, PA, 2002, pp. 123–143.

[9] A RIEH ISERLES, On the discretization of double-bracket flows, Found. Comput. Math., 2 (2002), pp. 305–329.


0 500 1000 1500 2000

−1

0

1

x 10−9

FIG. 3.5.Global error of the fourth order FM method with end points only and multiplicities all2, for the Airyequation y′′(t) =−ty(t) with [1,0]⊺ initial conditions,0≤ t ≤ 2000and step-size h= 1

4 .

0 500 1000 1500 2000−4

−2

0

2

4 x 10−11

FIG. 3.6.Global error of the fourth order FM method with end points only and multiplicities all2, for the Airyequation y′′(t) =−ty(t) with [1,0]⊺ initial conditions,0≤ t ≤ 2000and step-size h= 1

10.

[10] A. I SERLES, On the global error of discretization methods for highly-oscillatory ordinary differential equa-tions, BIT, 42 (2002), pp. 561–599.

[11] , On the method of Neumann series for highly oscillatory equations, BIT, 44 (2004), pp. 473–488.[12] , Magnus expansions and beyond, to appear in Proc. MPIM, Bonn, (2009).[13] A. I SERLES ANDS. P. NØRSETT, On quadrature methods for highly oscillatory integrals andtheir imple-

mentation, BIT, 44 (2004), pp. 755–772.[14] , Efficient quadrature of highly oscillatory integrals usingderivatives, Proc. R. Soc. Lond. Ser. A Math.

Phys. Eng. Sci., 461 (2005), pp. 1383–1399.[15] ARIEH ISERLES ANDANTONELLA ZANNA , Efficient computation of the matrix exponential by generalized

polar decompositions, SIAM J. Numer. Anal., 42 (2005), pp. 2218–2256 (electronic).[16] M. K HANAMIRYAN , Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary

differential equations. I, BIT, 48 (2008), pp. 743–761.[17] , Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equa-

tions. II, Submitted to BIT, (2009).[18] W. MAGNUS, On the exponential solution of differential equations for alinear operator, Comm. Pure Appl.

Math., 7 (1954), pp. 649–673.[19] P. C. MOAN AND J. NIESEN, Convergence of the Magnus series, Found. Comput. Math., 8 (2008), pp. 291–

301.[20] H. MUNTHE-KAAS,Runge-Kutta methods on Lie groups, BIT, 38 (1998), pp. 92–111.


[21] A. ZANNA , The method of iterated commutators for ordinary differential equations on lie groups, Tech. Rep.DAMTP 1996/NA12, University of Cambridge, (1996).

modified magnus expansion in application to highly oscillatory differential equations ·...

Documents