Exponential Integrators

John C. Bowman (University of Alberta)

May 22, 2007

www.math.ualberta.ca/∼bowman/talks

1

Outline

•Exponential Integrators

– Exponential Euler

– History

•Generalizations

– Stationary Green Function

– Higher-Order

– Vector Case

– Conservative Exponential Integrators

– Lagrangian Discretizations

•Charged Particle in Electromagnetic Fields

•Embedded Exponential Runge–Kutta (3,2) Pair

•Conclusions

2

Exponential Integrators

•Typical stiff nonlinear initial value problem:

dx

dt+ η x = f (t, x), x(0) = x0.

• Stiff: Nonlinearity f varies slowly in t compared with the valueof the linear coefficient η:

∣∣∣∣1

f

df

dt

∣∣∣∣≪ |η|

•Goal: Solve on the linear time scale exactly; avoid the lineartime-step restriction ητ ≪ 1.

• In the presence of nonlinearity, straightforward integratingfactor methods (cf. Lawson 1967) do not remove the explicitrestriction on the linear time step τ .

• Instead, discretize the perturbed problem with a scheme that isexact on the time scale of the solvable part.

3

Exponential Euler Algorithm

•Express exact evolution of x in terms of P (t) = eηt:

x(t) = P−1(t)

(x0 +

∫ t

0

fP dt

).

•Change variables: P dt = η−1dP ⇒

x(t) = P−1(t)

(x0 + η−1

∫ P (t)

1

f dP

).

•Rectangular approximation of integral ⇒ Exponential Euler:

xi+1 = P−1

(xi +

P − 1

ηfi

),

where P = eητ and τ is the time step.

•The discretization is now with respect to P instead of t.

4

Exponential Euler Algorithm (E-Euler)

xi+1 = P−1xi +1 − P−1

ηf (xi),

•Also called Exponentially Fitted Euler, ETD Euler, filteredEuler, Lie–Euler.

•As τ → 0 the Euler method is recovered:

xi+1 = xi + τf (xi).

• If E-Euler has a fixed point, it must satisfy x =f (x)

η; this is

then a fixed point of the ODE.

• In contrast, the popular Integrating Factor method (I-Euler).

xi+1 = P−1i (xi + τfi)

can at best have an incorrect fixed point: x =τf (x)

eητ − 1.

5

Comparison of Euler Integrators

dx

dt+ x = cos x, x(0) = 1.

−0.03

−0.02

−0.01

0

error

0 1 2 3 4 5

t

euler

i euler

e euler

6

History

•Certaine [1960]: Exponential Adams-Moulton

•Nørsett [1969]: Exponential Adams-Bashforth

•Verwer [1977] and van der Houwen [1977]: Exponential linearmultistep method

•Friedli [1978]: Exponential Runge-Kutta

•Hochbruck et al. [1998]: Exponential integrators up to order 4

•Beylkin et al. [1998]: Exact Linear Part (ELP)

•Cox & Matthews [2002]: ETDRK3, ETDRK4; worst case: stifforder 2•Lu [2003]: Efficient Matrix Exponential

•Hochbruck & Ostermann [2005a, 2005b]: Explicit ExponentialRunge-Kutta; stiff order conditions.

7

Generalization

•Let L be a linear operator with a stationary Green’s functionG(t, t′) = G(t − t′):

∂G(t, t′)

∂t+ LG(t, t′) = δ(t − t′).

•Let f be a continuous function of x. Then the ODE

dx

dt+ Lx = f (x), x(0) = x0,

has the formal solution

x(t) = e−∫

t

0L dt′x0 +

∫ t

0

G(t − t′)f (x(t′)) dt′.

8

•Letting s = t − t′:

x(t) = e−∫

t

0L dt′x0 +

∫ t

0

G(s)f (x(t − s)) ds.

•Change integration variable to h = H(s) =∫ s

0G(s) ds:

x(t) = e−∫

t

0L dt′x0 +

∫ H(t)

1

f(x(t − H−1(h))

)dh.

•Rectangular rule ⇒ Predictor (Euler):

x(t) ≈ e−∫

t

0L dt′x0 + f (x(0))H(t).

•Trapezoidal rule ⇒ Corrector:

x(t) ≈ e−∫

t

0L dt′x0 +

f (x(0)) + f (x(t))

2H(t).

9

Other Generalizations

•Higher-order exponential integrators: Hochbruck et al. [1998],Cox & Matthews [2002], Hochbruck & Ostermann [2005a],Bowman et al. [2006].

•Vector case (matrix exponential P = eηt).

•Exponential versions of Conservative Integrators [Bowmanet al. 1997, Shadwick et al. 1999, Kotovych & Bowman 2002].

•Lagrangian discretizations of advection equations are alsoexponential integrators:

∂u

∂t+ v

∂

∂xu = f (x, t, u), u(x, 0) = u0(x).

• η now represents the linear operator v ∂∂x

and

P−1u = e−tv ∂

∂x u

corresponds to the Taylor series of u(x − vt).

10

Higher-Order Integrators

•General s-stage Runge–Kutta scheme:

xi = x0 + τ

i−1∑

j=0

aijf (xj, t + bjτ ), (i = 1, . . . , s).

•Butcher Tableau (s=4):

b0 a10

b1 a20 a21

b2 a30 a31 a32

b3 a40 a41 a42 a43

11

Bogacki–Shampine (3,2) Pair

•Embedded 4-stage pair [Bogacki & Shampine 1989]:

01

2

1

20

3

4

3

4

2

9

1

3

4

9

17

24

1

4

1

3

1

8

3rd order

2nd order

• Since f (x3) is just f at the initial x0 for the next time step, noadditional source evaluation is required to compute x4 [FSAL].

•Also: 6-stage (5,4) pair [Bogacki & Shampine 1996]. 12

Vector Case

•When x is a vector, ν is typically a matrix:

dx

dt+ νx = f (x).

•Let z = −ντ . Discretization involves

ϕ1(z) = z−1(ez − 1).

•Higher-order exponential integrators require

ϕj(z) = z−j

(ez −

j−1∑

k=0

zk

k!

).

•Exercise care when z has an eigenvalue near zero!

•Although a variable time step requires re-evaluation of thematrix exponential, this is not an issue for problems where theevaluation of the nonlinear term dominates the computation.

•Pseudospectral turbulence codes: diagonal matrix exponential.13

Charged Particle in Electromagnetic Fields

•Lorentz force:

m

q

dv

dt=

1

cv×B + E.

•Efficiently compute the matrix exponential exp(Ω), where

Ω = −q

mcτ

0 Bz −By

−Bz 0 Bx

By −Bx 0

.

•Requires 2 trigonometric functions, 1 division, 1 square root,and 35 additions or multiplications.

•The other necessary matrix factor, Ω−1[exp(Ω) − 1] requires

care, since Ω is singular. Evaluate it as

limλ→0

[(Ω + λ1)−1(eΩ − 1)].

14

Motion Under Lorentz Force

10

5

20

x

0

20

40

y

10

20

z

exactPCE-PC

15

An Embedded 4-Stage (3,2) Exponential Pair•Letting z = −ντ and b4 = 1:

xi = e−biντx0 + τ

i−1∑

j=0

aijf (xj, t + bjτ ), (i = 1, . . . , s).

a10=1

2ϕ1

(1

2z

),

a20=3

4ϕ1

(3

4z

)− a21, a21=

9

8ϕ2

(3

4z

)+

3

8ϕ2

(1

2z

),

a30=ϕ1(z) − a31 − a32, a31=1

3ϕ1(z), a32=

4

3ϕ2(z) −

2

9ϕ1(z),

a40=ϕ1(z) −17

12ϕ2(z), a41=

1

2ϕ2(z), a42=

2

3ϕ2(z), a43=

1

4ϕ2(z). (1)

•x3 has stiff order 3 [Hochbruck and Ostermann 2005] (order ispreserved even when ν is a general unbounded linear operator).

•x4 provides a second-order estimate for adjusting the time step.

•ν → 0: reduces to [3,2] Bogacki–Shampine Runge–Kutta pair. 16

Application to GOY Turbulence Shell Model

10−2110−2010−1910−1810−1710−1610−1510−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1100101

E(k

)

100 101 102 103 104 105 106 107 108 109 1010

k

17

Conclusions

•Exponential integrators are explicit schemes for ODEs with astiff linearity.

•When the nonlinear source is constant, the time-steppingalgorithm is precisely the analytical solution to thecorresponding first-order linear ODE.

•Unlike integrating factor methods, exponential integrators havethe correct fixed point behaviour.

•We present an efficient adaptive embedded 4-stage (3,2)exponential pair.

•A similar embedded 6-stage (5,4) exponential pair also exists.

•Care must be exercised when evaluating ϕj(x) near 0.Accurate optimized double precision routines for evaluatingthese functions are available atwww.math.ualberta.ca/∼bowman/phi.h

18

Asymptote: The Vector Graphics Language

symptotesymptotesymptotesymptotesymptotesymptotesymptotesymptotesymptotesymptotesymptote

http://asymptote.sf.net

(freely available under the GNU public license)

19

References

[Beylkin et al. 1998] G. Beylkin, J. M. Keiser, &L. Vozovoi, J. Comp. Phys.,147:362, 1998.

[Bogacki & Shampine 1989] P. Bogacki & L. F. Shampine, Appl.Math. Letters, 2:1, 1989.

[Bogacki & Shampine 1996] P. Bogacki & L. F. Shampine,Comput. Math. Appl., 32:15, 1996.

[Bowman et al. 1997] J. C. Bowman, B. A. Shadwick,& P. J. Morrison, “Exactlyconservative integrators,” in15th IMACS World Congress

on Scientific Computation,

Modelling and Applied

Mathematics, edited by A. Sydow,

volume 2, pp. 595–600, Berlin,1997, Wissenschaft & Technik.

[Bowman et al. 2006] J. C. Bowman, C. R. Doering,B. Eckhardt, J. Davoudi,M. Roberts, & J. Schumacher,Physica D, 218:1, 2006.

[Certaine 1960] J. Certaine, Math. Meth. Dig.Comp., p. 129, 1960.

[Cox & Matthews 2002] S. Cox & P. Matthews, J. Comp.Phys., 176:430, 2002.

[Friedli 1978] A. Friedli, Lecture Notes inMathematics, 631:214, 1978.

[Hochbruck & Ostermann 2005a] M. Hochbruck & A. Ostermann,SIAM J. Numer. Anal., 43:1069,2005.

[Hochbruck & Ostermann 2005b] M. Hochbruck & A. Ostermann,Appl. Numer. Math., 53:323, 2005.

[Hochbruck et al. 1998] M. Hochbruck, C. Lubich, &H. Selfhofer, SIAM J. Sci. Comput.,19:1552, 1998.

[Kotovych & Bowman 2002] O. Kotovych & J. C. Bowman, J.Phys. A.: Math. Gen., 35:7849,2002.

[Lu 2003] Y. Y. Lu, J. Comput. Appl. Math.,161:203, 2003.

[Nørsett 1969] S. Nørsett, Lecture Notes inMathematics, 109:214, 1969.

[Shadwick et al. 1999] B. A. Shadwick, J. C. Bowman,& P. J. Morrison, SIAM J. Appl.Math., 59:1112, 1999.

[van der Houwen 1977] P. J. van der Houwen,Construction of integration

formulas for initial value

problems, North-HollandPublishing Co., Amsterdam, 1977,North-Holland Series in AppliedMathematics and Mechanics, Vol.19.

[Verwer 1977] J. Verwer, Numer. Math., 27:143,1977.