Efficient Integration of Large Stiff Systems of ODEs Using Exponential

IntegratorsM. Tokman,M. Tokman, University of California,

Merced

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2 hrs 1.5 hrs

Outline:Outline:

Motivation

Constructing Exponential Integrators

Numerical Examples

Astrophysical and Laboratory Plasmas:Astrophysical and Laboratory Plasmas:

Resistive MHD equations:Resistive MHD equations:

Momentum Conservation:

Vt - (V)V (B)B

1

R2V

Mass Conservation :

t (V)

Induction Equation :

Bt (VB) 1

S2B

Large scale evolution of plasma configurations can be described by equations of magnetohydrodynamics (MHD). This system is difficult to integrate numerically due to inherent 3-D nature of the problem and presence of widely varying time and spatial scales.

Alfven Speed VA B0

Lundquist Number S l0

2/l0 /VA

S ~ 102 104 (experiments)

~ 1015 (corona)

Reynolds Number Rl0

2/l0 /VA

Since these MHD equations discretized in space Since these MHD equations discretized in space yield a system which isyield a system which is

stiff

large (typical run =1.6Million unknowns)

difficult to construct efficient preconditioners for,

to integrate it in time we prefer a numerical to integrate it in time we prefer a numerical method whichmethod which

allows for large time steps

minimizes number of computations per time step

allows automatic time step control

00

N

U)U(t

UF(U),td

Ud

R

Implicit schemes

Need to solve nonlinear system

using Newton iteration:

with Jacobian

Integrator must be competitive with explicit and Integrator must be competitive with explicit and implicitimplicit methods: methods:

U(t n t)Un F(U(s))dst n

t n t

Un +1 Un a i

in k

n

tF(U i)tF(Un1)

G(Un +1)Un +1 -tF(Un1) Zn 0

Wm +1 = Wm - G (Wm ) 1G(Wm )

G (U)I tDF

DU(U)

Each Newton iteration a product of the inverse of the Jacobian and a vector has to be approximated, i.e. need f(A)b where f(x)=1/(1-x)!

Implicit Schemes Exponential Integrators

Implicit vs. Exponential Integrators:Implicit vs. Exponential Integrators:

For large stiff systems both can use Krylov projections to compute f(A)b. The number of Krylov vectors needed to approximate f(A)b accurately depends on

(i) function f(x)

(ii) norm ||b||

f(x)=1/(1-x) f(x) = exp(x) or functions of exp(x)

no control over vector b can be designed with ||b|| small

Exponential integrator can be constructed in many ways, e.g. Exponential Propagation Iterative Methods(EPI) (Tokman’06):

dU

dtF(U) = F(Un ) +

DF

DU(Un )(U - Un ) + R(U), URN

U(t n )Un

Integral form of the solution

Develop quadrature formula to approximate the nonlinear integral and use Krylov subspace projections (Arnoldi iterations) to estimate products of a matrix functions and vectors. GOAL: construct quadrature such that (i) the Arnoldi iterations converge fast and (ii) an adaptive time stepping scheme can be obtained.

U(t n + t) = Un +exp(tAn ) I

tAn

tF(Un ) + exp(An (t n t t))t n

t n t

R(U(t))dt

Exponential Propagation Iterative (EPI) Methods:Exponential Propagation Iterative (EPI) Methods:

These functions arise when we interpolate R(U(t))

and approximate nonlinear integral with quadrature rule, i.e. on nodes

t n +t

, t n +2t

, ... , t n +( -1)t

R(U(t)) = R(U(t n st))Rn sk

k1

1

kRn

Consider functions

k (z) ez(1 s) sk

0

1

ds or gk (z) ez(1 s) sk

0

1

ds

exp(An (t n t t))t n

t n t

R(U(t))dt

U(t n + t) = Un +exp(tAn ) I

tAn

tF(Un ) + exp(An (t n t t))t n

t n t

R(U(t))dt

Brusselator example:

Test convergence of Arnoldi iteration:Test convergence of Arnoldi iteration:

Jdiag(2uiv i 4) diag(ui

2 )

diag(3 2u iv i ) diag( u i2 )

(x)2

L 0

0 L

L

-2 1 0

1 1

0 1 -2

Jacobian matrix:Ni

x

uuu vu-u

dt

dv

x

uuu4u - vu

dt

du

iiii

2ii

i

iiiiii

i

,..,

)(

)(

1

23

21

211

2112

Comparison of Krylov Approximations toComparison of Krylov Approximations to andand ::

Problem size, 2N

GMRES FOM

200 92 85 35 27 19

400 187 174 70 55 38

800 382 359 140 112 78

b)tA( 20 b)tA( 21be tA

510Tolerance

k (At)b

(I At) 1b

The 2-norm of the approximation error is also smaller for the functions . Similar result also holds for Jacobian calculated at different times and for other examples.

20,21

(I tA) 1bbtAI 1 )(

åå

Methods to compare:Methods to compare:

AB2 Un +1 Un +t

2(3Fn - Fn-1)

AM2 Un +1 Un +t

2(Fn + Fn +1)

AM2IN Inexact Newton iteration

EPI2 Un +1 Un + g0(Ant)tFn, g0(z)(ez 1) /z

Solve G(x)0 with xk +1 xk sk

where

G'(xk )sk = -G(xk ) +rk ,

G(xk ) +G'(xk )sk k G(xk )

k G(xk ) - G(xk -1) - G'(xk -1)sk -1

G(xk -1)

EPIRK3 r1 Un + 220(tA n

2)t

2F(Un )

U n +1 Un + 20(tA n )tF(Un ) +1

321(tAn )tR(r1)

where 20(z)ez 1

z, 21(z)2

ez (1 z)

z2

EPI3 Un +1 Un + g0(Ant)tFn +2

3g1(Ant)tRn-1, g1(z)

ez (1 z)

z2

Comparison of integration times for Brusselator Comparison of integration times for Brusselator example over time interval [0,1]:example over time interval [0,1]:

EPI methods of order 3 and 4 (can be embedded):EPI methods of order 3 and 4 (can be embedded):

r1 Un + a1130(tAn

3)t

3F(Un )

r2 Un + a2130(2tAn

3)2t

3F(Un ) + a2231(

2tAn

3)2t

3R(r1)

Un +1 Un + b130(tAn )tF(Un ) + b231(tAn )tR(r1)

+ b332(tAn )t -2R(r1) + R(r2)

where 30(z)ez 1

z

31(z)3ez (1 z)

z2

32(z)3

2

ez(6 z) (6 5z 2z2)

z3

9/4 9/4

9/8 9/8

160/243 32/81

128/243 0

a11

a21

b1

b2

Conclusions & Future Work:Conclusions & Future Work: EPI methods provide an efficient alternative to standard explicit and implicit schemes for integrating large stiff systems of ODEs

Design exponential integrators which take advantage of approximating “optimal” products f(A)v

Parallel implementation and testing of EPI methods as part of such frameworks as SUNDIALS (LLNL)

Non-uniform grids

Further study of 3D MHD models and other applications