Efficient Integration of Large Stiff Systems of ODEs Using Exponential Integrators

1. Efficient Integration of Large Stiff Systems of ODEs Using Exponential Integrators M. Tokman,University of California, Merced 2 hrs 1.5 hrs

2. Motivation • Constructing Exponential Integrators • Numerical Examples Outline:

3. Astrophysical and Laboratory Plasmas:

4. Resistive MHD equations: 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.

5. Since these MHD equations discretized in space yield 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 method which • allows for large time steps • minimizes number of computations per time step • allows automatic time step control

6. Integrator must be competitive with explicit and implicit methods: Implicit schemes Need to solve nonlinear system using Newton iteration: with Jacobian 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)!

7. 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|| Implicit SchemesExponential Integrators f(x)=1/(1-x) f(x) = exp(x) or functions of exp(x) no control over vector bcan be designed with ||b|| small

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

9. Exponential Propagation Iterative (EPI) Methods:

10. Test convergence of Arnoldi iteration: Brusselator example: Jacobian matrix:

11. Comparison of Krylov Approximations toand : 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.

12. Methods to compare: å

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

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

15. 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