Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Saturday, January 25th, 1997

Discretizations of multi-scale problems in ODEs

The talk will be about numerical methods for differential equations with different time scales which integrate the equations directly (without analytical preparation).

(a) Stiff integrators have been extensively studied in the past decades for the solution of problems where fast variables are damped out quickly, e.g., parabolic and singularly perturbed problems [1].

(b) Multirate methods are concerned with problems where fast and slow motions are present throughout the integration interval, and a small number (or inexpensive portion) of fast components restricts the step size of standard integrators. The talk presents recent approaches to adaptive inactivation of scales of slow variables [2],[3].

(c) For highly oscillatory problems like Schroedinger or wave equations, where oscillations are generated by the linear part of the equation, we have recently made good experiences with novel integration methods that use the exponential of the Jacobian, with Krylov subspace approxi- mation to the product of the matrix exponential with a vector [4].

1. HAIRER, E., WANNER, G.: Solving Ordinary Differential Equations II, Springer, 1991 (2nd rev. ed. 1996).
2. ENGSTLER, CH., LUBICH, CH.: Multirate extrapolation for differential equations with different time scales, SFB 382 Preprint Nr. 29, Univ. Tübingen, 1995.
3. ENGSTLER, CH., LUBICH, CH.: MUR8: a multirate extension of the eighth-order Dormand-Prince method, SFB 382 Preprint Nr. 51, Univ. Tübingen, 1996.
4. HOCHBRUCK, M., LUBICH, CH., SELHOFER, H.: Exponential integrators for large systems of differential equations, SFB 382 Preprint Nr. 31, Univ. Tübingen, 1995.