A Parallel Matrix-free Implementation of a Runge-kutta Code

It is known that matrix-free numerical implementations for solving stii ordinary diierential equations (ODEs) can be considerably more eeective than implementations which rely on direct linear algebra techniques to solve the implicit equations governing the stage values. In this paper it will be shown how fully implicit, high order Runge-Kutta methods can be eeciently implemented in a matrix-free, parallel environment. The advantage of this is that no new parallel algorithms need be developed and that existing sequential methods that are adpated using these techniques need have no special structure (such as singly implicitness). This is demonstrated by the conversion of an existing Radau IIA method (RADAU5) to a matrix-free implementation using a dynamically pre-conditioned GMRES algorithm to solve the appropriate linear systems. Numerical results are presented for an implementation on a shared memory SGI Power Challenge and show the eecacy of this approach.