THE SOLUTION OF IMPLICIT DIFFERENTIAL-EQUATIONS ON PARALLEL COMPUTERS

We construct and analyze parallel iterative solvers for the solution o f the linear systems arising in the application of Newton's method to s-stage implicit Runge-Kutta (RK) type discretizations of implicit dif ferential equations (IDEs). These linear solvers are partly iterative and partly direct. Each linear system iteration again requires the sol ution of linear subsystems, but now only of IDE dimension, which is s times less than the dimension of the linear system in Newton's method. Thus, the effective costs on a parallel computer system are only one LU-decomposition of IDE dimension for each Jacobian update, yielding a considerable reduction of the effective LU-costs. The method paramete rs can be chosen such that only a few iterations by the linear solver are needed. The algorithmic properties are illustrated by solving the transistor problem (index 1) and the car axis problem (index 3) taken from the CWI test set. (C) 1997 Elsevier Science B.V.