ON THE USE OF PARALLEL PROCESSORS FOR IMPLICIT RUNGE-KUTTA METHODS

An iteration scheme, for solving the non-linear equations arising in t he implementation of implicit Runge-Kutta methods, is proposed. This s cheme is particularly suitable for parallel computation and can be app lied to any method which has a coefficient matrix A with all eigenvalu es real (and positive). For such methods, the efficiency of a modified Newton scheme may often be improved by the use of a similarity transf ormation of A but, even when this is the case, the proposed scheme can have advantages for parallel computation. Numerical results illustrat e this. The new scheme converges in a finite number of iterations when applied to linear systems of differential equations, achieving this b y using the nilpotency of a strictly lower triangular matrix S-1 AS - LAMBDA, with LAMBDA a diagonal matrix. The scheme reduces to the modif ied Newton scheme when S-1 AS is diagonal. A convergence result is obt ained which is applicable to nonlinear stiff systems.