Bit ? ? ( 199 ? ) , ? ? ? { ? ? ? . Approximating Runge { Kutta Matricesby Triangular

The implementation of implicit Runge{Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modiied Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers 5] substitute the Runge{Kutta matrix A in the Newton process for a triangular matrix T that approximates A, hereby making the method suitable for parallel implementation. The matrix T is constructed according to a simple procedure, such that the stii error components in the numerical solution are strongly damped. In this paper we proof for a large class of Runge{Kutta methods that this procedure can be carried out and that the diagonal entries of T are positive. This means that the linear systems that are to be solved have a non-singular matrix.