Approximating Runge-Kutta matrices by triangular matrices

The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [HSw95] 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 stiff error components in the numerical solution are strongly damped. In this paper we prove 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.