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 [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 stiff 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. AMS subject classification: Primary: 65L06, Secondary: 15A23. Key words: Numerical analysis, Runge--Kutta methods,..