On the implementation of singly implicit Runge-Kutta methods

Abstract

A modified Newton method is often used to solve the algebraic equations that arise in the application of implicit Runge-Kutta methods. When the Runge-Kutta method has a coefficient matrix A with a single point spectrum (with eigenvalue λ \lambda ), the efficiency of the modified Newton method is much improved by using a similarity transformation of A. Each iteration involves vector transformations. In this article an alternative iteration scheme is obtained which does not require vector transformations and which is simpler in other respects also. Both schemes converge in a finite number of iterations when applied to linear systems of differential equations, but the new scheme uses the nilpotency of A − λ I A - \lambda I to achieve this. Numerical results confirm the predicted convergence for nonlinear systems and indicate that the scheme may be a useful alternative to the modified Newton method for low-dimensional systems. The scheme seems to become less effective as the dimension increases. However, it has clear advantages for parallel computation, making it competitive for high-dimensional systems.