Towards efficient implementation of singly-implicit methods

Abstract

It has been observed that for problems of low dimension the transformations used in the implementation of singly-implicit Runge-Kutta methods consume an unreasonable share of the total computational costs. Two proposals for reducing these costs are presented here. The first makes use of an alternative transformation for which the combined operation counts of the transformations together with the iterations themselves are lower than for the standard implementation scheme for singly-implicit methods. The second proposal is to use a Runge-Kutta method for which the first row of the coefficient matrix is zero but which still possesses acceptable stability properties. It is hoped that by combining these two proposals increased efficiency in the implementation of Runge-Kutta methods for stiff problems can be achieved.