Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

Abstract

Several iteration schemes have been proposed to solve the nonlinear equations arising in the implementation of implicit Runge-Kutta methods. As an alternative to the modified Newton scheme, some iteration schemes with reduced linear algebra costs have been proposed A scheme of this type proposed in [9] avoids expensive vector transformations and is computationally more efficient. The rate of convergence of this scheme is examined in [9] when it is applied to the scalar test differential equation x ′ = qx and the convergence rate depends on the spectral radius of the iteration matrix M (z), a function of z = hq, where h is the step-length. In this scheme, we require the spectral radius of M (z) to be zero at z = 0 and at z = ∞ in the z-plane in order to improve the rate of convergence of the scheme. New schemes with parameters are obtained for three-stage and four-stage Gauss methods. Numerical experiments are carried out to confirm the results obtained here.

Improving Rate of Convergence of an Iterative Scheme With Extra Sub-Steps for Two Stage Gauss Method

“…Cooper and Vignesvaran [9] developed an alternative scheme which is computationally more efficient. Improved rates of convergence of this scheme were obtained in [22], [23], [24]. Another scheme was proposed by Cooper and Vignesvaran [10] in order to obtain improved rate of convergence, by adding extra sub-steps.…”