SOME EFFICIENT IMPLEMENTATION SCHEMES FOR IMPLICIT RUNGE-KUTTA METHODS
S. Kajanthan · International Journal of Pure and Apllied Mathematics · 2014
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.