Improving rate of convergence of an iterative scheme with extra sub-steps for two stage gauss method.

Abstract

The non-linear equations, when implementing implicit Runge-Kutta methods, may be solved by a modified Newton scheme and by several linear iteration schemes which sacrificed superlinear convergence for reduced linear algebra costs. A linear scheme of this type was proposed, which requires some additional computation in each iteration step. The rate of convergence of this scheme is examined when it is applied to the scalar test problem x ′ = qx and the convergence rate depends on the spectral radius ρ[M(z)] of the iteration matrix M, a function of z = hq, where h is a step size. The supremum of the lower bound for ρ[M(z)] is minimized over left-half plane of the z-complex plane and over the negative real axis of the z-plane in order to improve the rate of convergence of that scheme. Two new schemes are obtained for the two stage Gauss method and numerical results are given.