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

A Class of S-Step Non-Linear Iteration Scheme Based on Projection Method for Gauss Method

“…Cooper and Vigneswaran [6] proposed another scheme, which is a generalization of the basic scheme (5), to obtain improved rate of convergence, by adding extra sub-steps. Further improvement in the rate of convergence of this scheme has been obtained in [7].…”

“…DY  Here G and m E have to be chosen so that the scheme can be efficiently implemented and performs well. In each step of the iteration (6) the scalar m  has to be calculated by using (7)…”

“…is the block diagonal matrix and each diagonal block is the Jacobian off at one of Hence from (7), we obtain…”