Analysis of the Convergence of More General Linear Iteration Scheme on the Implementation of Implicit Runge-Kutta Methods to Stiff Differential Equations

Abstract

A modified Newton scheme is typically used to solve large sets of non-linear equations arising in the implementation of implicit Runge-Kutta methods. As an alternative to this scheme, iteration schemes, which sacrifice superlinear convergence for reduced linear algebra costs, have been proposed. A more general linear iterative scheme of this type proposed by Cooper and Butcher in 1983 for implicit Runge-Kutta methods, and he has applied the successive over relaxation technique to improve the convergence rate. In this paper, we establish the convergence result of this scheme by proving some theoretical results suitable for stiff problems. Also these convergence results are verified by two and three stage Gauss method and Radue IIA method.