Linear stability analysis of more general linear iteration scheme in the implementation of implicit runge kutta methods

Abstract

more general linear iterative scheme solves non-linear equations arising in the implementation of implicit Runge-Kutta methods proposed by Cooper and Butcher is of the form

where B and S are real non-singular matrices and L is strictly lower triangular matrix of order s, and is a real constant. They showed that successive over relaxation technique applied to improve the convergence rate of this scheme. Later, convergence result of this scheme established by proving some theoretical results suitable for stiff problems. This article examines stability properties of this linear iterative scheme with the alternate approximation It is better to use because it requires less evaluation of f and is more accurate for stiff problem. For a fixed starting value 0 Y