Some Efficient Schemes with Improving Rate of Convergence for Two-stage Gauss Method

Abstract

Various iteration schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. A modified Newton scheme is typically used to solve these equations. As an alternative to this scheme, iteration schemes, which sacrifice super-linear convergence for reduced linear algebra costs, have been proposed. A scheme of this type proposed in [11] avoids expensive vector transformations and is computationally more efficient. The rate of convergence of this scheme is examined in [11] when it is applied to the scalar test differential equation 􀢞′ 􀵌 􀢗􀢞 and the convergence rate depends on the spectral radius of the iteration matrix 􀡹􁈺􀢠􁈻, a function of 􀢠 􀵌 􀢎􀢗, where 􀢎 is the step-length. In this scheme, some conditions are imposed on the spectral radius of its iteration matrix in order to get super-linear convergence at two points in the 􀢠 -plane. Then the supremum of the spectral radius of each scheme is minimized over the left-half 􀢠 -plane in order to improve the convergence rate of the scheme. Two new schemes with optimal parameters are obtained for the two-stage Gauss method and some numerical experiments are reported.