Some Efficient Schemes with Improving Rate of Convergence for Two-stage Gauss Method
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  avoids expensive vector transformations and is computationally more
efficient. The rate of convergence of this scheme is examined in  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.