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.