Abstract:
Several authors proposed variety of linear iteration schemes to solve non-linear equations
arising in the implementation of implicit Runge-Kutta methods. A linear scheme of this type
with some additional computation in each iteration step was proposed. The rate of
convergence of this scheme was examined when it is applied to the scalar test problem
𝑥′ = 𝑞𝑥 and the convergence rate depends on the spectral radius M(z) of the iteration
matrix M(z), a function of 𝑧 = ℎ𝑞, where ℎ is a step size. The spectral radius M(z) was
minimized over left-half of the complex plane for the case 𝑟 = 𝑠 + 1. Improved
convergence rates are obtained for the case 𝑟 = 2𝑠 for two, three and four stage Gauss
methods and numerical results are given.
