Improving Rates of Convergence of Iterative Schemes for Implicit Runge‐Kutta Methods

Abstract

AbstractVarious iterative schemes have been proposed to solve the non‐linear equations arising in the implementation of implicit Runge‐Kutta methods. In one scheme, when applied to an s‐stage Runge‐Kutta method, each step of the iteration still requires s function evaluations but consists of r(>s) sub‐steps. Improved convergence rate was obtained for the case r = s + 1 only. This scheme is investigated here for the case r = ks, k = 2, 3, …, and superlinear convergence is obtained in the limit k→∞. Some results are obtained for Gauss methods and numerical results are given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)