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

Various 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)