R. Vigneswaran, S. Kajanthan
Department of Mathematics and Statistics
Faculty of Science
University of Jaffna
SRI LANKA

Abstract. Several iteration schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. As an alternative to the modified Newton scheme, some iteration schemes with reduced linear algebra costs have been proposed A scheme of this type proposed in [9] avoids expensive vector transformations and is computationally more efficient. The rate of convergence of this scheme is examined in [9] 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, we require the spectral radius of to be zero at and at in the -plane in order to improve the rate of convergence of the scheme. New schemes with parameters are obtained for three-stage and four-stage Gauss methods. Numerical experiments are carried out to confirm the results obtained here.

Received: January 3, 2014

AMS Subject Classification: 65L04, 65L05

Key Words and Phrases: implementation, Gauss methods, rate of convergence, stiff systems

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