IJPAM: Volume 93, No. 4 (2014)

SOME EFFICIENT IMPLEMENTATION SCHEMES
FOR IMPLICIT RUNGE-KUTTA METHODS

R. Vigneswaran$^1$, S. Kajanthan$^2$
$^{1,2}$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 $x' = qx$ and the convergence rate depends on the spectral radius of the iteration matrix $M(z)$, a function of $z=hq$, where $h$ is the step-length. In this scheme, we require the spectral radius of $M(z)$ to be zero at $z=0$ and at $z=\infty$ in the $z$-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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DOI: 10.12732/ijpam.v93i4.4 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 93
Issue: 4
Pages: 525 - 540

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).