IJPAM: Volume 27, No. 1 (2006)

STABILIZED LINEARLY IMPLICIT SIMPSON-TYPE
SCHEMES FOR NONLINEAR DIFFERENTIAL EQUATIONS

M.M. Chawla
Department of Mathematics and Computer Science
Kuwait University
P.O. Box 5969, Safat, 13060, KUWAIT
e-mail: chawla@mcs.sci.kuniv.edu.kw


Abstract.The classical Simpson rule is an optimal fourth order two-step integration scheme for first-order initial-value problems; however, it is unconditionally unstable. An A-stabilized version of Simpson rule was given by Chawla et al [3] and an L-stable version was given by Chawla et al [2]. These rules are functionally implicit, and when applied for the time integration of nonlinear differential equations, require an iterative method such as Newton's method for the solution of resulting nonlinear systems at each time step of integration. In the present paper, we present a new class of linearly implicit Simpson-type rules which are A- and L-stable. For the time integration of nonlinear differential equations, our linearly implicit schemes obviate the need to solve a nonlinear system at each time step of integration. The obtained schemes are computationally illustrated for stability and accuracy by considering a nonlinear initial value problem in ODEs and the diffusion equation with a nonlinear reaction term.

Received: February 18, 2006

AMS Subject Classification: 65L05, 65M06

Key Words and Phrases: first order initial-value problem, Simpson rule, linearly implicit Simpson rules, A- and L-stability, unconditional stability, nonlinear reaction-diffusion

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 27
Issue: 1