Asymptotic stability of numerical methods for linear delay parabolic differential equations

Authors:
Hongjiong Tian
Affiliations:
Department of Mathematics, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, PR China and Division of Computational Science, E-Institute of Shanghai Universities, 100 Guilin Road, Shan ...
Venue:
Computers & Mathematics with Applications
Year:
2008

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Abstract

This paper is concerned with the asymptotic stability property of some numerical processes by discretization of parabolic differential equations with a constant delay. These numerical processes include forward and backward Euler difference schemes and Crank-Nicolson difference scheme which are obtained by applying step-by-step methods to the resulting systems of delay differential equations. Sufficient and necessary conditions for these difference schemes to be delay-independently asymptotically stable are established. It reveals that an additional restriction on time and spatial stepsizes of the forward Euler difference scheme is required to preserve the delay-independent asymptotic stability due to the existence of the delay term. Numerical experiments have been implemented to confirm the asymptotic stability of these numerical methods.