Asymptotic stability of numerical methods for linear delay parabolic differential equations

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.