Linear θ-Method and Compact θ-Method for Generalised Reaction-Diffusion Equation with Delay

Abstract

This paper is concerned with the analysis of the linear $\theta$-method and compact $\theta$-method for solving delay reaction-diffusion equation. Solvability, consistence, stability, and convergence of the two methods are studied. When $\theta \in [\mathrm{0},\mathrm{1}/\mathrm{2})$, sufficient and necessary conditions are given to show that the two methods are asymptotically stable. When $\theta \in [\mathrm{1}/\mathrm{2},\mathrm{1}]$, the two methods are proven to be unconditionally asymptotically stable. Finally, several examples are carried out to confirm the theoretical results.