Higher-Order Linearized Multistep Finite Difference Methods for Non-Fickian Delay Reaction-Diffusion Equations

Abstract

In this paper, two types of higher-order linearized multistep finite difference schemes are proposed to solve non-Fickian delay reaction-diffusion equations. For the first scheme, the equations are discretized based on the backward differentiation formulas in time and compact finite difference approximations in space. The global convergence of the scheme is proved rigorously with convergence order $\mathcal{O}(\tau^2 + h^4)$ in the maximum norm. Next, a linearized noncompact multistep finite difference scheme is presented and the corresponding error estimate is established. Finally, extensive numerical examples are carried out to demonstrate the accuracy and efficiency of the schemes, and some comparisons with the implicit Euler scheme in the literature are presented to show the effectiveness of our schemes.