Extended multi-step high-order numerical methods for the nonlinear convection-diffusion-reaction equation with vanishing delay

Abstract

In this paper, we propose two multi-step, linearized numerical schemes for a nonlinear convection-diffusion-reaction (CDR) equation with vanishing delay, a temporally nonlocal partial differential equation. These semi-implicit numerical schemes use a combination of explicit Adams–Bashforth extrapolation for the nonlinear term and implicit Adams–Moulton interpolation for the diffusion term. A long stencil finite difference approximation is employed for the spatial discretization, and a boundary extrapolation is used to prescribe the solution at “ghost” points lying outside of the computational domain. The numerical stability and convergence analysis is provided, and the discrete ℓ² convergence estimate is obtained, with fourth-order spatial accuracy and high-order (third- or fourth-order) temporal accuracy. A few numerical experiments are also presented to confirm the theoretical results.