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.