Optimal convergence rate of the explicit Euler method for convection–diffusion equations II: High dimensional cases

Abstract

Abstract This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection–diffusion equations [Zhang et al. Appl. Math. Lett. 131 (2022), 108048] which focuses on high‐dimensional linear/nonlinear cases under Dirichlet/Neumann boundary conditions. Several new difference schemes are proposed based on the explicit Euler discretization in temporal derivative and centered difference discretization in spatial derivatives. The priori estimate of the improved difference scheme with application to the constant convection coefficients is performed under the maximum norm and the optimal convergence rate four is achieved when the step‐ratios along each direction equal to . Also we give partial results for the three‐dimensional case. The improved difference schemes have essentially improved the CFL condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two‐/three‐dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee–Infante equation and the Burgers' equation substantiate the good properties claimed for the improved difference scheme.