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

TL;DR

This paper develops corrected explicit Euler schemes for high-dimensional convection-diffusion equations, achieving optimal convergence rates and improved stability and accuracy over classical methods, validated through numerical experiments.

Contribution

It introduces new corrected difference schemes for high-dimensional convection-diffusion equations that enhance stability and accuracy, with proven optimal convergence rates.

Findings

Corrected schemes improve CFL conditions.

Optimal convergence rate of four is achieved.

Numerical examples confirm enhanced accuracy.

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 [Appl. Math. Lett. \textbf{131} (2022) 108048] which focuses on high-dimensional linear/nonlinear cases under Dirichlet or Neumann boundary conditions. Several new corrected difference schemes are proposed based on the explicit Euler discretization in temporal derivative and central difference discretization in spatial derivatives. The priori estimate of the corrected scheme with application to constant convection coefficients is provided at length by the maximum principle and the optimal convergence rate four is proved when the step ratios along each direction equal to $1/6$. The corrected difference schemes have essentially improved {\rm \textbf{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, the Burgers' equation and classification to name a few substantiate the good properties claimed for the corrected difference scheme.