Strang splitting structure-preserving high-order compact difference schemes for nonlinear convection diffusion equations

TL;DR

This paper introduces high-order, efficient compact difference schemes for nonlinear convection diffusion equations that preserve bounds and mass, with proven accuracy and demonstrated effectiveness through numerical examples.

Contribution

It develops a fourth-order accurate Strang splitting scheme with bound and mass preservation, incorporating Lagrange multipliers and ADI methods for 2D problems.

Findings

Schemes achieve high-order accuracy in space and second-order in time.

Numerical examples confirm the schemes' effectiveness in preserving bounds and mass.

Error estimates validate the theoretical accuracy of the methods.

Abstract

In this paper, we present a class of high-order and efficient compact difference schemes for nonlinear convection diffusion equations, which can preserve both bounds and mass. For the one-dimensional problem, we first introduce a high-order compact Strang splitting scheme (denoted as HOC-Splitting), which is fourth-order accurate in space and second-order accurate in time. Then, by incorporating the Lagrange multiplier approach with the HOC-Splitting scheme, we construct two additional bound-preserving or/and mass-conservative HOC-Splitting schemes that do not require excessive computational cost and can automatically ensure the uniform bounds of the numerical solution. These schemes combined with an alternating direction implicit (ADI) method are generalized to the two-dimensional problem, which further enhance the computational efficiency for large-scale modeling and simulation. Besides, we present an optimal-order error estimate for the bound-preserving ADI scheme in the discrete $L_2$ norm. Finally, ample numerical examples are presented to verify the theoretical results and demonstrate the accuracy, efficiency, and effectiveness in preserving bounds or/and mass of the proposed schemes.