A general fourth-order mesoscopic multiple-relaxation-time lattice Boltzmann model and equivalent macroscopic finite-difference scheme for two-dimensional diffusion equations

TL;DR

This paper develops a general mesoscopic MRT-LB model for 2D diffusion equations, derives equivalent finite-difference schemes, and demonstrates fourth-order accuracy and unconditional stability through analysis and numerical experiments.

Contribution

It introduces a comprehensive MRT-LB model for diffusion equations and establishes its equivalence to high-order finite-difference schemes with proven stability.

Findings

Both models achieve fourth-order spatial accuracy.

Models are unconditionally stable.

Numerical results confirm theoretical predictions.

Abstract

In this work, we first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the equivalent macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain an equivalent macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis of the finite-difference scheme and the mesoscopic MRT-LB model. It is found that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis.