A general multiple-relaxation-time lattice Boltzmann model for nonlinear anisotropic convection-diffusion equations

TL;DR

This paper introduces a versatile multiple-relaxation-time lattice Boltzmann model for nonlinear anisotropic convection-diffusion equations, demonstrating improved accuracy and stability over traditional models through theoretical analysis and numerical tests.

Contribution

It develops a general MRT lattice Boltzmann model for NACDEs, showing enhanced stability and accuracy, and addresses boundary slip issues present in BGK models.

Findings

MRT model accurately recovers NACDEs via Chapman-Enskog analysis.

Numerical results align well with analytical solutions.

MRT model exhibits better stability and reduced boundary slip than BGK model.

Abstract

In this paper, based on the previous work [B. Shi, Z. Guo, Lattice Boltzmann model for nonlinear convection-diffusion equations, Phys. Rev. E 79 (2009) 016701], we develop a general multiple-relaxation-time (MRT) lattice Boltzmann model for nonlinear anisotropic convection-diffusion equation (NACDE), and show that the NACDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the MRT model through some classic CDEs, and find that the numerical results are in good agreement with analytical solutions or some available results. Besides, the numerical results also show that similar to the single-relaxation-time (SRT) lattice Boltzmann model or so-called BGK model, the present MRT model also has a second-order convergence rate in space. Finally, we also perform a comparative study on the accuracy and stability of the MRT model and BGK model by using two examples. In terms of the accuracy, both the theoretical analysis and numerical results show that a \emph{numerical} slip on the boundary would be caused in the BGK model, and cannot be eliminated unless the relaxation parameter is fixed to be a special value, while the \emph{numerical} slip in the MRT model can be overcome once the relaxation parameters satisfy some constrains. The results in terms of stability also demonstrate that the MRT model could be more stable than the BGK model through tuning the free relaxation parameters.