An automatic approach to develop the fourth-order and L^2-stable lattice Boltzmann model for diagonal-anisotropic diffusion equations

TL;DR

The paper develops a unified, high-order MRT-LB model for the $d$-dimensional diagonal-anisotropic diffusion equation, using a natural transformation matrix and the $\text{DdQ}(2d^2+1)$ lattice to achieve fourth-order accuracy. By performing a direct Taylor expansion, it derives fourth-order consistency conditions and a fourth-order initialization scheme, and it establishes explicit stability structure conditions to guarantee $L^2$ stability. An automatic procedure, implemented in Matlab, determines the relaxation parameters and weight coefficients by solving the coupled fourth-order and stability equations for each direction pair, enabling practical deployment in higher dimensions. Numerical benchmarks, including Gauss Hill problems and diffusion with linear sources, confirm fourth-order convergence and illustrate the generality and stability advantages over alternative lattices, particularly highlighting the broader applicability of the $\text{DdQ}(2d^2+1)$ lattice in anisotropic diffusion settings.

Abstract

This paper discusses how to develop a high-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the general d(>=1)-dimensional diagonal-anisotropic diffusion equation. Such an MRT-LB model considers the transformation matrix constructed in a natural way and the DdQ(2d^2+1) lattice structure. A key step in developing the high-order MRT-LB model is to determine the adjustable relaxation parameters and weight coefficients, which are used to eliminate the truncation errors at certain orders of the MRT-LB model, while ensuring the stability of the MRT-LB model. In this work, we first present a unified MRT-LB model for the diagonal-anisotropic diffusion equation. Then, through the direct Taylor expansion, we analyze the macroscopic modified equations of the MRT-LB model up to fourth-order, and further derive the fourth-order consistent conditions of the MRT-LB model. Additionally, we also construct the fourth-order initialization scheme for the present LB method. After that, the condition which guarantees that the MRT-LB model can satisfy the stability structure is explicitly given, and from a numerical perspective, once the stability structure is satisfied, the MRT-LB model must be L^2 stable. In combination with the fourth-order consistent and L^2 stability conditions, the relaxation parameters and weight coefficients of the MRT-LB model can be automatically given by a simple computer code. Finally, we perform numerical simulations of several benchmark problems, and find that the numerical results can achieve a fourth-order convergence rate, which is in agreement with our theoretical analysis. In particular, for the isotropic diffusion equation, we also make a comparison between the fourth-order MRT-LB models with the DdQ(2d^2+1) and DdQ(2d+1) lattice structures, and the numerical results show that the MRT-LB model with the DdQ(2d^2+1) lattice structure is more general.

Figures (6)

• Figure 1: $L^2$ stability regions of the MRT-LB model for the two-dimensional (top) and three-dimensional (bottom) cases under different values of the weight coefficient $\tilde{\omega}$.
• Figure 2: Regions of the parameter pair $(\tilde{\epsilon}_x,\tilde{\epsilon}_y)$ represent that (\ref{['fourth-conition']}) and (\ref{['condition-stability']}) can be solved under different values of the weight coefficient $\tilde{\omega}$.
• Figure 3: Contour lines of the numerical and analytical solutions under different values of the parameter pair $(\epsilon_{x_1},{\epsilon}_{x_2})$.
• Figure 4: Convergence rates of the MRT-LB model with different initialization schemes at the final time $t=2$.
• Figure 5: Profiles of the numerical and analytical solutions under different values of the final time $t$.