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

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.