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.

翻译：本文讨论了如何为一般d(>=1)维对角线各向异性扩散方程构建高阶多松弛时间格子Boltzmann模型。该MRT-LB模型采用自然方式构建的变换矩阵和DdQ(2d^2+1)格子结构。构建高阶MRT-LB模型的关键步骤是确定可调松弛参数和权重系数，这些参数用于消除MRT-LB模型特定阶数的截断误差，同时保证模型的稳定性。本工作首先提出了针对对角线各向异性扩散方程的统一MRT-LB模型。随后，通过直接泰勒展开，我们分析了该模型直至四阶的宏观修正方程，并进一步推导出模型的四阶相容性条件。此外，我们还为当前LB方法构建了四阶初始化格式。之后，明确给出了保证MRT-LB模型满足稳定性结构的条件，并从数值角度证明：一旦满足该稳定性结构，MRT-LB模型必定是L^2稳定的。结合四阶相容性条件和L^2稳定性条件，MRT-LB模型的松弛参数和权重系数可通过简单的计算机代码自动确定。最后，我们对若干基准问题进行了数值模拟，发现数值结果能达到四阶收敛精度，这与理论分析一致。特别地，对于各向同性扩散方程，我们还比较了采用DdQ(2d^2+1)和DdQ(2d+1)格子结构的四阶MRT-LB模型，数值结果表明采用DdQ(2d^2+1)格子结构的MRT-LB模型具有更广泛的适用性。