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.

翻译：本文首先针对具有恒定扩散系数和源项的二维扩散方程，发展了一个通用的介观多重弛豫时间格子玻尔兹曼（MRT-LB）模型，其中考虑了D2Q5（二维空间中的五个离散速度）晶格结构。随后，我们精确推导了该MRT-LB模型的等价宏观有限差分格式。此外，我们还针对带有线性源项的扩散方程提出了一个适当的MRT-LB模型，并得到了一个等价的宏观六层有限差分格式。之后，我们对有限差分格式和介观MRT-LB模型进行了精度和稳定性分析。研究发现，在扩散尺度下，基于泰勒展开，两者均能在空间上达到四阶精度。稳定性分析还表明它们都是无条件稳定的。最后，进行了一些数值实验，数值结果与我们的理论分析一致。