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In this work, we first develop a general mesoscopic multiple-relaxation-timelattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equationwith the constant diffusion coefficient and source term, where the D2Q5 (fivediscrete velocities in two-dimensional space) lattice structure is considered.Then we exactly derive the equivalent macroscopic finite-difference scheme ofthe MRT-LB model. Additionally, we also propose a proper MRT-LB model for thediffusion equation with a linear source term, and obtain an equivalentmacroscopic six-level finite-difference scheme. After that, we conduct theaccuracy and stability analysis of the finite-difference scheme and themesoscopic MRT-LB model. It is found that at the diffusive scaling, both ofthem can achieve a fourth-order accuracy in space based on the Taylorexpansion. The stability analysis also shows that they are both unconditionallystable. Finally, some numerical experiments are conducted, and the numericalresults are also consistent with our theoretical analysis.

A general fourth-order mesoscopic multiple-relaxation-time lattice  Boltzmann model and equivalent macroscopic finite-difference scheme for  two-dimensional diffusion equations