  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.
