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The Allen-Cahn equation (ACE) inherently possesses two crucial properties:the maximum principle and the energy dissipation law. Preserving these twoproperties at the discrete level is also necessary in the numerical methods forthe ACE. In this paper, unlike the traditional top-down macroscopic numericalschemes which discretize the ACE directly, we first propose a novel bottom-upmesoscopic regularized lattice Boltzmann method based macroscopic numericalscheme for d (=1, 2, 3)-dimensional ACE, where the DdQ(2d+1) [(2d+1) discretevelocities in d-dimensional space] lattice structure is adopted. In particular,the proposed macroscopic numerical scheme has a second-order accuracy in space,and can also be viewd as an implicit-explicit finite-difference scheme for theACE, in which the nonlinear term is discretized semi-implicitly, the temporalderivative and dissipation term of the ACE are discretized by using theexplicit Euler method and second-order central difference method, respectively.Then we also demonstrate that the proposed scheme can preserve the maximumbound principle and the original energy dissipation law at the discrete levelunder some conditions. Finally, some numerical experiments are conducted tovalidate our theoretical analysis.

Regularized lattice Boltzmann method based maximum principle and energy stability preserving finite-difference scheme for the Allen-Cahn equation