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

Abstract

The Allen-Cahn equation (ACE) inherently possesses two crucial properties: the maximum principle and the energy dissipation law. How to preserve these two properties at the discrete level is of significant importance in the numerical methods for the ACE. In this paper, unlike the traditional macroscopic numerical schemes that directly discretize the ACE, we first propose a novel mesoscopic regularized lattice Boltzmann method based macroscopic numerical scheme for the d (=1, 2, 3)-dimensional ACE, where the DdQ$(2d+1)$ [($2d+1$) discrete velocities in d-dimensional space] lattice structure is employed. In particular, the proposed numerical scheme has a second-order accuracy in space, and can also be regarded as an implicit-explicit finite-difference scheme for the ACE. In this scheme, the nonlinear term is discretized semi-implicitly, the temporal derivative term and the dissipation term are discretized via the explicit Euler and second-order central difference methods, respectively. Compared to the implicit schemes for the ACE, the present scheme proves to be more efficient as it is actually explicit and avoids the use of iterative methods when dealing with the nonlinear term. In addition, we also demonstrate that under some certain conditions, the proposed scheme can preserve the maximum bound principle and the original energy dissipation law at the discrete level. Finally, we conduct numerical simulations of several benchmark problems, and find that the numerical results are not only in agreement with our theoretical analysis, but also show that in comparison with the mesoscopic lattice Boltzmann method, the proposed macroscopic scheme has a great advantage in reducing the memory usage.