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

TL;DR

This paper introduces a novel mesoscopic lattice Boltzmann-based finite-difference scheme for the Allen-Cahn equation that preserves maximum principle and energy dissipation at the discrete level, ensuring stability and accuracy.

Contribution

A new bottom-up lattice Boltzmann method-based numerical scheme for the Allen-Cahn equation that maintains key physical properties discretely, differing from traditional macroscopic discretizations.

Findings

The scheme achieves second-order spatial accuracy.

It preserves the maximum bound principle.

It maintains the energy dissipation law discretely.

Abstract

The Allen-Cahn equation (ACE) inherently possesses two crucial properties: the maximum principle and the energy dissipation law. Preserving these two properties at the discrete level is also necessary in the numerical methods for the ACE. In this paper, unlike the traditional top-down macroscopic numerical schemes which discretize the ACE directly, we first propose a novel bottom-up mesoscopic regularized lattice Boltzmann method based macroscopic numerical scheme for d (=1, 2, 3)-dimensional ACE, where the DdQ(2d+1) [(2d+1) discrete velocities 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 the ACE, in which the nonlinear term is discretized semi-implicitly, the temporal derivative and dissipation term of the ACE are discretized by using the explicit Euler method and second-order central difference method, respectively. Then we also demonstrate that the proposed scheme can preserve the maximum bound principle and the original energy dissipation law at the discrete level under some conditions. Finally, some numerical experiments are conducted to validate our theoretical analysis.