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.

翻译：Allen-Cahn方程（ACE）天然具备两个关键性质：最大值原理和能量耗散定律。在ACE的数值方法中，保持这些性质在离散层面上的成立同样必要。本文不同于传统自上而下的宏观数值方案直接离散ACE，首先提出一种新颖的自下而上的介观正则化格子玻尔兹曼方法，用于d(=1,2,3)维ACE的宏观数值方案，其中采用DdQ(2d+1) [d维空间中的(2d+1)个离散速度] 格子结构。特别地，所提出的宏观数值方案在空间上具有二阶精度，也可视为ACE的隐式-显式有限差分格式：其中非线性项采用半隐式离散，时间导数和ACE的耗散项分别通过显式欧拉法和二阶中心差分法离散。随后证明，该方案在特定条件下能在离散层面保持最大值有界原理和原始能量耗散定律。最后通过数值实验验证理论分析结果。