Regularized Lattice Boltzmann Method Based Maximum Principle and Energy Stability Preserving Finite-Difference Scheme for the Allen-Cahn Equation

作     者：Chen, Ying Liu, Xi Chai, Zhenhua Shi, Baochang 

作者机构：School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China 

出 版 物：《SSRN》 

年 卷 期：2024年

核心收录：

主　　题：Maximum principle 

摘      要：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 traditionally 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 viewed 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 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. © 2024, The Authors. All rights reserved.