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

TL;DR

This paper addresses the challenge of preserving the maximum principle and energy dissipation law for the Allen–Cahn equation in numerical simulations. It introduces a regularized lattice Boltzmann–based two-level macroscopic implicit–explicit finite-difference scheme (RLB-MIE-FD) for $d=1,2,3$ on a $D_dQ(2d+1)$ lattice, with $\\varepsilon^2=\\epsilon\\Delta x^2/\\Delta t$, and a semi-implicit treatment of the nonlinearity. Theoretical results establish first-order accuracy in time and second-order accuracy in space, along with discrete maximum-principle and energy-dissipation properties under specific time-step constraints; these are validated by comprehensive numerical experiments in 1D, 2D, and 3D, including traveling waves and mean-curvature-flow–like dynamics. The findings demonstrate that the RLB-MIE-FD scheme is more stable and memory-efficient than explicit FD and other LB-based approaches, while accurately capturing interface dynamics and preserving the core physical principles of the ACE.

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.

Figures (13)

• Figure 1: Schematic of the D$d$Q$(2d+1)$ lattice structure.
• Figure 2: The initial, numerical and analytical solutions under different values of interfacial parameter $\varepsilon$ (symbols: numerical solutions).
• Figure 3: The discrete maximum norms (a-b) and energies (c-d) of the numerical solutions under different values of interfacial parameter $\varepsilon$.
• Figure 4: The discrete maximum norms (a-b) and energies (c-d) of the numerical solutions of the RLB-MIE-FD and FEX-FD schemes.
• Figure 5: The initial solution at time $T_t=0$ (a) and numerical solution at time $T_t=40$ (b).