A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

2022/11/02 by Dianming Hou, Lili Ju, Hou, Dianming +3 · 4 citations
Materials Science · Mathematics · #41A05 #41A25 #65M06 #65M15 #Differential Equations and Numerical Methods #FOS: Mathematics #G.1.8 #Numerical Analysis (math.NA) #Numerical methods for differential equations #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2211.00852

openalex publication_date 2022/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete H¹ error estimate and energy stability for the classic constant mobility case and the L^∞ error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy.

Cited by

• A linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation
• Dynamic-stabilization-based linear schemes for the Allen-Cahn equation with degenerate mobility: MBP and energy stability
• High-order nonuniform time-stepping and MBP-preserving linear schemes for the time-fractional Allen-Cahn equation
• A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions

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