Arbitrarily High-order Maximum Bound Preserving Schemes with Cut-off Postprocessing for Allen-Cahn Equations

2021/02/26 by Yang Jiang, Yang, Jiang, Zhaoming Yuan +3 · 2 citations
Engineering · Materials Science · Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #Fluid Dynamics and Thin Films #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2102.13271

openalex publication_date 2021/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop and analyze a class of maximum bound preserving schemes for approximately solving Allen--Cahn equations. We apply a kth-order single-step scheme in time (where the nonlinear term is linearized by multi-step extrapolation), and a lumped mass finite element method in space with piecewise rth-order polynomials and Gauss--Lobatto quadrature. At each time level, a cut-off post-processing is proposed to eliminate extra values violating the maximum bound principle at the finite element nodal points. As a result, the numerical solution satisfies the maximum bound principle (at all nodal points), and the optimal error bound O(τ^k+h^r+1) is theoretically proved for a certain class of schemes. These time stepping schemes under consideration includes algebraically stable collocation-type methods, which could be arbitrarily high-order in both space and time. Moreover, combining the cut-off strategy with the scalar auxiliary value (SAV) technique, we develop a class of energy-stable and maximum bound preserving schemes, which is arbitrarily high-order in time. Numerical results are provided to illustrate the accuracy of the proposed method.

Cited by

• Dynamic-stabilization-based linear schemes for the Allen-Cahn equation with degenerate mobility: MBP and energy stability
• A bound-preserving and conservative enriched Galerkin method for elliptic problems

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