High-order, unconditionally maximum-principle preserving finite element method for the Allen-Cahn equation

In this paper, based on the mass-lumping finite element space discretization, we incor-porate the integrating factor Runge-Kutta method and stabilization technique to develop a class of temporal up to the fourth-order unconditionally structure-preserving schemes for the Allen-Cahn equation and its conservative forms. The proposed methods are lin-ear, without requiring any post-processing or limiters, and unconditionally preserve the maximum principle and mass conservation law. Several numerical experiments verify the high-order temporal accuracy of the proposed schemes, as well demonstrate the ability to preserve the maximum principle, mass conservation, and energy stability over long peri-ods. Moreover, by the aid of numerical simulation, we show that the proposed schemes also have good performances in terms of structure-preserving with high order finite ele-ment method. (c) 2023 IMACS. Published by Elsevier B.V. All rights reserved.

Automated summary

Based on the mass-lumping finite element space discretization, this paper introduces a class of temporal up to the fourth-order unconditionally structure-preserving schemes for the Allen-Cahn equation and its conservative forms, by incorporating the integrating factor Runge-Kutta method and stabilization technique. The proposed methods are linear, do not require any post-processing or limiters, and unconditionally preserve the maximum principle and mass conservation law. Numerical experiments verify the high-order temporal accuracy and the ability to preserve the maximum principle, mass conservation, and energy stability over long periods. Additionally, numerical simulation demonstrates the good performance of the proposed schemes in terms of structure-preserving with high-order finite element method.