Maximum principle preserving and unconditionally stable scheme for a conservative Allen–Cahn equation

Abstract

In this study, we present a novel conservative Allen–Cahn (CAC) equation and its maximum principle preserving and unconditionally stable numerical method. There have been many research works of the numerical methods for the CAC equation. To conserve the total mass, many mathematical models for the CAC equation introduced Lagrange multipliers which are added to the original Allen–Cahn equation. Therefore, some of the methods do not preserve the maximum principle, i.e., it is possible to have values greater than the maximum and smaller than the minimum values of the admissible solutions. In this study, we propose a novel CAC equation with a new Lagrange multiplier which is a power exponent to the concentration so that the maximum principle strictly holds. Furthermore, we describe the proposed numerical algorithm in detail and present several computational experiments to validate the superior performance of the proposed scheme.