Novel mass-conserving Allen–Cahn equation for the boundedness of an order parameter

Abstract

There are several theoretically well-posed models for the Allen–Cahn equation under mass conservation. The conservative property is a gift from the additional nonlocal term playing a role of a Lagrange multiplier. However, the same term destroys the boundedness property that the original Allen–Cahn equation presents: The solution is bounded by 1 with an initial datum bounded by 1. In this paper, we propose a novel mass-conserving Allen–Cahn equation and prove the existence and uniqueness of a classical solution in the context of the theory of analytic semigroups as well as the boundedness property of the solution. From the numerical point of view, we investigate a linear unconditionally energy stable splitting scheme of the proposed model for the boundedness of numerical solutions. Various numerical experiments are presented to demonstrate the validity of the proposed method and to make distinctions from a few closely related methods.