The numerical solutions for the energy-dissipative and mass-conservative Allen–Cahn equation

Abstract

There are three criteria when we model with phase-field equations. The first one is the evolution of phase separation. The second one is energy dissipation law. The last one is whether we have the law of conservation of mass or not. If we minimize the Ginzburg–Landau free-energy functional, the kinetics of phase separation inherently follows. However, the other properties such as energy dissipation and mass conservation are more difficult to model. Hitherto, there have been a few proposals to conserve mass using the Allen–Cahn (AC) equation, but we do not have the energy dissipation law in these models when considered in light of local effects of mass-correction. For the reasons, certain researchers still prefer to use the Cahn–Hilliard equation, e.g. biharmonic equation instead of using the Allen–Cahn equation when we consider a mass conserving case. This paper is concerned with numerical solutions of the AC equation in the mass-conserving space, where the solution is corrected at each iteration for mass conservation. We introduce two different methods for solving the conservative AC equation with periodic boundary conditions. In this work, we have the AC equation in common, but use two different mass correction operators. Note that both of mass correction operators satisfy the discrete energy dissipation property. First, we project the solution of the AC equation to the mass conserving space, and then recover the linear convex splitting scheme for the conventional mass-conserving equation. The linear convex splitting scheme is well-known but the convergence of numerical solutions is elaborately studied in terms of gradient-projection. Next, we propose the other energy-minimizing method on the mass-conserving space. It translates the solution of the AC equation to the mass conserving space via an energy-dissipation operator. The dissipation of discrete energy functional for the proposed method is also established. As far as the author knows, no report has been presented for the numerical scheme having discrete energy dissipation property of the mass-conserving AC equation without using the nonlocal and constant multiplier. To verify the common properties of mass-projection and proposed methods numerically, we simulate the evolutions for phase separation, dissipation of energy, and mass conservation. We moreover consider the short and long time stages of evolutions to see the difference between two methods. These several tests are provided for its applications of the proposed method for modeling natural phenomena.