Review of the conservative Allen-Cahn equations

초록

The conservative Allen-Cahn (CAC) equation extends the classical Allen-Cahn (AC) equation by enforcing mass conservation. The CAC equation has been used as a computational model in materials science, fluid mechanics, biology, and environmental modeling. Unlike the classical AC equation, which does not preserve mass, the CAC equation uses Lagrange multipliers to guarantee mass conservation. This property makes the CAC equation suitable for the simulation of physical systems where conservation laws are important. This article reviews the development of the CAC equation with a focus on its mathematical formulation and numerical solution methods. The numerical approaches are finite difference methods, spectral methods, and operator splitting techniques. Several different forms of the CAC equation are presented, and their theoretical properties, including mass conservation and energy dissipation, are considered. Furthermore, the performance of different numerical methods in capturing interface motion is evaluated. Applications to phase separation and multiphase flow problems are discussed to illustrate the strengths and limitations of each numerical method. This review demonstrates that the CAC equation is an effective model for describing interface dynamics under mass conservation constraints.

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