Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

TL;DR

This paper introduces linear, second order, energy stable numerical schemes for the Allen-Cahn equation with nonlocal constraints, preserving phase volume and demonstrating slower dynamics compared to classical models, suitable for phase evolution in immiscible systems.

Contribution

It develops unconditionally energy stable, volume-preserving schemes for the Allen-Cahn equation using energy quadratization and finite difference methods, with practical implementation strategies.

Findings

Schemes preserve total phase volume.

Demonstrates slower dynamics with volume constraints.

Effective as alternatives to Cahn-Hilliard model.

Abstract

We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn equation with nonlocal constraints that preserves the total volume of each phase in a binary material system. The energy quadratization strategy is employed to derive the energy stable semi-discrete numerical algorithms in time. Solvability conditions are then established for the linear systems resulting from the semi-discrete, linear schemes. The fully discrete schemes are obtained afterwards by applying second order finite difference methods on cell-centered grids in space. The performance of the schemes are assessed against two benchmark numerical examples, in which dynamics obtained using the volumepreserving Allen-Cahn equations with nonlocal constraints is compared with those obtained using the classical Allen-Cahn as well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics when volume constraints are imposed as well as their usefulness as alternatives to the Cahn-Hilliard equation in describing phase evolutionary dynamics for immiscible material systems while preserving the phase volumes. Some performance enhancing, practical implementation methods for the linear energy stable schemes are discussed in the end.