Error Estimates of Energy Stable Numerical Schemes for Allen-Cahn Equations with Nonlocal Constraints

TL;DR

This paper provides rigorous error estimates and demonstrates the effectiveness of four energy stable, second-order numerical schemes for Allen-Cahn equations with nonlocal constraints, emphasizing accuracy, solvability, and physical property preservation.

Contribution

It introduces four linear, second-order schemes based on EQ and SAV methods with proven error bounds and solvability, advancing numerical analysis for constrained Allen-Cahn equations.

Findings

Error estimates for all schemes are rigorously established.

Numerical experiments confirm accuracy and physical property preservation.

Schemes are unconditionally energy stable and uniquely solvable.

Abstract

We present error estimates for four unconditionally energy stable numerical schemes developed for solving Allen-Cahn equations with nonlocal constraints. The schemes are linear and second order in time and space, designed based on the energy quadratization (EQ) or the scalar auxiliary variable (SAV) method, respectively. In addition to the rigorous error estimates for each scheme, we also show that the linear systems resulting from the energy stable numerical schemes are all uniquely solvable. Then, we present some numerical experiments to show the accuracy of the schemes, their volume-preserving as well as energy dissipation properties in a drop merging simulation.