Error Estimates of Energy Stable Numerical Schemes for Allen-Cahn Equations with Nonlocal Constraints
This work provides rigorous error analysis for energy stable schemes in a specific class of phase-field equations, which is an incremental contribution for researchers working on numerical methods for nonlocal Allen-Cahn equations.
The paper presents error estimates for four unconditionally energy stable numerical schemes for solving Allen-Cahn equations with nonlocal constraints, proving unique solvability and demonstrating accuracy, volume preservation, and energy dissipation in drop merging simulations.
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.