The stabilized exponential-SAV approach preserving maximum bound principle for nonlocal Allen-Cahn equation

TL;DR

This paper develops unconditionally energy stable and maximum bound principle-preserving numerical schemes for the nonlocal Allen-Cahn equation using a stabilized exponential scalar auxiliary variable approach, with rigorous proofs and efficient FFT-based solvers.

Contribution

It introduces first- and second-order accurate schemes that preserve energy dissipation and maximum bounds, with rigorous proofs and improved computational efficiency.

Findings

Schemes are unconditionally energy stable and MBP-preserving.

Rigorous proofs of stability and MBP at the fully discrete level.

Numerical experiments demonstrate the schemes' effectiveness.

Abstract

The nonlocal Allen-Cahn equation with nonlocal diffusion operator is a generalization of the classical Allen-Cahn equation. It satisfies the energy dissipation law and maximum bound principle (MBP), and is important for simulating a series of physical and biological phenomena involving long-distance interactions in space. In this paper, we construct first- and second-order (in time) accurate, unconditionally energy stable and MBP-preserving schemes for the nonlocal Allen-Cahn type model based on the stabilized exponential scalar auxiliary variable (sESAV) approach. On the one hand, we have proved the MBP and unconditional energy stability carefully and rigorously in the fully discrete levels. On the other hand, we adopt an efficient FFT-based fast solver to compute the nearly full coefficient matrix generated from the spatial discretization, which improves the computational efficiency. Finally, typical numerical experiments are presented to demonstrate the performance of our proposed schemes.