Discrete Maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation

TL;DR

This paper develops a high-order finite difference scheme for a generalized Allen-Cahn equation with passive convection, proving a discrete maximum principle under certain mesh and time step constraints, ensuring stability and bound preservation.

Contribution

It introduces a fourth-order accurate finite difference scheme based on spectral element formulation and proves a discrete maximum principle for the scheme under specific conditions.

Findings

The scheme is stable under mesh and time step constraints.

The maximum principle holds for passive convection with incompressible velocity.

The method ensures bound preservation in numerical solutions.

Abstract

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.