An energy stable bound-preserving finite volume scheme for the Allen-Cahn equation based on operator splitting method

Abstract

In this paper, an energy stable bound-preserving finite volume scheme is constructed for the Allen-Cahn equation. The first-order operator splitting method is used to split the original equation into a nonlinear equation and a heat equation in each time interval. The nonlinear equation is solved by the explicit scheme, and the heat equation is discretized by the extremum-preserving scheme. The harmonic averaging points on cell facets are employed to define auxiliary unknowns, which enable our discrete scheme to be applicable to unstructured meshes. The energy stable and bound-preserving analysis of the finite volume scheme are also presented. Numerical experiments illustrate that this linear numerical scheme is practical and accurate in solving the Allen-Cahn equation.