SAV Method Applied to Fractional Allen-Cahn Equation

Xiaolan Zhou,Chuanju Xu,Mejdi Azaiez

doi:10.1007/978-3-030-39647-3_39

Abstract

AbstractFractional differential equations are becoming increasingly used as a modeling tool for processes associated with anomalous diffusion or spatial heterogeneity. We consider in this paper a nonlocal phase transition model, in particular described by the Allen-Cahn equation. Precisely, we deal with a fractional Allen-Cahn equation (FACE) withspectraldefinition of the fractional Laplacian. For the space discretization, we first use spectral Galerkin method to solve the eigenvalue problem (EVP) for the Laplacian operator with homogeneous Neumann boundary condition, then we employ the computed eigenfunctions as trial and test bases to solve the FACE. For the time discretization, based on the scalar auxiliary variable (SAV) approach, we construct unconditionally second-order energy stable BDF scheme (SAV/BDF2). Finally, the developed algorithm is used to simulate the phase evolution of a benchmark problem.

Highlights

The Allen-Cahn equation was originally introduced to describe the motion of antiphase boundaries in crystalline solids [1]
There have been a large body of work on numerical analysis of Allen-Cahn equations
We aim in this paper to use the scalar auxiliary variable (SAV) scheme, recently introduced and analyzed by a number of researchers; see, e.g., [5] and the references therein, to approximate the solution of the fractional version of the Allen-Cahn model

Summary

Introduction

The Allen-Cahn equation was originally introduced to describe the motion of antiphase boundaries in crystalline solids [1]. There have been a large body of work on numerical analysis of Allen-Cahn equations (cf [2–5] and the references therein). We aim in this paper to use the SAV scheme, recently introduced and analyzed by a number of researchers; see, e.g., [5] and the references therein, to approximate the solution of the fractional version of the Allen-Cahn model.

Azaiez School of Mathematical Sciences, Xiamen University, Xiamen, China

Spatial Discretizations

Implementation

Test of the Convergence Order

Benchmark Test