Highly efficient and energy dissipative schemes for the time fractional Allen-Cahn equation

TL;DR

This paper introduces a new, unconditionally stable time-stepping scheme for the time fractional Allen-Cahn equation that preserves the system's energy dissipation law, verified through theoretical analysis and numerical experiments.

Contribution

It develops and analyzes a novel energy dissipative scheme for the fractional Allen-Cahn equation that is unconditionally stable on general meshes, including graded meshes.

Findings

The scheme is unconditionally stable for general meshes.

The scheme preserves the nonlocal free energy dissipation law.

Numerical experiments confirm the efficiency and stability of the method.

Abstract

In this paper, we propose and analyze a time-stepping method for the time fractional Allen-Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for this type of equations. The unconditional stability is proved through establishing a discrete nonlocal free energy dispassion law, which is also true for the continuous problem. The main idea used in the analysis is to split the time fractional derivative into two parts: a local part and a history part, which are discretized by the well known L1, L1-CN, and $L1^{+}$-CN schemes. Then an extended auxiliary variable approach is used to deal with the nonlinear and history term. The main contributions of the paper are: First, it is found that the time fractional Allen-Chan equation is a dissipative system related to a nonlocal free energy. Second, we construct efficient time stepping schemes satisfying the same dissipation law at the discrete level. In particular, we prove that the proposed schemes are unconditionally stable for quite general meshes. Finally, the efficiency of the proposed method is verified by a series of numerical experiments.