Highly efficient schemes for time fractional Allen-Cahn equation using extended SAV approach

TL;DR

This paper develops high-order, unconditionally stable numerical schemes for the time fractional Allen-Cahn equation using an extended SAV approach, improving robustness and efficiency in simulating coarsening dynamics.

Contribution

It introduces new high-order, unconditionally stable schemes based on the extended SAV approach for the time fractional Allen-Cahn equation, with comprehensive stability analysis and numerical validation.

Findings

Schemes are unconditionally stable for uniform meshes.

Numerical experiments confirm efficiency and robustness.

Fractional order significantly influences coarsening behavior.

Abstract

In this paper, we propose and analyze high order efficient schemes for the time fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist in: 1) constructing first and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; 2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time fractional Allen-Cahn equation. Particularly, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials.