Second-Order Linear Relaxation Schemes for Time-Fractional Phase-Field Models

TL;DR

This paper introduces a linear relaxation scheme for efficiently solving time-fractional phase-field models, achieving second-order accuracy and unconditional energy stability, with demonstrated numerical effectiveness.

Contribution

It develops a novel linear relaxation method using auxiliary variables for time-fractional Allen-Cahn and Cahn-Hilliard equations, improving efficiency and stability.

Findings

Scheme is linear and second-order accurate in time.

Unconditionally energy stable scheme.

Numerical results confirm effectiveness.

Abstract

This work uses a linear relaxation method to develop efficient numerical schemes for the time-fractional Allen-Cahn and Cahn-Hilliard equations. The L1+-CN formula is used to discretize the fractional derivative, and an auxiliary variable is introduced to approximate the nonlinear term by solving an algebraic equation rather than a differential equation as in the invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches. The proposed semi-discrete scheme is linear, second-order accurate in time, and the inconsistency between the auxiliary and the original variables does not deteriorate over time. Furthermore, we prove that the scheme is unconditionally energy stable. Numerical results demonstrate the effectiveness of the proposed scheme.