Energy Stable L2 Schemes for Time-Fractional Phase-Field Equations

TL;DR

This paper establishes the energy stability of two high-order L2 numerical schemes for time-fractional phase-field equations, introducing new properties and reformulations to ensure boundedness and fractional energy laws.

Contribution

It introduces a reformulation of the L2 operator and new stability proofs for high-order schemes applied to time-fractional phase-field equations.

Findings

Proves energy boundedness of an L2 scalar auxiliary variable scheme.

Establishes fractional energy law for an implicit-explicit L2 Adams--Bashforth scheme.

Develops a new Cholesky decomposition for stability analysis.

Abstract

In this article, the energy stability of two high-order L2 schemes for time-fractional phase-field equations is established. We propose a reformulation of the L2 operator and also some new properties on it. We prove the energy boundedness (by initial energy) of an L2 scalar auxiliary variable scheme for any phase-field equation and the fractional energy law of an implicit-explicit L2 Adams--Bashforth scheme for the Allen--Cahn equation. The stability analysis is based on a new Cholesky decomposition proposed recently by some of us.