Numerical Energy Dissipation for Time-Fractional Phase-Field Equations (Journal of Computational Mathematics, 2024)
The numerical integration of phase-field equations is a delicate task which needs torecover at the discrete level intrinsic properties of the solution such as energy dissipationand maximum principle. Although the theory of energy dissipation for classical phase fieldmodels is well established, the corresponding theory for time-fractional phase-field modelsis still incomplete. In this article, we study certain nonlocal-in-time energies using thefirst-order stabilized semi-implicit L1 scheme. In particular, we will establish a discretefractional energy law and a discrete weighted energy law. The extension for a $(2−α)$-orderL1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate thatthe energy bound is preserved for the L1 schemes with nonuniform time steps. Severalnumerical experiments are carried to verify our theoretical analysis.