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The numerical integration of phase-field equations is a delicate task whichneeds to recover at the discrete level intrinsic properties of the solutionsuch as energy dissipation and maximum principle. Although the theory of energydissipation for classical phase field models is well established, thecorresponding theory for time-fractional phase-field models is stillincomplete. In this article, we study certain nonlocal-in-time energies usingthe first-order stabilized semi-implicit L1 scheme. In particular, we willestablish a discrete fractional energy law and a discrete weighted energy law.The extension for a $(2-{\alpha})$-order L1 scalar auxiliary variable schemewill be investigated. Moreover, we demonstrate that the energy bound ispreserved for the L1 schemes with nonuniform time steps. Several numericalexperiments are carried to verify our theoretical analysis.

Numerical Energy Dissipation for Time-Fractional Phase-Field Equations