On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

Abstract

For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter α reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and timefractional molecular beam epitaxy model, while there exists a −α/3 power law coarsening stage.

A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations

“…Up to now we are unable to build up a discrete energy dissipation law for the second-order scheme (3.6)-(3.7). As seen in [28], the key issue is to prove the positive semi-definite of the quadratic form n k=1 w k k j=1 A (k) k−j w j . In fact, we can show the energy stability under uniform mesh using similar arguments as in [28].…”

“…As a generalization of the classical Allen-Cahn equation [2,6,8,25], the above time-fractional Allen-Cahn equation (1.1) has been widely investigated in recent years [11,15,22,28], In particular, it was first shown in [28] that the time-fractional Allen-Cahn equation admits the following energy law…”

“…From the numerical scheme point of view, first order schemes that combine the L1 formula [21,27] and the stabilization technique [29] were proposed in [28] for the time-fractional Allen-Cahn equation. Furthermore, it is shown in [28] that the stabilization L1 scheme preserves the energy law (1.5) and the maximum principle (1.7) in the discrete level. More recently, sharp regularity analysis of the time-fractional Allen-Cahn equation and numerical analysis for a class of numerical schemes under limited regularity were presented in [5].…”

Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation

“…The backward Euler scheme (2.11)-(2.12) is a fully nonlinear implicit scheme and some inner iteration will be needed. To accelerate the time-stepping process, we build a linearized scheme here by using the well-known stabilized technique via a stabilized term S(u n − u n−1 ) for a properly large scalar parameter S > 0 , see also the recent work [17]. The resulting stabilized semi-implicit scheme for the problem (1.1)-(1.2) reads…”

“…Boundary conditions are set to be periodic so as not to complicate the analysis with unwanted details. Very recently, the energy decay laws of time-fractional phase field models, involving timefractional Allen-Cahn equation, time-fractional Cahn-Hilliard equation and time-fractional molecular beam epitaxy models, are reported in [17]. In comparison to the classical physical model, the energy dissipation law of the time-fractional Allen-Cahn equation (1.1) is…”