A decreasing upper bound of energy for time-fractional phase-field equations

Abstract

In this article, we study the energy dissipation property of time-fractional Allen-Cahn equation. We propose a decreasing upper bound of energy that decreases with respect to time and coincides with the original energy at t = 0 and as t tends to ∞. This upper bound can also be viewed as a nonlocal-in-time modified energy, the summation of the original energy and an accumulation term due to the memory effect of time fractional derivative. In particular, this indicates that the original energy indeed decays w.r.t. time in a small neighborhood at t = 0. We illustrate the theory mainly with the time-fractional Allen-Cahn equation, but it could be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation.On the discrete level, the first-order L1 and second-order L2 schemes for timefractional Allen-Cahn equation have similar decreasing modified energies, so that the stability can be established. Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.

Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models

“…These models exhibit multi-scaling time behaviour, which makes them suitable for the description of different diffusive regimes and characteristic crossover dynamics in complex systems [21,25,30]. They are always formulated in the integral form, including the Riemann-Liouville fractional integral (I β t w)(t) := In capturing the multi-scale behaviors in many of integro-differential equations, such as the timefractional phase field models [8,9,14,15,[31][32][33]35] and nonlinear fractional wave models [1,2,[4][5][6]19,20,26], adaptive time-stepping strategies, namely, small time steps are utilized when the solution varies rapidly and large time steps are employed otherwise, are practically useful [3,15,18,20,22,23,[26][27][28][29]. It requires practically and theoretically reliable (stable and convergent) time-stepping methods on general setting of time step-size variations [8,9,14,15,18].…”

Fast High Order and Energy Dissipative Schemes with Variable Time Steps for Time-Fractional Molecular Beam Epitaxial Growth Model

“…It seems rather difficult to theoretically establish the stabilities of the proposed L1 and L2 based schemes on the nonuniform mesh even for a modified free energy. Very recently, another type of nonlocal-in-time modified energy dissipation law was obtained in [39] for the time-fractional Allen-Cahn equation. The proposed nonlocal modified energy is an upper bound of the traditional original free energy, and coincides the original energy at initial time and t → ∞.…”