Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods

Abstract

In this work, we consider a time-fractional Allen-Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order α ∈ (0, 1). First, the well-posedness and (limited) smoothing property are studied, by using the maximal L p regularity of fractional evolution equations and the fractional Grönwall's inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall's inequality and maximal p regularity, we prove that the convergence rates of those time-stepping schemes are O(τ α ) without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen-Cahn dynamics.

On Energy Stable, Maximum-Principle Preserving, Second-Order BDF Scheme with Variable Steps for the Allen--Cahn Equation

“…For simplicity, we consider periodic solution u along the boundary. The above time fractional Allen-Cahn equation has been studied both theoretically and numerically in recent years [3,6,8,9,16,[22][23][24]. When α → 1, the TFAC equation recovers the classical Allen-Cahn equation [1]:…”

“…The remainder [T 0 , T ] is tested by two types of time meshes: (Graded-uniform mesh) Uniform step size τ = 0.01;(Graded-adaptive mesh) Adaptive time-stepping with τ max = 10 −1 and τ min = 10 −3 . step Graded−adaptive step (κ = 10)Graded−adaptive step (κ = 10 2 )Graded−adaptive step (κ = 10 3 step Graded−adaptive step (κ = 10)Graded−adaptive step (κ = 10 2 )Graded−adaptive step (κ = 103 The energies E(t), E α (t) and adaptive steps of Example 2.…”

Approximation of an optimal control problem for the time-fractional Fokker-Planck equation

“…This refers to the trajectory of a single particle modeled by the non-Markovian stochastic process where the mean square displacement is no longer linear [5,6,24,23,31]. The nonlocal character of fractional time derivative makes it very useful to describe memory effects and derive the power law decay structure of energy in a diffusion-transport PDE system [13,17,18,29].…”

“…The natural functional structure would be that of vector-valued fractional Sobolev space, and well-posedness is usually established via a priori estimates based on energy dissipation. This approach has been adopted in a large number of works, [13,12,15,16,18,27,28,29] for examples.…”

“…We use a classical scheme, called L1 approximation, which is naturally derived from the approximation of the fractional integral as a Riemann sum and long known to be consistent (for a concise illustration, we refer to section 3 of [29]). Moreover, the L1 approximation scheme stands out by being able to preserve at the discrete level certain desirable features of the original PDEs, such as maximum principle [14] and energy stability [13,27,29]. Therefore, approximations can be constructed to converge to suitable notions of weak solutions, which may be viscosity solutions to Hamilton-Jacobi-Bellman equations [14], or distributional solutions in Sobolev spaces to diffusion (phase field) equations [13,27,29].…”