Abstract
In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time-fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist of (1) constructing first- and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; (2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time-fractional Allen-Cahn equation. In particular, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials.
Highlights
As a useful modelling tool, gradient flow has been used to model many physical problems, dissipative systems, which are systems driven by dissipation of free energy
We have proposed two schemes for the α order time-fractional Allen-Cahn equation: a first-order scheme and a 2 − α order scheme, both constructed on the uniform mesh and graded mesh
Such fractional gradient flows can be regarded as a variation of the phase field models, used to describe the memory effect of some materials, which have attracted attention in recent few years
Summary
As a useful modelling tool, gradient flow has been used to model many physical problems, dissipative systems, which are systems driven by dissipation of free energy. There appeared two novel strategies: the invariant energy quadratization (IEQ) method [41, 45] and the so-called scalar auxiliary variable (SAV) approach [34, 35] Based on these two approaches, second-order unconditionally stable schemes have been successfully constructed for a large class of gradient flow models. When long-time simulation is needed for the physical understanding of interface dynamics and far-from-equilibrium phase transitions, construction of highly stable methods will be indispensable This task becomes achievable only very recently thanks to the new developed SAV approach. The first is to make use of the abovementioned results to construct efficient numerical schemes for the time-fractional Allen-Cahn equation, and analyze the stability properties of the proposed methods.
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