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Highly Efficient and Energy Dissipative Schemes for the Time Fractional Allen--Cahn Equation
SIAM Journal on Scientific Computing  (IF2.373),  Pub Date : 2021-09-21, DOI: 10.1137/20m135577x
Dianming Hou, Chuanju Xu

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3305-A3327, January 2021.
In this paper, we propose and analyze a time stepping method for the time fractional Allen--Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for this type of equation. The unconditional stability is proved through establishing a discrete nonlocal free energy dissipation law, which is also true for the continuous problem. The main idea used in the analysis is to split the time fractional derivative into two parts: a local part and a history part, which are discretized by the well known L1, L1-CN, and $L1^{+}$-CN schemes. Then an extended auxiliary variable approach is used to deal with the nonlinear and history term. The main contributions of the paper are first, it is found that the time fractional Allen--Cahn equation is a dissipative system related to a nonlocal free energy. Second, we construct efficient time stepping schemes satisfying the same dissipation law at the discrete level. In particular, we prove that the proposed schemes are unconditionally stable for quite general meshes. Finally, the efficiency of the proposed method is verified by a series of numerical experiments.