【6月26日】Time-Fractional Allen-Cahn Equations: Analysis and Numerical

】【打印】【关闭窗口 来源：本站原创 作者：统计与数学学院 编辑：王凝 发布时间：2019-06-17
报告题目：Time-Fractional Allen-Cahn Equations: Analysis and Numerical
主讲人：

时间：2019年6月26日（周三）10:30 a.m.
地点：北院卓远楼305
主办单位：统计与数学学院

摘要：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 \$/alpha/in(0,1)\$. First, the well-posedness and (limited) smoothing property are systematically analyzed, by using the maximal \$L^p\$ regularity of fractional evolution equations and the fractional Gr/"onwall's inequality. We also show the maximum principle like their conventional local-in-time counterpart. Precisely, 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, i.e., convex splitting scheme, weighted convex splitting scheme and 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. Finally, by using a discrete version of fractional Gr/"onwall's inequality and maximal \$/ell^p\$ regularity, we prove that the convergence rates of those time-stepping schemes are \$O(/tau^/alpha)\$ 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.

主讲人简介：
杨将博士，于2010年在浙江大学数学系获数学学士学位，2014年在香港浸会大学数学系获得博士学位，现任教于南方科技大学。他的研究方向包括微分方程数值解，相场模型的数值算法及应用，非局部模型的数值算法及应用。