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Talk of Prof. Lili Ju

Lili Ju


鞠立力,南卡莱罗纳州立大学教授,中国海洋大学数学科学学院绿卡教授。主要从事数值计算方法与分析,网格优化,图像处理,非局部模型,高性能科学计算,及其在材料与地球科学中的应用等方面的研究工作。至今已发表科研论文80余篇,Google学术引用2400多次。主持了多项由美国国家科学基金会(NSF)和美国能源部(DOE)等联邦机构资助的科研项目。2012至2017年任数值分析领域国际重要学术期刊SIAM Journal on Numerical Analysis的编委。多次受邀担任美国国家科学基金会计算数学领域基金会审评议组成员。


Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen-Cahn Equation


The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the maximum principle as its local counterpart. In this talk, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete maximum principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadrature-based finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotic compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relationship between the discontinuities and the nonlocal parameters.