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Maximum Bound Principles for Semilinear Parabolic Equations and Exponential Time Differencing Schemes

Wednesday, July 15th, 2020, 8:00 PM  Zoom会议 房间号:924 6976 5319
主持人:郑海标 副教授
报告时间:2020年7月15日晚上20:00- 21:00
报告平台:Zoom会议 房间号:924 6976 5319 密码 123456

报告人介绍:鞠立力教授1995年毕业于武汉大学数学系获数学学士学位,1998年在中国科学院计算数学与科学工程计算研究所获得计算数学硕士学位,2002年在美国爱荷华州立大学获得应用数学博士学位。2002-2004年在美国明尼苏达大学数学与应用研究所从事博士后研究。随后进入美国南卡罗莱纳大学工作,历任数学系助理教授(2004年8月-2008年8月),副教授(2008年8月-2012年12月),及教授(2013年1月-现在)。主要从事科学计算与数值分析,网格优化,非局部模型, 图像处理,深度学习, 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100余篇,Google学术引用3000多次。自2006年起连续主持了多项由美国国家科学基金会(NSF)和美国能源部(DOE)资助的科研项目。美国工业与应用数学学会(SIAM)成员,2008-2009年期间担任其东南大西洋分会主席。2012至2017年任国际数值分析领域重要学术期刊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。与合作者关于合金微结构演化在“神威?太湖之光”超级计算机上的相场模拟工作入围2016年国际高性能计算应用最高奖?“戈登?贝尔”奖提名。

报告内容摘要:The ubiquity of semilinear parabolic equations has been illustrated in their numerous applications ranging from physics, biology, to materials and social sciences. In this talk, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $u_t = Lu + f[u]$ with $L$ being a linear dissipative operator and $f$ being a nonlinear operator in space, namely a time-invariant maximum bound principle, in the sense that the time-dependent solution $u$ preserves for all time a uniform pointwise bound in absolute value imposed by its initial and boundary conditions. We first study an analytical framework for some sufficient conditions on $L$ and $f$ that lead to such a maximum bound principle for the time-continuous dynamic system of infinite or finite dimensions. Then, we utilize a suitable exponential time differencing approach with a properly chosen generator of contraction semigroup to develop first- and second-order accurate temporal discretization schemes, that satisfy the maximum bound principle unconditionally in the time-discrete setting. Error estimates of the proposed schemes are derived along with their energy stability. Extensions to vector- and matrix-valued systems are also discussed. We demonstrate that the abstract framework and analysis techniques developed here offer an effective and unified approach to study the maximum bound principle of the abstract evolution equation, that covers a wide variety of well-known models and their numerical discretization schemes. Some numerical experiments are also carried out to verify the theoretical results.
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