報告題目：Maximum Bound Principles for Semilinear Parabolic Equations and Exponential Time Differencing Schemes
報 告 人：鞠立力 教授 Department of Mathematics，University of South Carolina
報告時間：2020年6月23日，上午9:00
報告地點：騰訊會議ID：933 232 131
會議鍊接：https://meeting.tencent.com/s/cl1LksnfNMBO
校内聯系人：張然 zhangran@jlu.edu.cn
報告摘要：
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.
報告人簡介：
鞠立力教授1995年畢業于武漢大學數學系獲數學學士學位, 1998年在中國科學院計算數學與科學工程計算研究所獲得計算數學碩士學位，2002年在美國愛荷華州立大學獲得應用數學博士學位。2002-2004年在美國明尼蘇達大學數學與應用研究所從事博士後研究。随後進入美國南卡羅萊納大學工作，曆任數學系助理教授（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-現在）。主要從事科學計算與數值分析，網格優化，非局部模型, 圖像處理，深度學習, 高性能科學計算，及其在材料與地球科學中的應用等方面的研究工作。至今已發表科研論文100餘篇，Google學術引用3000多次。自2006年起連續主持了多項由美國國家科學基金會（NSF）和美國能源部（DOE）資助的科研項目。美國工業與應用數學學會（SIAM）成員，2008-2009年期間擔任其東南大西洋分會主席。2012至2017年任國際數值分析領域重要學術期刊SIAM Journal on Numerical Analysis的編委。多次受邀擔任美國國家科學基金會計算數學領域基金會審評議組成員。與合作者關于合金微結構演化在“神威·太湖之光”超級計算機上的相場模拟工作入圍2016年國際高性能計算應用最高獎—“戈登·貝爾”獎提名。
