Efficient positivity/bound/length preserving schemes for complex nonlinear systems and application to PKS-NS equations
科研大讨论系列报告
报告题目(Title)：Efficient positivity/bound/length preserving schemes for complex nonlinear systems and application to PKS-NS equations
报告人(Speaker)：沈捷 教授（宁波东方理工大学）
地点(Place)：后主楼1124
时间(Time)：2023年11月22日（星期三），14:00-15:00
邀请人(Inviter)：蔡勇勇
报告摘要
Solutions of a large class of partial differential equations (PDEs) arising from sciences and engineering applications are required to preserve positivity/bound or length, and also energy dissipative. It is of critical importance that their numerical approximations preserve these structures at the discrete level, as violation of these structures may render the discrete problems ill posed or inaccurate.
I will review the existing approaches for constructing positivity/bound preserving schemes, and then present several efficient and accurate approaches: (i) through reformulation as Wasserstein gradient flows; (ii) through a suitable functional transform; and (iii) through a Lagrange multiplier, which can also be used to construct length preserving schemes. These approaches have different advantages and limitations, are all relatively easy to implement and can be combined with most spatial discretizations.
As an application, I shall consider the Patlak-Keller-Segel-Navier-Stokes (PKS-NS) system which couples the Patlak-Keller-Segel equations with a positive density function and the Navier-Stokes equations. I shall present an efficient, totally decoupled, energy stable scheme for the PKS-NS system, and use it to investigate its blowup phenomena.
主讲人简介
沈捷教授于1982年毕业于北京大学计算数学专业, 1983年公派赴法国巴黎十一大学留学，于1987年获得博士学位后赴美国 Indiana University 从事博士后研究。 1991年至2001年先后任美国 Pennsylvania State University 数学系助理教授，副教授，教授。 2002年后任美国 Purdue University 数学系教授，2012-2022年任普度大学计算与应用数学中心主任，2023年被评为普度大学杰出教授。 2023年五月起，任宁波东方理工大学讲席教授和数学科学学院院长。沈捷教授主要从事偏微分方程数值解研究工作，具体研究方向包括谱方法数值分析理论, 计算流体, 以及计算材料科学。在国际杂志上发表了两百多篇论文，两本专著，在谷歌学术的引用逾2万五千余次。 他于2009年入选教育部长江讲座教授，2017年当选 AMS fellow，2020当选 SIAM fellow。
