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Title: Efficient Structure Preserving Schemes for Complex Nonlinear Systems
Abstract: Solutions for a large class of partial differential equations (PDEs) arising from sciences and engineering applications are required to be positive or within a specified bound, and/or 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. These approaches have different advantages and limitations, are all relatively easy to implement and can be combined with most spatial discretization.
Biography: Jie Shen is a professor and director at Center for Computational and Applied Mathematics Department of Mathematics at Purdue University. Received B.S. in Computational Mathematics at Peking University, China, in 1982, and Ph.D. in Numerical Analysis, at University of Paris-Sud (currently University of Paris-Saclay), France, 1987. Research interest include numerical analysis, spectral methods, scientific computing, computational fluid dynamics and computational materials science.
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