Abstract:  

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.

Short bio:

Professor Jie Shen received his B.S. in Computational Mathematics from Peking University in 1982, and his Ph.D in Numerical Analysis from Universite de Paris-Sud (currently Paris Saclay) at Orsay in 1987. Before joining Eastern Institute of Technology, Ningbo in May 2023, he was a Distinguished Professor of Mathematics at Purdue University. He is an elected Fellow of AMS, SIAM and CSIAM.

His main research interests are numerical analysis, spectral methods and scientific computing, with applications in computational fluid dynamics and materials science. He has authored/coauthored over 280 peer-reviewed research articles and three books with over 30,000 citations in Google Scholar.