Past Event: Oden Institute Seminar

Efficient structure preserving schemes for complex nonlinear systems

Jie Shen, Professor, Department of Mathematics, Purdue University

Abstract

Solutions for a large class of partial differential equations (PDEs) arising from sciences and engineering applications  are required to be positive 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 discretizations.

Biography

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 University) at Orsay in 1987. Before joining the Purdue Faculty in Fall 2002, he was a faculty member at Penn State University (1991-2001) and University of Central Florida (2001-2002). 

Since Jan. 2012 he serves as the Director of Center for Computational and Applied Mathematics at Purdue University.  He is a recipient of the Fulbright “Research Chair” Award in 2008 and the Inaugural Research Award of the College of Science at Purdue University in 2013, and  an elected Fellow of AMS and SIAM.

His main research interests are numerical analysis, spectral methods and scientific computing with applications in computational fluid dynamics and materials science.