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. It is of critical importance that their numerical approximations preserve the positivity/bound at the discrete level, as violation of the positivity/bound preserving may render the discrete problems ill posed.

I will review the existing approaches for constructing positivity/bound preserving  schemes, and then present  several efficient and accurate approaches which are relative easy to implement and can be combined with most spatial discretization.

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 the Purdue Faculty in Fall 2002, he served as Professor of Mathematics at Penn State University and University of Central Florida.  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.

He serves on editorial boards for several leading international research journals, and has authored/coauthored over 200 peer-reviewed research articles and two books with nearly 20,000 citations in Google Scholar.

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