In scientific and engineering computing, we encounter time-dependent partial differential equations (PDEs) frequently.  When designing high order schemes for solving these time-dependent PDEs, we often first develop semi-discrete schemes paying attention only to spatial discretizations and leaving time $t$ continuous. It is then important to have a high order time discretization to main the stability properties of the semi-discrete schemes.  In this talk we discuss several classes of high order time discretization, including the strong stability preserving (SSP) time discretization, which preserves strong stability from a stable spatial discretization with Euler forward, the implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the more stiff term (e.g. diffusion term in a convection-diffusion equation) implicitly and the less stiff term (e.g. the convection term in such an equation) explicitly, for which strong stability can be proved under the condition that the time step is upper-bounded by a constant under suitable conditions, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes.  Numerical examples will be given to demonstrate the performance of these schemes.

About the speaker

Professor Shu received his B.S. degree in Mathematics from the University of Science and Technology of China, Hefei, in 1982. In 1986 he received his Ph.D. degree in Mathematics from the Mathematics Department of the University of California at Los Angeles with Professor Stanley Osher as his advisor. He then spent a year at the Institute for Mathematics and Its Applications (IMA) in University of Minnesota as a post doctoral fellow. Since 1987 he has been with the Division of Applied Mathematics, Brown University, as an Assistant Professor (1987-91), Associate Professor (1992-96), Professor (1996- ), Chairman (1999-2005), and Theodore B. Stowell University Professor (2008- ). In 1992 he received the NASA Public Service Group Achievement Award for the pioneering work in Computational Fluid Dynamics as part of the ICASE algorithm team. In 1995 he received the first Feng Kang Prize of Scientific Computing from the Chinese Academy of Sciences. Since 2004 he has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge, Thomson Scientific Company. In 2007 he received the SIAM/ACM Prize in Computational Science and Engineering (SIAM/ ACM CSE Prize) "for the development of numerical methods that have had a great impact on scientific computing, including TVD temporal discretization, ENO and WENO finite difference schemes, discontinuous Galerkin methods, and spectral methods" (from the prize citation). See SIAM News Article. In 2009 he was selected as one of the first 183 members of the inaugural class of Fellows of the Society for Industrial and Applied Mathematics (SIAM). In 2012 he was selected as one of the inaugural class of Fellows of the American Mathematical Society (AMS). In 2014 he was an invited 45-minute speaker in the International Congress of Mathematicians (ICM) in Seoul. In 2019 he was elected as a Fellow of the Association for Women in Mathematics (AWM) "for his exceptional dedication and contribution to mentoring, supporting, and advancing women in the mathematical sciences; for his incredible role in supervising many women Ph.D.s, bringing them into the world of research to which he has made fundamental contributions, and nurturing their professional success" (from the election citation). In 2021 he received the SIAM John von Neumann Prize.