Theodore B. Stowell University Professor of Applied Mathematics, Division of Applied Mathematics, Brown University
Title: Stability of time discretizations for semi-discrete high order schemes for time-dependent PDEs
Abstract: When designing high order schemes for solving 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 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.
Biography: Professor Chi-Wang Shu received his B.S. degree in Mathematics from the University of Science and Technology of China in 1982 and his Ph.D. degree in Mathematics from University of California at Los Angeles in 1986. Since 1987 he has been at the Division of Applied Mathematics of Brown University, as Professor since 1996, as Chair of the Division of Applied Mathematics between 1999 and 2005, and as the Theodore B. Stowell University Professor since 2008. In 1995 he received the first Feng Kang Prize of Scientific Computing from the Chinese Academy of Sciences. In 2007 he received the SIAM/ACM Prize in Computational Science and Engineering 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). Professor Shu was selected in 2009 as an inaugural Fellow of the Society for Industrial and Applied Mathematics (SIAM), and in 2012 as an inaugural Fellow of the American Mathematical Society (AMS). In 2014 he was an Invited Speaker of the International Congress of Mathematicians (ICM) held in Seoul. In 2019 he was elected as a Fellow of the Association for Women in Mathematics "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). Currently Professor Shu is the Chief Editor of the Journal of Scientific Computing, the Chief Editor of Communications on Applied Mathematics and Computation, and a Co-Chief Editor, Editor or Associate Editor of several other research journals including Mathematics of Computation and Journal of Computational Physics