CAM Colloquium: Chi-Wang Shu (Brown University) - Bound-preserving high order schemes for hyperbolic equations: survey and recent developments


Frank H. T. Rhodes Hall 655


Abstract: Solutions to many hyperbolic equations have convex invariant regions, for example solutions to scalar conservation laws satisfy maximum principle, solutions to compressible Euler equations satisfy positivity-preserving property for density and internal energy, etc. It is, however, a challenge to design schemes whose solutions also honor such invariant regions. This is especially the case for high order accurate schemes. In this talk, we will first survey strategies in the literature to design high order bound-preserving schemes, including the general framework in constructing high order bound-preserving finite volume and discontinuous Galerkin schemes for scalar and systems of hyperbolic equations through a simple scaling limiter and a convex combination argument based on first order bound-preserving building blocks and various flux limiters to design high order bound-preserving finite difference schemes. We will then discuss a few recent developments, including high order bound-preserving schemes for relativistic hydrodynamics, high order discontinuous Galerkin Lagrangian schemes, high order discontinuous Galerkin methods for radiative transfer equations, and implicit bound-preserving schemes. Numerical tests demonstrating the good performance of these schemes will be reported. Bio: Chi-Wang Shu obtained his BS degree from the University of Science and Technology of China in 1982 and his PhD degree from the University of California at Los Angeles in 1986. He came to Brown University as an Assistant Professor in 1987, moving up to Associate Professor in 1992 and Full Professor in 1996. He was the Chair of the Division of Applied Mathematics between 1999 and 2005 and is now the Theodore B. Stowell University Professor of Applied Mathematics. His research interest includes high order finite difference, finite element and spectral methods for solving hyperbolic and other convection dominated partial differential equations, with applications to areas such as computational fluid dynamics, semi-conductor device simulations and computational cosmology. He served as the Managing Editor of Mathematics of Computation between 2002 and 2012, is now the the Chief Editor of Journal of Scientific Computing and serves in the editorial boards of several other journals. His honors include the First Feng Kang Prize of Scientific Computing in 1995 and the SIAM/ACM Prize in Computational Science and Engineering in 2007. He is a SIAM Fellow and an AMS Fellow, and an invited speaker at the International Congress of Mathematicians at Seoul in 2014.