High order numerical methods are known to be unstable when applied to nonlinear conservation laws with shocks and turbulence, and traditionally require additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality even in the presence of under-resolved solution features and inexact quadrature. The construction of entropy stable methods has traditionally relied on equivalences between “collocation” discontinuous Galerkin (DG) methods and summation-by-parts (SBP) finite differences. We present a more general framework for constructing entropy stable schemes based instead on a “modal” finite element formulation and demonstrate its flexibility for several numerical settings and nonlinear conservation laws governing fluid flows.
Jesse Chan is an assistant professor in the Department of Computational and Applied Mathematics at Rice University. After receiving his PhD from the University of Texas at Austin in 2013, he served as postdoc at Rice University (2013-2015) and at Virginia Tech (2015-2016) before returning to Rice as faculty in 2016. His research focuses on accurate and efficient numerical solutions of time-dependent partial differential equations. His recent work has focused on the construction and efficient implementation of provably stable high order methods for compressible fluid flow and wave propagation.
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