Entropy Stable Schemes for Nonlinear Conservation Laws: High Order Discontinuous Galerkin Methods and Reduced Order Modeling

Jesse Chan

Abstract: High order 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 apply it towards the construction of stable high order DG methods and reduced order models for the compresssible Euler and Navier-Stokes equations.

Biography: Jesse Chan is an assistant professor in the Department of Computational and Applied Mathematics at Rice University. He received his PhD in Computational and Applied Mathematics from the University of Texas at Austin in 2013 working on high order adaptive finite element methods for steady compressible fluid flows. He served as a Pfieffer postdoctoral instructor at Rice University from 2013-2015, and as a postdoctoral researcher at Virginia Tech from 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 of provably stable high order methods for wave propagation and fluid dynamics and their implementation on Graphics Processing Units (GPUs).

Website: https://caamweb.rice.edu/jesse-chan

Last Updated: May 28, 2020 - 4:06 pm