Past Event: Oden Institute Seminar

Joint ASE/Oden Institute Seminar: High order entropy stable discretizations for nonlinear conservation laws

Jesse Chan, Assistant Professor, Department of Computational Applied Mathematics and Operations Research, Rice University

Abstract

High order numerical methods for computational fluid dynamics (CFD) provide a path towards high fidelity simulations of unsteady flows which are computationally intractable using traditional low order (e.g., first and second order) numerical methods. However, while high order methods are

attractive in theory, they often suffer from instability and a lack of robustness in practice, especially in the presence of shocks and under-resolved solution features. These instabilities are typically suppressed by adding numerical dissipation through slope limiting, artificial viscosity, or shock capturing; however, these techniques introduce heuristically tuned parameters which force users to choose between accuracy and stability. 

In this talk, we will introduce high order entropy stable discretizations for nonlinear conservation laws, which address instability and improve robustness by ensuring that physically relevant solutions satisfy a semi-discrete cell entropy inequality independently of the numerical resolution and in the presence of discretization errors (e.g., inexact quadrature or aliasing). Moreover, because these discretizations do not rely on numerical dissipation to maintain an entropy balance, they are formally high order accurate for arbitrary orders of approximation.

We begin by reviewing the construction of entropy stable high order nodal discontinuous Galerkin methods for systems of nonlinear conservation laws. We then describe approaches for enforcing physically relevant constraints on the solution (i.e., positivity of thermodynamic variables) with applications to problems in viscous compressible flow. Finally, we generalize entropy stable discretizations to more flexible "modal" formulations, which we apply towards the construction of entropy stable projection-based reduced order models.

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).