Entropy Stable Spectral Collocation Schemes for the Navier--Stokes Equations: Discontinuous Interfaces

Abstract

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation element methods of arbitrary order for the compressible Navier-Stokes equations. The new methods are similar to strong form, nodal discontinuous Galerkin spectral elements but conserve entropy for the Euler equations and are entropy stable for the Navier-Stokes equations. Shock capturing follows immediately by combining them with a dissipative companion operator via a comparison approach. Smooth and discontinuous test cases are presented that demonstrate their efficacy. Introduction.Next generation numerical algorithms for use in large eddy simulations (LES) and hybrid Reynolds-averaged Navier-Stokes (RANS)-LES simulations will undoubtedly rely on efficient high-order formulations. Although highorder techniques are well suited for LES, most lack robustness when the solution contains discontinuities or even underresolved physical features. Although a variety of stabilization techniques have been developed for second-order methods (e.g., total variation diminishing (TVD) limiters [35], and entropy stability [40]), extending these techniques to high-order formulations has been problematic. High-order essentially nonoscillatory (ENO) [20,36] and weighted ENO (WENO) [31,25] schemes provide a partial remedy to the problem; they achieve high-order design accuracy away from captured discontinuities and maintain sharp "nearly monotone" captured shocks. Unfortunately, nonoscillatory schemes experience instabilities in less than ideal circumstances (e.g., curvilinear mapped grids or expansion of flows into vacuum). Because nonoscillatory schemes are largely based on stencil biasing heuristics rather than stability analysis, there is little theory to guide further development efforts focused on

A Roadmap to Well Posed and Stable Problems in Computational Physics

“…The fully nonlinear theory, necessary for problems with discontinuities, is incomplete. Entropy estimates can be used to bound the solution, see for example [12,17,18], but neither uniqueness nor existence follows. In this paper we do not consider problems with discontinuities.…”

“…We stress that any discretization technique that can be formulated on SBP form such as for example finite difference [7,8], finite volume [9,10], spectral element [11,12], discontinuous Galerkin [13,14] and flux reconstruction schemes [15,16] will lead to the same analysis and principal results.…”

“…It has been shown previously that a weak imposition of well-posed boundary conditions for finite difference [7,8], finite volume [9,10], spectral element [11,12], discontinuous Galerkin [13,14] and flux reconstruction schemes [15,16] on summation-by-parts (SBP) form can lead to energy stability. We will show that the continuous analysis of well posed boundary conditions implemented with weak boundary procedures together with schemes on Summation-by-parts (SBP) form automatically leads to stability.…”

“…a roadmap for the whole computational chain. In order not to be tangled up in technical and yet unresolved theoretical difficulties, we do not deal with discontinuous problems, such as for example in [12,17,18], only sufficiently smooth solutions with related smooth and compatible data are considered. The subscripts t, x, y, z denotes partial differentiation with respect to time and space.…”

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

“…Recent work has appeared on the use of high-order discontinuous Galerkin (DG) approximations to create robust numerical methods for the solution of systems of conservation laws, e.g., [8,9,10]. These robust high-order DG methods may be derived from the perspective of mathematical entropy conservation, e.g.…”