On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows

Abstract

High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.

Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting

“…We define the left and right state variables at the boundary as . Furthermore, by discretizing the continuous and conservative integral terms with the generalized SBP operators, the high-order DGSEM scheme based on LGL solution points (DGSEM-LGL) [ 11 ] can be obtained where J is the Jacobian matrix of the mapping of local cells. contains the interpolation matrix of the interior and boundary of the form .…”

“…The high-order entropy-stable scheme based on LGL solution points (ESDGSEM-LGL) [ 11 ] is expressed as where is the numerical flux of the two-point entropy conservative displayed in Appendix A . The solution obtained by the ESDGSEM-LGL scheme has a physical meaning, in contrast to the DGSEM-LGL scheme.…”

“…The precision of the high-order scheme based on the LG solution points is also higher. Therefore, it is necessary to establish the high-order entropy-stable scheme on the LG solution points [ 11 ]. Instead of approximating the variables at the boundary directly by interpolation of the original variables, establishing the high-order entropy-stable scheme at the LG solution points requires converting the original variables into entropy variables and using boundary interpolation and L2 projection to re-estimate the variables at the boundary.…”

“…This process is defined as “entropy projected” [ 11 ]. Here, contains the conservative variable values on the cell interface and inside, and are the entropy variables values at the solution points.…”

“…The instability caused by shock waves occurs mainly because a high-order scheme generally adopts high-order distribution approximation, which tends to generate many “non-physically relevant” solutions. In response to this phenomenon, we mainly adopt the following methods: the entropy-stable scheme [ 11 ], the hybrid scheme [ 12 ], and the limiter [ 13 ].…”