A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler Equations

TL;DR

This paper introduces a provably entropy stable hybrid shock capturing method for high order split form DG schemes applied to the compressible Euler equations, enabling seamless blending of low and high order discretizations with proven stability.

Contribution

It presents a novel hybrid low order/high order entropy stable scheme with a blending strategy and shock indicator, extendable to 3D unstructured curvilinear meshes, validated through multiple shock test cases.

Findings

Proven entropy stability for blended schemes.

Demonstrated convergence and conservation on curvilinear grids.

Effective shock capturing with minimal parameter tuning.

Abstract

The main result in this paper is a provably entropy stable shock capturing approach for the high order entropy stable DGSEM based on a hybrid blending with a subcell low order variant. Since it is possible to rewrite a high order SBP operator into an equivalent conservative finite volume form, we were able to design a low order scheme directly with the LGL nodes that is compatible to the discrete entropy analysis used for the proof of the entropy stable DGSEM. Furthermore, we present a hybrid low order/high order discretisation where it is possible to seamlessly blend between the two approaches, while still being provably entropy stable. We are able to extend the approach to three spatial dimensions on unstructured curvilinear hexahedral meshes. We validate our theoretical findings and demonstrate convergence order for smooth problems, conservation of the primary quantities and discrete entropy stability for an arbitrary blending on curvilinear grids. In practical simulations, we connect the blending factor to a local troubled element indicator that provides the control of the amount of low order dissipation injected into the high order scheme. We modified an existing shock indicator, which is based on the modal polynomial representation, to our provably stable hybrid scheme. The aim is to reduce the impact of the parameters as good as possible. We describe our indicator in detail and demonstrate its robustness in combination with the hybrid scheme, as it is possible to compute all the different test cases without changing the indicator. The test cases include e.g. the double Mach reflection setup, forward and backward facing steps with shock Mach numbers up to 100. The proposed approach is relatively straight forward to implement in an existing entropy stable DGSEM code as only modifications local to an element are necessary.