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

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.

35 pages, 10 figures

References in corpus (6)

• Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations
• Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations
• On discretely entropy conservative and entropy stable discontinuous Galerkin methods
• A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations
• Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
• An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

Cited by in corpus (23)

• Subcell limiting strategies for discontinuous Galerkin spectral element methods
• A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics
• On thermodynamically compatible finite volume schemes for continuum mechanics
• An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing
• Toward Discretization-Consistent Closure Schemes for Large Eddy Simulation Using Reinforcement Learning
• High-Order Methods for Hypersonic Flows with Strong Shocks and Real Chemistry
• A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations
• Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements
• A new thermodynamically compatible finite volume scheme for magnetohydrodynamics
• GALÆXI: Solving complex compressible flows with high-order discontinuous Galerkin methods on accelerator-based systems
• On error-based step size control for discontinuous Galerkin methods for compressible fluid dynamics
• Entropy-Stable Gauss Collocation Methods for Ideal Magneto-Hydrodynamics
• An entropy stable spectral vanishing viscosity for discontinuous Galerkin schemes: application to shock capturing and LES models
• A priori subcell limiting based on compact nonuniform nonlinear weighted schemes of high-order CPR method for hyperbolic conservation laws
• A priori subcell limiting approach for the FR/CPR method on unstructured meshes
• On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws
• A flux-differencing formulation with Gauss nodes
• Adaptive Compressible Smoothed Particle Hydrodynamics
• Multirate Time-Integration based on Dynamic ODE Partitioning through Adaptively Refined Meshes for Compressible Fluid Dynamics
• Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration
• An entropy stable discontinuous Galerkin method for the two-layer shallow water equations on curvilinear meshes
• Spectral Difference method with a posteriori limiting: II- Application to low Mach number flows
• Compact Runge-Kutta flux reconstruction methods for non-conservative hyperbolic equations