Comparison of some Entropy Conservative Numerical Fluxes for the Euler Equations

Abstract

Entropy conservation and stability of numerical methods in gas dynamics have received much interest. Entropy conservative numerical fluxes can be used as ingredients in two kinds of schemes: Firstly, as building blocks in the subcell flux differencing form of Fisher and Carpenter (2013) and secondly (enhanced by dissipation) as numerical surface fluxes in finite volume like schemes. The purpose of this article is threefold. Firstly, the flux differencing theory is extended, guaranteeing high-order for general symmetric and consistent numerical fluxes and investigating entropy stability in a generalised framework of summation-by-parts operators applicable to multiple dimensions and simplex elements. Secondly, a general procedure to construct affordable entropy conservative fluxes is described explicitly and used to derive several new fluxes. Finally, robustness properties of entropy stable numerical fluxes are investigated and positivity preservation is proven for several entropy conservative fluxes enhanced with local Lax-Friedrichs type dissipation operators. All these theoretical investigations are supplemented with numerical experiments.

55 pages, 1 figure

Cited by in corpus (29)

• General Relaxation Methods for Initial-Value Problems with Application to Multistep Schemes
• Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization
• Fully-Discrete Explicit Locally Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations
• Entropy stable, robust and high-order DGSEM for the compressible multicomponent Euler equations
• Global and local conservation of mass, momentum and kinetic energy in the simulation of compressible flow
• Entropy Symmetrization and High-Order Accurate Entropy Stable Numerical Schemes for Relativistic MHD Equations
• Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes
• Entropy Stable Discontinuous Galerkin Methods for Balance Laws in Non-Conservative Form: Applications to the Euler Equations with Gravity
• Analysis and entropy stability of the line-based discontinuous Galerkin method
• High-Order Methods for Hypersonic Flows with Strong Shocks and Real Chemistry
• Numerical treatment of the energy equation in compressible flows simulations
• Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws
• Mimetic Properties of Difference Operators: Product and Chain Rules as for Functions of Bounded Variation and Entropy Stability of Second Derivatives
• Asymptotically entropy-conservative and kinetic-energy preserving numerical fluxes for compressible Euler equations
• Error Boundedness of Discontinuous Galerkin Methods with Variable Coefficients
• Entropy stable spectral collocation schemes for the 3-D Navier-Stokes equations on dynamic unstructured grids
• Efficient Entropy-Stable Discontinuous Spectral-Element Methods Using Tensor-Product Summation-by-Parts Operators on Triangles and Tetrahedra
• A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model
• Novel Pressure-Equilibrium and Kinetic-Energy Preserving fluxes for compressible flows based on the harmonic mean
• On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws
• Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods
• Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts
• Multirate Time-Integration based on Dynamic ODE Partitioning through Adaptively Refined Meshes for Compressible Fluid Dynamics
• Entropy conservative discretization of compressible Euler equations with an arbitrary equation of state
• Finite-difference compatible entropy-conserving schemes for the compressible Euler equations
• On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems II: Entropy Bounded Formulations for Nonlinear Conservation Laws
• Stability of step size control based on a posteriori error estimates
• Multiderivative time integration methods preserving nonlinear functionals via relaxation
• A Note on Numerical Fluxes Conserving a Member of Harten's One-Parameter Family of Entropies for the Compressible Euler Equations