Finite-difference compatible entropy-conserving schemes for the compressible Euler equations

Abstract

This paper introduces a family of entropy-conserving finite-difference discretizations for the compressible flow equations. In addition to conserving the primary quantities of mass, momentum, and total energy, the methods also preserve kinetic energy and pressure equilibrium. The schemes are based on finite-difference (FD) representations of the logarithmic mean, establishing and leveraging a broader link between linear and nonlinear two-point averages and FD forms. The schemes are locally conservative due to the summation-by-parts property and therefore admit a local flux form, making them applicable also in finite-volume and finite-element settings. The effectiveness of these schemes is validated through various test cases (1D Sod shock tube, 1D density wave, 2D isentropic vortex, 3D Taylor-Green vortex) that demonstrate exact conservation of entropy along with conservation of the primary quantities and preservation of pressure equilibrium.

37 pages, 13 figures

References in corpus (20)

• Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations
• Numerically stable formulations of convective terms for turbulent compressible flows
• Summation-By-Parts Operators and High-Order Quadrature
• A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes
• Comparison of some Entropy Conservative Numerical Fluxes for the Euler Equations
• Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization
• On thermodynamically compatible finite volume schemes for continuum mechanics
• Global and local conservation of mass, momentum and kinetic energy in the simulation of compressible flow
• Formulation of Entropy-Stable schemes for the multicomponent compressible Euler equations
• Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes
• High-Order Methods for Hypersonic Flows with Strong Shocks and Real Chemistry
• Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements
• 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
• Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part I: The one-dimensional case
• Asymptotically entropy-conservative and kinetic-energy preserving numerical fluxes for compressible Euler equations
• Efficient Entropy-Stable Discontinuous Spectral-Element Methods Using Tensor-Product Summation-by-Parts Operators on Triangles and Tetrahedra
• 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
• Entropy conservative discretization of compressible Euler equations with an arbitrary equation of state