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Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equations

Zelalem Arega Worku, David W. Zingg

TL;DR

The paper develops a diagonal-$ extsf{E}$ multidimensional SBP-SAT framework that extends the entropy-split discretization to element-type unstructured grids for the Euler and compressible Navier–Stokes equations. It demonstrates high-order accuracy and entropy conservation for the Euler system on periodic curved grids, and proposes a viscous SAT scheme that yields entropy stability for NS when heat fluxes are neglected, while also enabling a locally conservative hybrid scheme to handle discontinuities. A matrix-type interface dissipation operator is introduced to ensure entropy dissipation across shocks, and a fully discrete RRK4 time integrator preserves entropy behavior. Numerical experiments (shock-tube, isentropic vortex, manufactured solutions, Taylor-Green vortex) show competitive accuracy and significant efficiency gains relative to Hadamard-form discretizations, with robustness improvements through the hybrid approach and interface dissipation. The work provides a practical, scalable path toward efficient high-order entropy-stable discretizations on unstructured meshes for compressible flow simulations.

Abstract

High-order Hadamard-form entropy stable multidimensional summation-by-parts discretizations of the Euler and compressible Navier-Stokes equations are considerably more expensive than the standard divergence-form discretization. In search of a more efficient entropy stable scheme, we extend the entropy-split method for implementation on unstructured grids and investigate its properties. The main ingredients of the scheme are Harten's entropy functions, diagonal-$ \mathsf{E} $ summation-by-parts operators with diagonal norm matrix, and entropy conservative simultaneous approximation terms (SATs). We show that the scheme is high-order accurate and entropy conservative on periodic curvilinear unstructured grids for the Euler equations. An entropy stable matrix-type interface dissipation operator is constructed, which can be added to the SATs to obtain an entropy stable semi-discretization. Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta method. Entropy stable viscous SATs, applicable to both the Hadamard-form and entropy-split schemes, are developed for the compressible Navier-Stokes equations. In the absence of heat fluxes, the entropy-split scheme is entropy stable for the compressible Navier-Stokes equations. Local conservation in the vicinity of discontinuities is enforced using an entropy stable hybrid scheme. Several numerical problems involving both smooth and discontinuous solutions are investigated to support the theoretical results. Computational cost comparison studies suggest that the entropy-split scheme offers substantial efficiency benefits relative to Hadamard-form multidimensional SBP-SAT discretizations.

Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equations

TL;DR

The paper develops a diagonal- multidimensional SBP-SAT framework that extends the entropy-split discretization to element-type unstructured grids for the Euler and compressible Navier–Stokes equations. It demonstrates high-order accuracy and entropy conservation for the Euler system on periodic curved grids, and proposes a viscous SAT scheme that yields entropy stability for NS when heat fluxes are neglected, while also enabling a locally conservative hybrid scheme to handle discontinuities. A matrix-type interface dissipation operator is introduced to ensure entropy dissipation across shocks, and a fully discrete RRK4 time integrator preserves entropy behavior. Numerical experiments (shock-tube, isentropic vortex, manufactured solutions, Taylor-Green vortex) show competitive accuracy and significant efficiency gains relative to Hadamard-form discretizations, with robustness improvements through the hybrid approach and interface dissipation. The work provides a practical, scalable path toward efficient high-order entropy-stable discretizations on unstructured meshes for compressible flow simulations.

Abstract

High-order Hadamard-form entropy stable multidimensional summation-by-parts discretizations of the Euler and compressible Navier-Stokes equations are considerably more expensive than the standard divergence-form discretization. In search of a more efficient entropy stable scheme, we extend the entropy-split method for implementation on unstructured grids and investigate its properties. The main ingredients of the scheme are Harten's entropy functions, diagonal- summation-by-parts operators with diagonal norm matrix, and entropy conservative simultaneous approximation terms (SATs). We show that the scheme is high-order accurate and entropy conservative on periodic curvilinear unstructured grids for the Euler equations. An entropy stable matrix-type interface dissipation operator is constructed, which can be added to the SATs to obtain an entropy stable semi-discretization. Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta method. Entropy stable viscous SATs, applicable to both the Hadamard-form and entropy-split schemes, are developed for the compressible Navier-Stokes equations. In the absence of heat fluxes, the entropy-split scheme is entropy stable for the compressible Navier-Stokes equations. Local conservation in the vicinity of discontinuities is enforced using an entropy stable hybrid scheme. Several numerical problems involving both smooth and discontinuous solutions are investigated to support the theoretical results. Computational cost comparison studies suggest that the entropy-split scheme offers substantial efficiency benefits relative to Hadamard-form multidimensional SBP-SAT discretizations.
Paper Structure (31 sections, 17 theorems, 132 equations, 8 figures, 4 tables)

This paper contains 31 sections, 17 theorems, 132 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SBP operators and notation
  4. Governing equations
  5. Homogeneity
  6. Symmetrization
  7. Continuous entropy stability analysis
  8. Entropy-split discretization of the Euler equations
  9. High-order accuracy and consistency
  10. Entropy conservation
  11. Primary conservation error
  12. Entropy conservative hybrid scheme
  13. Interface dissipation
  14. Extension to the compressible Navier-Stokes equations
  15. Entropy stable viscous SATs
  16. ...and 16 more sections

Key Result

Lemma 1

Consider a function, $\bm{\mathcal{F}} = \bm{\mathcal{F}}\left(\bm{\mathcal{U}},\bm{\mathcal{V}}\right)$, that is degree $n$ homogeneous in its first argument, and let $\mathsf{A}=\frac{\partial\bm{\mathcal{F}}}{\partial\bm{\mathcal{U}}}$. Then, the the following identity holds, Furthermore, if $\bm{\mathcal{F}} = \bm{\mathcal{F}}\left(\bm{\mathcal{U}}\right)$, then, in addition to eq:eq1 prop ho

Figures (8)

  • Figure 1: Density profile for the Sod shock tube problem at $t=0.2$ using the locally conservative hybrid scheme and the nonconservative entropy-split scheme.
  • Figure 2: Change in entropy for the Sod shock tube problem at $t=0.2$ using the hybrid and entropy-split schemes with and without dissipation.
  • Figure 3: Density profile for the Shu-Osher shock tube problem at $t=1.8$ using the locally conservative hybrid scheme and the nonconservative entropy-split scheme.
  • Figure 4: Density profile for the Shu-Osher shock tube problem at $t=1.8$ using the locally conservative hybrid scheme.
  • Figure 5: Long time entropy conservation and error behavior of the entropy-split method applied for the two dimensional isentropic vortex problem using $n_e = 800$ curved elements and $\text{CFL} = 0.5$.
  • ...and 3 more figures

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • ...and 27 more