Comparison of some Entropy Conservative Numerical Fluxes for the Euler Equations

Contributions

• High-order accuracy is proven for consistent, symmetric numerical fluxes, and generalized SBP operators are used to investigate entropy conservation and stability in multiple dimensions and on simplices.The high-order result extends the earlier result limited to Tadmor’s flux.Source: §1 · p. 1
• A general procedure for constructing affordable entropy-conservative fluxes is presented and used to derive several new fluxes.Source: §1 · p. 1

Motivation

• Entropy conservation or stability alone does not ensure gas-dynamics robustness; positivity of density and internal energy or pressure may require dissipation or limiting.Such additions may reduce scheme order, and general convergence results remain unknown.Source: §1 · p. 2

Contributions

• Most entropy-conservative fluxes enhanced with local Lax-Friedrichs-type dissipation are proven to preserve nonnegative density under a non-vanishing CFL condition.Source: §1 · p. 2

• The two-dimensional Euler equations are introduced to fix notation and support later reference.Source: §2 · p. 2
• For a perfect gas, total energy splits into internal and kinetic energy, and air is assigned γ = 1.4 unless stated otherwise.Source: §2 · p. 2
• The mathematical entropy is U = −ϱs, with specific entropy defined from pressure and density; smooth solutions satisfy an entropy conservation law, while an entropy inequality is used as an admissibility criterion for weak solutions.Source: §2 · p. 2
• For positive density and pressure, the entropy U(u) is strictly convex.Source: §2 · p. 2

3.1 Summation-by-Parts Operators

• SBP operators combine discrete derivatives and quadrature with compatible boundary operators, yielding a discrete analogue of the fundamental theorem of calculus or divergence theorem.Nodal bases represent solutions, with nonlinear operations performed pointwise.Source: §3.1 · p. 3

3.2 Flux Differencing Form

• Flux differencing uses two-point numerical fluxes and SBP operators to build high-order semidiscretizations, with distinct volume fluxes for interior divergence terms and surface fluxes coupling elements.Source: §3.2 · p. 4

3.3 Order of Accuracy

• Smooth, consistent, symmetric volume fluxes yield volume terms with the same order of accuracy as the SBP derivative matrix.The result also extends to multiple dimensions when coordinate-direction contributions are summed.Source: §3.3 · p. 5

3.4 Entropy Conservation

• With consistent, symmetric, entropy-conservative volume fluxes and diagonal mass and boundary operators, entropy-conservative surface fluxes give entropy conservation across elements, while entropy-stable surface fluxes give entropy stability.Source: §3.4 · p. 7
• The multidimensional simplex extension relies on diagonal boundary operators that are not generally known for sufficiently accurate simplex SBP schemes, and discontinuities generally require additional dissipation.The resulting scheme is an entropy-stable, high-order baseline; adding boundary nodes may reduce efficiency.Source: §3.4 · p. 8

4 Entropy Conservative Fluxes

• Entropy conservation constrains the numerical flux through jumps in entropy variables and flux potentials, corresponding to a mean of the physical flux; integral means can be difficult to compute.Source: §4 · p. 8

4.1 Using

• The flux derived using the variables z is consistent and entropy conservative, but pressure enters its numerical density flux and may cause problems across pressure discontinuities.Source: §4.1 · p. 10

4.2 General Procedure to Construct Affordable Entropy Conservative Fluxes

• A general construction procedure expresses flux potentials and entropy variables in chosen variables, then rewrites their jumps as mean values multiplied by jumps.This approach is used to construct affordable entropy-conservative fluxes, including the flux of [3].Source: §4.2 · p. 10

4.3 Kinetic Energy Preservation

• Kinetic-energy preservation is proposed as a flux design criterion, but accepting fnum ϱ as a numerical pressure means the definition alone does not provide a useful construction criterion.The article uses the criterion but does not judge it or use it to prove convergence.Source: §4.3 · pp. 10–11

4.4 Using ϱ, v, β as Variables

• For this variable choice, the section introduces Chandrashekar [3]’s flux potential and entropy variables.The section begins by using the inverse of temperature.Source: §4.4 · p. 11

4.4.1 Variant 1

• Chandrashekar [3]’s variant 1 is entropy-conservative, consistent, and kinetic-energy-preserving.Its numerical pressure flux is also specified.Source: §4.4.1 · p. 12

4.4.2 Variant 2

• Chandrashekar [3]’s variant 2 is entropy-conservative, but its kinetic-energy-preserving status is uncertain because that property is not well-defined.The authors say it could possibly be considered kinetic-energy-preserving.Source: §4.4.2 · p. 13

4.5 Using ϱ, v, 1 p as Variables

• The ϱ, v, 1/p formulation yields fluxes that are entropy-conservative and consistent.The pressure influences the numerical density flux, leading to problems.Source: §4.5 · p. 15

4.6 Using ϱ, v, p as Variables

• The ϱ, v, p formulation also yields fluxes described as entropy-conservative and consistent.As in the preceding formulation, pressure influences the numerical density flux and leads to problems.Source: §4.6 · p. 17

4.7.1 Variant 1

• Variant 1 is entropy-conservative, consistent, and kinetic-energy-preserving, with the same numerical pressure flux as Chandrashekar [3]’s flux.The density flux is also discussed in comparison with the Chandrashekar flux.Source: §4.7.1 · p. 18

4.7.2 Variant 2

• Variant 2 is entropy-conservative, but its kinetic-energy-preserving status is uncertain because that property is not well-defined.The authors say it could possibly be considered kinetic-energy-preserving.Source: §4.7.2 · p. 20

4.8 Using ϱ, v, g−1(ϱ/p) as Variables

• Choosing ϱ, v, χ with ϱ/p = g(χ) permits a kinetic-energy-preserving flux whose density flux depends only on ϱ and v.Different expressions for the jump of log(ϱ/p) yield energy fluxes that produce entropy-conservative, kinetic-energy-preserving schemes.Source: §4.8 · pp. 20–21

4.8.1 Variant 1

• Variant 1 uses a mean value to construct a flux described as entropy-conservative, kinetic-energy-preserving, and consistent.The arithmetic and geometric means are adapted to the variable transformation.Source: §4.8.1 · p. 22

4.8.2 Variant 2

• Variant 2 likewise uses a mean value to construct a flux described as entropy-conservative, kinetic-energy-preserving, and consistent.A numerically stable procedure for computing the exponential mean is required.Source: §4.8.2 · pp. 22–23

4.9 Using Other Variables

• The authors leave other variable sets unexplored because they see no clear intuition for which choice is preferable; pressure dependence in the numerical density flux should be avoided.They cite Derigs, Winters, Gassner, and Walch [6] and point to Remark 6.2 and the numerical tests in section 7.Source: §4.9 · p. 23

5 Reversing the Role of Energy and Entropy

• The authors consider an entropy conservation law with an energy inequality, and for smooth solutions its entropy-conserving fluxes correspond to those for the usual Euler entropy condition.They state that exchanging another conserved quantity with entropy generally does not yield a convex quantity with the required properties.Source: §5 · p. 23
• The reversed formulation defines numerical energy fluxes from the entropy-variable average dotted with the numerical flux, minus the entropy-potential average.The resulting general momentum flux form is the one proposed by Jameson for a kinetic-energy-preserving flux in one dimension.Source: §5 · pp. 24–25

5.1 Using ϱ, v, T as Variables

• Using density, velocity, and temperature as variables, the paper expresses the entropy in terms of temperature and density.The stated relation is s = log RT − (γ − 1) log ϱ.Source: §5.1 · p. 25, §5.1.1 · p. 25

5.1.1 Variant 1

• Variant 1 yields fluxes described as consistent, entropy conservative, and kinetic-energy preserving for the Euler equations.The energy fluxes are computed from the average entropy variables dotted with the numerical flux, minus the average flux potential.Source: §5.1.1 · p. 26

5.2 Other Variables

• Other variable choices, including ϱ, v, χ with p/ϱ = g(χ), are possible but are not pursued because no preferable choice is apparent.The authors give no further comparison among these alternatives.Source: §5.2 · p. 27

6 Numerical Surface Fluxes / Riemann Solvers

• Surface numerical fluxes are important to stability and robustness, and in a first-order finite-volume setting they determine the method completely.They appear in the semidiscretization's surface terms, distinct from volume fluxes in its volume terms.Source: §6 · p. 27

6.1 Adding Dissipation to Entropy Conservative Fluxes

• An entropy-stable flux is formed by adding dissipation to an entropy-conservative central flux; the simplest stated choice is local Lax-Friedrichs-type dissipation.The paper also describes scalar and matrix dissipation forms based on entropy variables and eigenvalue magnitudes.Source: §6.1 · p. 28

6.2 Preserving Positivity of the Density

• Local Lax-Friedrichs-type dissipation preserves non-negativity of density for the specified entropy-conservative fluxes under the stated CFL condition, without requiring a vanishing time step as density approaches zero.This applies when the density flux has no pressure contribution; the scalar dissipation operator is also covered.Source: §6.2 · pp. 28–29

6.3 Preserving Positivity of the Pressure

• Positivity preservation for pressure or internal energy is more complicated than for density, and further investigation is needed for the numerical fluxes considered.Reversing entropy and energy roles is mentioned as a possible route to fluxes that preserve internal-energy positivity, but is not pursued.Source: §6.3 · p. 30

6.4 Suliciu Relaxation Solver

• The Suliciu relaxation flux is stated to preserve entropy stability and positivity of density and pressure, and to satisfy a maximum principle on specific entropy.It also resolves stationary contact discontinuities with zero x-velocity and constant pressure exactly.Source: §6.4 · p. 31

7.1 Isentropic Vortex; 7.6 Near Vacuum Rarefaction

• In the isentropic vortex test, entropy-stable formulations show no advantage, with density-error variations up to about 15% for p = 2 and p = 3; near vacuum, no finite-volume flux is clearly superior.Source: §7.1 · p. 32, Table 1 · p. 33, §7.6 · p. 45

7.2 Sod’s Shock Tube: Subcell Flux Differencing

• For Sod’s shock tube with subcell flux differencing, Suliciu beats LLF at adequate resolution, but volume fluxes show no consistent winner; central and Morinishi are unstable, and Ducros crashes at p = 5.Stable calculations using the other simple splittings show little error variance.Source: §7.2 · p. 33, Table 2 · p. 34, Table 3 · p. 35

7.3 Modified Version of Sod’s Shock Tube: Subcell Flux Differencing

• For modified Sod with subcell flux differencing, Suliciu again outperforms LLF at adequate resolution, with no clearly superior volume flux; stability varies with splitting and resolution.All calculations are stable at low resolution, but Morinishi with Suliciu and central with LLF initially blow up; Kennedy–Gruber and Pirozzoli also blow up at higher polynomial degrees.Source: §7.3 · p. 38, Table 6 · p. 39, Table 7 · p. 40

7.4 Sod’s Shock Tube: Finite Volume Setting; 7.5 Modified Version of Sod’s Shock Tube: Finite Volume Setting

• In finite-volume Sod tests, matrix dissipation and Suliciu perform equally well and have lower errors than other fluxes; modified Sod shows the same ranking, with little variance for matrix or LLF dissipation.There is also nearly no variance across solvers using LLF or scalar dissipation in the classical Sod test.Source: §7.4 · p. 43, Table 10 · p. 43, §7.5 · p. 43

7.7 Left Half of the Blast Wave Problem of Woodward and Colella; 7.9 Right Half of the Blast Wave Problem of Woodward and Colella

• For Woodward–Colella blast waves, pressure-dependent density fluxes are unstable; Suliciu performs best on the left half, while matrix dissipation and Suliciu do slightly better on the right.On the left half, matrix dissipation has similar errors to Suliciu until it crashes at 12,800 elements.Source: §7.7 · p. 46, Table 13 · p. 46, §7.9 · p. 48

7.8 Slowly Moving Contact Discontinuity

• For the slowly moving contact discontinuity, pressure-dependent density fluxes are unstable, scalar and LLF dissipation have similar errors, and Suliciu is superior, usually beating scalar dissipation.Source: §7.8 · p. 47, Table 14 · p. 47

7.10 Left Half of the Blast Wave Problem of Derigs, Winters, Gassner and Walch; 7.11 Right Half of the Blast Wave Problem of Derigs, Winters, Gassner and Walch

• In Derigs et al.’s blast-wave halves, pressure influence in the density flux is unstable, while other stable methods have similar errors; LLF suffices and specialized dissipation offers no improvement for Euler.For the left half, the problem requires very high resolution; the specially designed operator did not improve on LLF.Source: §7.10 · pp. 48–49, Table 16 · p. 49, §7.11 · p. 50

7.12 Another Blast Wave Problem

• The MHD-derived scalar and matrix dissipation operators [6] and produce negative pressures in the blast-wave test, while simple LLF dissipation remains stable.The authors note that [6] had reported LLF as less stable than those operators for MHD equations.Source: §7.12 · pp. 50–51, Figure 1 · p. 51
• Fluxes incorporating pressure in the density flux are unstable, whereas the remaining fluxes with LLF dissipation or Suliciu are stable and yield errors equal to two-digit precision.The comparison is reported for the tested blast-wave problem.Source: §7.12 · p. 51
• For one explicit Euler step, scalar and matrix dissipation yield negative pressure with no reasonable CFL condition, while LLF keeps pressure positive.The test uses the entropy-conservative flux (52) of Chandrashekar [3].Source: §7.12 · p. 51, Figure 1 · p. 51

7.13 Summary of the Numerical Results

• No entropy-conservative flux excluding pressure from the density flux is clearly superior; such volume fluxes improve robustness for discontinuities but are insufficient for strong shocks without shock capturing.The reported tests used no additional shock-capturing mechanisms, which the authors say will be needed in practice.Source: §7.13 · p. 51
• Local Lax–Friedrichs dissipation is more robust for pressure positivity than MHD-derived dissipation operators transferred to the Euler equations.This comparison concerns entropy-conservative fluxes that do not use pressure in the density flux.Source: §7.13 · p. 51

8 Summary and Conclusions

• The generalized SBP analysis extends high-order flux-differencing results to consistent, symmetric fluxes and considers multiple dimensions and simplex elements.The authors note that suitable simplex SBP operators may require more nodes per element and therefore be less efficient.Source: §8 · p. 53
• For discontinuous solutions, entropy-conservative volume fluxes produced more stable schemes than alternatives, but dissipative surface fluxes alone were insufficient for strong shocks.No additional shock-capturing mechanisms were used, although the authors say these will be needed in practice.Source: §8 · p. 53
• No tested volume flux was clearly superior in every case; smooth-solution tests favored some non-entropy-conservative fluxes, unlike the discontinuous-solution tests.The discontinuous tests found entropy-conservative volume fluxes more stable, though additional shock capturing was still needed for strong shocks.Source: §8 · p. 53
• Most entropy-conservative fluxes preserve non-negative density when enhanced with local Lax-Friedrichs-type dissipation.Pressure positivity with LLF dissipation was observed in all test cases but has not been proved analytically.Source: §8 · p. 53
• The LLF dissipation was more stable than the scalar operator in this comparison, while hybrid and matrix operators were generally less stable than the Suliciu solver.Hybrid and matrix operators yielded similar stability and accuracy to Suliciu in some problems.Source: §8 · p. 53
• Suliciu was the best solver in the comparison when implementation costs were considered, since it was second-cheapest after LLF and entropy-baseline fluxes were significantly more expensive.The implementation was not optimized in detail for every flux.Source: §8 · p. 53
• Whether entropy-conservative fluxes offer advantages over tested split forms, whether superior or cheaper fluxes exist, and general convergence results remain unresolved.Entropy stability implies L2 bounds only when density and pressure are strictly positive.Source: §8 · p. 53