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A near-vacuum Euler state remains positive under symmetric HLL speeds

Example

Take $γ = 7 / 5$ , density $ρ = 10^{- 8}$ , pressure $p = 10^{- 12}$ , and velocities $u_{L} = - 1 / 100$ , $u_{R} = 1 / 100$ . The conserved states have $m_{L, R} = \mp 10^{- 10}$ and $E_{L, R} = 3 \times 10^{- 12}$ . Choose the symmetric HLL speed $a = 11 / 500$ at every face and Courant ratio $ν = Δ t / h = 125 / 11$ . For the four-cell periodic data $(U_{L}, U_{L}, U_{R}, U_{R})$ , one HLL forward-Euler update preserves positive density and pressure despite the very small initial values. The calculation illustrates a first-order positivity branch; it does not claim accuracy of the shock profile.

Facts & Assumptions

Given: The ideal-gas Euler flux and HLL update, with the parameters above. Verify the states, speed branch, CFL condition, and positivity after one update.

[F1]

Admissible states and pressure for one-dimensional ideal-gas Euler: $p = (γ - 1) (E - m^{2} / (2 ρ))$ and $c^{2} = γ p / ρ$ .

[F2]

Ideal-gas Euler states remain positive under Lax–Friedrichs flux splitting: if $a \geq ∣ u ∣ + c$ , then $U \pm F (U) / a \in G_{γ}$ .

[F3]

Euler positive states are convex and positive-margin sets are closed convex: $G_{γ}$ is convex.

[F4]

First-order HLL updates are averages of adjacent half-cell fan averages: under $2 ν a \leq 1$ , the HLL update is the mean of adjacent half-cell fan averages.

[F5]

Symmetric HLL speeds preserve ideal-gas density and internal-energy positivity: the symmetric HLL branch with $a \geq ∣ u ∣ + c$ and $2 ν a \leq 1$ preserves $G_{γ}$ .

[F6]

First-order finite-volume update using supplied HLL speeds: $U_{i}^{+} = U_{i} - ν ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2})$ .

[F7]

HLL two-wave approximate flux: the symmetric middle-branch HLL flux is used at every interface, with equal-state data substituted into its displayed formula.

Verification

1.1givenF1algebra

The kinetic energy density is $ρ u^{2} / 2 = 5 \times 10^{- 13}$ and $p / (γ - 1) = 5 / 2 \times 10^{- 12}$ , so $E = 3 \times 10^{- 12}$ . Hence [F1] gives the specified pressure and positive internal energy, so both states lie in $G_{γ}$ .

2.1givenF1F2F5step 1.1algebra

Their sound speed satisfies $c^{2} = 7 / 50000 < (3 / 250)^{2}$ , hence $∣ u ∣ + c < 1 / 100 + 3 / 250 = 11 / 500 = a$ . Also $ν a = 1 / 4$ , so $2 ν a = 1 / 2 \leq 1$ . The symmetric-speed hypotheses in [F2]–[F5] hold, with the positive admissibility verified in step 1.1.

3.1givenF1F2F3F4F6F7step 2.1algebra∎

At the discontinuity, substitution of $S_{L} = - a, S_{R} = a$ into [F7] and the HLL intermediate-state formula of [F4] gives $U_{*} = (U_{L} + F_{L} / a + U_{R} - F_{R} / a) / 2$ . Both split states are in $G_{γ}$ by [F2], so $U_{*} \in G_{γ}$ by [F3]. At equal-state faces, the same middle-state formula gives $U_{*} = U_{L}$ or $U_{R}$ . Substituting the shared face fluxes in [F6], or equivalently using [F4], yields $U_{i}^{+} = (3 / 4) U_{L} + (1 / 4) U_{*}$ and $U_{i + 1}^{+} = (3 / 4) U_{R} + (1 / 4) U_{*}$ . Convexity keeps both states in $G_{γ}$ , and [F1] gives positive density and pressure, using the CFL verification in step 2.1.

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Sources

D. I. Ketcheson, R. J. LeVeque, and M. J. del Razo, Riemann Problems and Jupyter Solutions (standard reference, not scraped)
X. Zhang and C.-W. Shu, On Positivity-Preserving High Order Discontinuous Galerkin Schemes for Compressible Euler Equations on Rectangular Meshes (standard reference, not scraped)