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Symmetric HLL speeds preserve ideal-gas density and internal-energy positivity

Statement

Fix $γ > 1$ and a periodic uniform mesh. Suppose every cell state $U_{i} \in G_{γ}$ . Apply the first-order HLL update of First-order finite-volume update using supplied HLL speeds. At each face $(i + 1 / 2)$ choose a symmetric HLL branch $S_{L} = - a_{i + 1 / 2}$ , $S_{R} = a_{i + 1 / 2}$ with $a_{i + 1 / 2} \geq \max (∣ u_{i} ∣ + c_{i}, ∣ u_{i + 1} ∣ + c_{i + 1}), 2 ν a_{i + 1 / 2} \leq 1.$ Then one first-order HLL update preserves $G_{γ}$ ; in particular, every updated density and pressure is strictly positive. This is the symmetric local Lax–Friedrichs branch with a stated speed bound, not a claim that arbitrary HLL speed estimates are valid or that the bound is necessary.

Facts & Assumptions

Given: The one-dimensional ideal-gas Euler flux, positive initial states, symmetric speeds satisfying the displayed bound, and the HLL update. Prove positivity after one step.

[F1]

Admissible states and pressure for one-dimensional ideal-gas Euler: $G_{γ}$ is the set $ρ > 0$ , $E - m^{2} / (2 ρ) > 0$ ; $p = (γ - 1) (E - m^{2} / (2 ρ))$ , $c^{2} = γ p / ρ$ , and $F = (m, m^{2} / ρ + p, u (E + p))$ .

[F2]

HLL two-wave approximate flux: on the middle branch, $\hat{F} = (S_{R} F_{L} - S_{L} F_{R} + S_{L} S_{R} (U_{R} - U_{L})) / (S_{R} - S_{L})$ .

[F3]

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

[F4]

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

[F5]

First-order HLL updates are averages of adjacent half-cell fan averages: the intermediate state is $U_{*} = (S_{R} U_{R} - S_{L} U_{L} - (F_{R} - F_{L})) / (S_{R} - S_{L})$ and the update is the average of its adjacent half-cell fan averages.

[F6]

First-order HLL update preserves a convex invariant domain under admissible fan averages: a first-order HLL update preserves a convex set if its cell states and HLL intermediate states belong to that set and the half-cell CFL bounds hold.

Proof

1.1givenF1F3choose

Fix a face with states $U_{L}, U_{R}$ and speed $a$ . By [F3] applied to $U_{L}$ and $U_{R}$ , both $U_{L} + F_{L} / a$ and $U_{R} - F_{R} / a$ lie in $G_{γ}$ .

2.1givenF2F4F5step 1.1algebra

Substituting $S_{L} = - a$ , $S_{R} = a$ into [F2] and using the intermediate-state formula in [F5] gives $U_{*} = \frac{1}{2} (U_{L} + F_{L} / a) + \frac{1}{2} (U_{R} - F_{R} / a)$ . The two weights are nonnegative and sum to one, so [F4] implies $U_{*} \in G_{γ}$ .

3.1givenF1F6step 2.1algebra∎

The assumed speed bound gives $a > 0$ and the displayed CFL gives $2 ν a \leq 1$ , so both HLL half-cell averages have nonnegative convex-combination coefficients. Thus every HLL intermediate state and cell state lies in $G_{γ}$ , and [F6] yields $U_{i}^{+} \in G_{γ}$ for each cell. By [F1], this means positive density and pressure. The proof makes no claim for the other HLL branches or for fully discrete entropy.

Depends on

Used by

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

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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)