Euler state, flux, positive set and numerical roles

Fix a constant $\gamma >1$. The one-dimensional ideal-gas Euler state is $U=(\rho ,m,E)\in {\mathcal{G}}_{\gamma }\subset {\mathbb{R}}^3$, where ${\mathcal{G}}_{\gamma }=\{(\rho ,m,E):\rho >0,E-\frac{m^2}{2\rho }>0\}\mathrm{.}$ For $U\in {\mathcal{G}}_{\gamma }$, define the pressure, velocity, sound speed, and flux by $p(U)=(\gamma -1)(E-\frac{m^2}{2\rho }),u(U)=\frac{m}{\rho },c(U)=\sqrt{\frac{\gamma p(U)}{\rho }},F(U)=(m,\frac{m^2}{\rho }+p(U),u(U)(E+p(U)))\mathrm{.}$ Thus ${\mathcal{G}}_{\gamma }$ is exactly the set with positive density and positive pressure; every displayed map is defined on that open set.

For a periodic uniform mesh with $N\ge 2$ cells, width $\mathrm{\Delta }x>0$, and time step $\mathrm{\Delta }t\ge 0$, index cells by $i\in \mathbb{Z}\mathrm{/}N\mathbb{Z}$ and put $\lambda =\mathrm{\Delta }t\mathrm{/}\mathrm{\Delta }x$. Given cell states $U_i\in {\mathcal{G}}_{\gamma }$, a shared-face flux is one vector ${\widehat{F}}_{i+1\mathrm{/}2}\in {\mathbb{R}}^3$ assigned to each face and used in both adjacent cell balances. When produced by a two-state numerical flux $\widehat{F}(U_L,U_R)$, it is consistent if $\widehat{F}(U,U)=F(U)$ for every $U\in {\mathcal{G}}_{\gamma }$. The associated one-step update in conservative finite-volume form is $U_i^{+}=U_i-\lambda ({\widehat{F}}_{i+1\mathrm{/}2}-{\widehat{F}}_{i-1\mathrm{/}2})\mathrm{.}$ This shared-face form means that the same face vector appears with opposite signs in the two neighboring cell updates. A step is admissible when every output state $U_i^{+}$ belongs to ${\mathcal{G}}_{\gamma }$; this is a property to prove under specific flux and time-step hypotheses, not part of the update definition.

When the HLL middle branch is used at a face, its supplied speeds satisfy $S_L<0<S_R$ and the flux is ${\widehat{F}}_{\mathrm{H}\mathrm{L}\mathrm{L}}(U_L,U_R)=\frac{S_{R}F(U_L)-S_{L}F(U_R)+S_L{S}_R(U_R-U_L)}{S_R-S_L}\mathrm{.}$ The speeds are supplied data; their appearance in this formula does not assert that they enclose the waves of an exact nonlinear Riemann solution. The ratio $\lambda$ is the time step divided by cell width and is not a physical viscosity.

For a specified convex entropy pair, write $v$ for its entropy variable and $F^{ec}$ for a consistent entropy-conservative two-point flux. If $D$ is a symmetric positive-semidefinite matrix, meaning $z^{T}Dz\ge 0$ for every vector $z$, and $F^{ec}$ satisfies $(v(U_R)-v(U_L){)}^T{F}^{ec}(U_L,U_R)=\psi (U_R)-\psi (U_L),$ where $\psi$ is the corresponding entropy potential, the entropy-stable dissipative flux is $F^{es}(U_L,U_R)=F^{ec}(U_L,U_R)-\frac{1}{2}D(U_L,U_R)(v(U_R)-v(U_L))\mathrm{.}$ Here $v=\mathrm{\nabla }\eta$ for the chosen entropy $\eta$, and the entropy-conservative flux satisfies the corresponding entropy-variable jump identity. This is a separate flux condition; it includes no fully discrete time-step entropy inequality. Conservation, admissibility, entropy stability, and convergence therefore name different properties and require their own hypotheses.

The following schematic view shows the orientation of one shared face flux. Positive flux points to the right; the same vector leaves cell $i$ and enters cell $i+1$. The diagram records the indexing convention only; the update formula supplies the definition used in proofs.

\begin{tikzpicture}[x=1.15cm,y=1cm]
  \draw (0,0) rectangle (2,0.7);
  \draw (2,0) rectangle (4,0.7);
  \draw (4,0) rectangle (6,0.7);
  \node at (1,0.35) {$U_{i-1}$};
  \node at (3,0.35) {$U_i$};
  \node at (5,0.35) {$U_{i+1}$};
  \draw[->] (1.55,1.1) -- (2.45,1.1);
  \draw[->] (3.55,1.1) -- (4.45,1.1);
  \node at (2,1.4) {$\widehat F_{i-1/2}$};
  \node at (4,1.4) {$\widehat F_{i+1/2}$};
\end{tikzpicture}

Admissible states and pressure for one-dimensional ideal-gas Euler — fixes the one-dimensional Euler state, pressure, velocity, sound speed, flux, and admissible-state set.

HLL two-wave approximate flux — fixes the three-branch HLL flux and treats its wave speeds as supplied data.

First-order finite-volume update using supplied HLL speeds — defines the periodic first-order HLL update and the ratio of time step to cell width.

Entropy-stable flux from positive-semidefinite entropy-variable dissipation — fixes the entropy-variable dissipation formula and states that it has no fully discrete time-step property.

De Sterck and Ullrich, Chapter 3 — independent course-note treatment of finite-volume cell averages, shared fluxes, and discrete conservation.

Tadmor, §§4.1–4.3 and 5 — full chapter treatment of entropy variables, dissipative fluxes, and fully discrete analysis.

Ketcheson, LeVeque, and del Razo, Euler approximate-solver chapter — full chapter treatment of HLL intermediate states and supplied speeds.