Entropy, invariant-domain and weak-limit hypotheses

The relevant numerical properties have separate hypotheses. A shared-face finite-volume update on a periodic uniform grid with $N\ge 2$ cells of common width $\mathrm{\Delta }x>0$ uses the same ratio $\lambda =\mathrm{\Delta }t\mathrm{/}\mathrm{\Delta }x$ in every cell, where $\mathrm{\Delta }t\ge 0$. Its balance is $U_i^{+}=U_i-\lambda ({\widehat{F}}_{i+1\mathrm{/}2}-{\widehat{F}}_{i-1\mathrm{/}2}),i\in \mathbb{Z}\mathrm{/}N\mathbb{Z},$ with one vector flux per face shared by the two neighboring cells. Finite telescoping conserves ${\sum }_i{U}_i$, and hence the total vector $\mathrm{\Delta }x{\sum }_i{U}_i$. That algebra alone gives no membership of the updated states in an admissible set, entropy inequality, or convergence result.

For the symmetric ideal-gas HLL/LLF branch, the stated one-step positivity result assumes $a\ge \max (\mathrm{\mid }{u}_L\mathrm{\mid }+c_L,\mathrm{\mid }{u}_R\mathrm{\mid }+c_R)$ and $2\lambda a\le 1$ at every face. For a general HLL middle branch, the convex invariant-domain result uses a different explicit hypothesis. Let $K$ be a convex state set, meaning every segment joining two members lies in $K$, and write $\nu =\mathrm{\Delta }t\mathrm{/}h$ for the time-step-to-cell-width ratio in this HLL formula. The approximate HLL fan at a face with $S_L<0<S_R$ has middle state $U_{\ast }=\frac{S_R{U}_R-S_L{U}_L-(F_R-F_L)}{S_R-S_L}\mathrm{.}$ This is the state in the two-wave approximation; its associated model flux is not assumed to be the physical flux $F(U_{\ast })$. The fan is $U_L$ to the left of $S_L$, $U_{\ast }$ between $S_L$ and $S_R$, and $U_R$ to the right of $S_R$. Its spatial averages over the left and right adjacent half-cells at time $\mathrm{\Delta }t$, respectively, are $A^{-}=(1-2\nu (-S_L))U_L+2\nu (-S_L)U_{\ast },A^{+}=(1-2\nu S_R)U_R+2\nu S_R{U}_{\ast },$ under the one-sided bounds $2\nu (-S_L)\le 1$ and $2\nu S_R\le 1$. The HLL invariant-domain result assumes both half-cell averages lie in $K$ at every face and every cell mean lies in $K$. Since each cell update is the average of its two adjacent half-cell averages, convexity then gives the updated mean in $K$. A sufficient way to verify the fan-average hypothesis is to show $U_L,U_R,U_{\ast }\in K$; the result does not infer $U_{\ast }\in K$ from supplied endpoint speed estimates alone.

The sketch fixes the fan and half-cell conventions used in those formulas. The horizontal axis is $x$, with $x=0$ at the face. The vertical arrow at the left marks increasing time $t$ and is offset for readability; the label $0$ marks the face at $t=0$. The wave lines have supplied speeds $S_L<0<S_R$, with their slopes representing $dx\mathrm{/}dt$. The dashed verticals mark the adjacent cell centers at $-h\mathrm{/}2$ and $h\mathrm{/}2$. The one-sided CFL bounds keep each wave inside its half-cell through time $\mathrm{\Delta }t$. The proof and averages are the displayed algebraic formulas, not the drawing.

\begin{tikzpicture}[x=2cm,y=1.15cm]
  \draw[->] (-1.35,0) -- (1.45,0) node[right] {$x$};
  \draw[->] (-1.3,0) -- (-1.3,1.2) node[above] {$t$};
  \draw[dashed] (-0.5,0) -- (-0.5,1.05);
  \draw[dashed] (0.5,0) -- (0.5,1.05);
  \draw[thick] (0,0) -- (-0.34,1);
  \draw[thick] (0,0) -- (0.34,1);
  \node at (-0.95,0.7) {$U_L$};
  \node at (0,0.68) {$U_*$};
  \node at (0.95,0.7) {$U_R$};
  \node[below] at (-0.5,0) {$-h/2$};
  \node[below] at (0.5,0) {$h/2$};
  \node[above right] at (-0.34,1) {$S_L$};
  \node[above left] at (0.34,1) {$S_R$};
  \node[below right] at (0,0) {$0$};
  \node[left] at (-1.3,1) {$\Delta t$};
\end{tikzpicture}

The entropy-stable dissipative-flux definition specifies a consistent entropy-conservative flux with positive-semidefinite entropy-variable dissipation, and explicitly supplies no fully discrete time-step inequality. Any semidiscrete entropy inequality for a selected semidiscrete evolution is a separate result; a fully discrete bound also requires an argument for the chosen time integrator. The ratio $\nu$ above is only the mesh time-step to width ratio and is not physical viscosity.

For the convex-scaling limiter, assume additionally that $K\subseteq {\mathbb{R}}^m$ is closed, meaning it contains every limit in ${\mathbb{R}}^m$ of a convergent sequence of its members; $K$ remains convex. The limiter applies to a finite nonempty set of node states $V_j\in {\mathbb{R}}^m$ with positive weights $w_j$ summing to one and an already admissible mean $\bar{V}={\sum }_j{w}_j{V}_j\in K$. It returns the largest $\theta \in [0,1]$ for which all scaled states $V_j^{\theta }=\bar{V}+\theta (V_j-\bar{V})$ belong to $K$. The feasible parameter set is closed in $[0,1]$ and contains $0$, so this maximum exists. Direct summation gives ${\sum }_j{w}_j{V}_j^{\theta }={\sum }_j{w}_j\bar{V}+\theta ({\sum }_j{w}_j{V}_j-{\sum }_j{w}_j\bar{V})=\bar{V}+\theta (\bar{V}-\bar{V})=\bar{V}\mathrm{.}$ Thus the construction preserves the supplied mean and constrains the chosen nodes; it has no admissibility branch for a mean outside $K$ and gives no pointwise-in-cell assertion between those nodes.

The conservative finite-volume weak-limit theorem is a separate scalar one-dimensional result on $\mathbb{R}$. It assumes $T>0$, a continuous scalar flux $f:\mathbb{R}\to \mathbb{R}$, initial data $u_0\in L^{\mathrm{\infty }}(\mathbb{R})\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathbb{R})$, a continuous consistent numerical flux $F:{\mathbb{R}}^2\to \mathbb{R}$, initial cell averages, $h_m,{\tau }_m\to 0$, and the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (for every family $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets, there is a function $f:\mathbb{N}\to {\bigcup }_{n\in \mathbb{N}}{X}_n$ with $f(n)\in X_n$ for every $n$). Strong local $L^1$ convergence of the reconstructed states and local uniform bounds on every bounded space-time cylinder through $t=0$ are also hypotheses. Under them the limit is a locally bounded scalar distributional weak solution. In its proof, ${\mathrm{A}\mathrm{C}}_{\omega }$ is used only in the local spatial-translation step of the face-flux convergence lemma; the discrete summation and limit passage require no further choice. The theorem assumes neither a CFL condition nor an entropy inequality, proves neither convergence nor compactness, and does not apply to the compressible Euler system.

Convex invariant domains for a one-step update — defines convexity as closure under line segments and states the one-step invariant-domain scope.

Shared-face flux telescoping conservation — proves periodic total-vector conservation by exact shared-face cancellation.

Symmetric HLL speeds preserve ideal-gas density and internal-energy positivity — supplies the symmetric ideal-gas branch and its endpoint speed and CFL hypotheses.

First-order HLL updates are averages of adjacent half-cell fan averages — defines the HLL intermediate state and exact adjacent half-cell average formulas.

First-order HLL update preserves a convex invariant domain under admissible fan averages — states the general convex-set result under the fan-average hypothesis.

Entropy-stable flux from positive-semidefinite entropy-variable dissipation — defines the entropy-variable dissipation flux and expressly excludes a fully discrete time-step property.

Maximal convex scaling toward an admissible cell mean — states the admissible-mean precondition and finite-node scaling output.

Conservative finite-volume strong-limit theorem — supplies the exact scalar weak-limit conclusion and its strong-convergence, local-bound, and no-CFL/no-entropy scope.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) — defines ${\mathrm{A}\mathrm{C}}_{\omega }$ as a choice function for every countable family of nonempty sets.

De Sterck and Ullrich, §§3.4.1–3.4.3 — full course notes treatment of finite-volume conservation and scalar-law context.

Zhang and Shu, §2 — full article treatment of Euler positivity, convex combinations, and finite-node limiting.

Tadmor, §§4.1–4.3 and 5 — full chapter treatment of entropy-variable dissipation and fully discrete analysis.

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