Positivity, invariant domains and entropy-stable systems fluxes

Let $m,N\ge 1$, let $\mathcal{A}\subseteq {\mathbb{R}}^m$ be a state domain, and let $\mathrm{\Phi }:{\mathcal{A}}^N\to {\mathcal{A}}^N$ be a one-step update on $N$ cell states. A set $K\subseteq \mathcal{A}$ is convex if $\theta U+(1-\theta )V\in K$ for every $U,V\in K$ and $\theta \in [0,1]$. It is a convex invariant domain for $\mathrm{\Phi }$ if it is convex and $\mathrm{\Phi }(K^N)\subseteq K^N$. The word invariant refers only to this stated one-step map and its stated admissible inputs; it asserts no invariance for a different flux, time integrator, or exact Riemann evolution.

Use the conservation-system convention of Strictly hyperbolic conservation systems and characteristic families: on an open convex state domain $\mathcal{A}\subseteq {\mathbb{R}}^m$, the flux is $F:\mathcal{A}\to {\mathbb{R}}^m$. A convex entropy pair is a pair $(\eta ,q)$ with $\eta \in C^2(\mathcal{A})$, $q\in C^1(\mathcal{A})$, positive-semidefinite Hessian $D^2\eta$, and $Dq(U)=D\eta (U)DF(U)(U\in \mathcal{A})\mathrm{.}$ The entropy variable is the column vector $v(U)=\mathrm{\nabla }\eta (U)$; gradients are represented as columns, so the compatibility equation above means $\mathrm{\nabla }q(U)=DF(U{)}^{T}v(U)$. The entropy potential is $\psi (U)=v(U{)}^{T}F(U)-q(U)$. This definition does not assert that every convex $\eta$ admits a flux $q$; the compatibility equation is part of the hypothesis.

Let $K\subseteq {\mathbb{R}}^m$ be closed and convex (convex in the sense of Convex invariant domains for a one-step update), let $J$ be a finite nonempty index set, let $w_j>0$ with ${\sum }_{j\in J}{w}_j=1$, and let input states $V_j\in {\mathbb{R}}^m$ have mean $\bar{V}={\sum }_j{w}_j{V}_j\in K$. The maximal convex scaling limiter returns the largest $\theta \in [0,1]$ for which every $V_j^{\theta }=\bar{V}+\theta (V_j-\bar{V})$ belongs to $K$, together with all the states $V_j^{\theta }$. Equivalently, form $S=\{\theta \in [0,1]:V_j^{\theta }\in K\text{ for every }j\in J\},$ and return ${\theta }_{\ast }:=\max S$ and $(V_j^{{\theta }_{\ast }}{)}_{j\in J}$. The mean is assumed admissible; implementation, cost, and any pointwise-in-cell property are not part of this definition.

Let $F:\mathcal{A}\to {\mathbb{R}}^m$ be the flux of the one-dimensional system in Strictly hyperbolic conservation systems and characteristic families. On a periodic uniform mesh of $N\ge 2$ cells of width $h>0$, let $U_i\in \mathcal{A}$ be the cell means and put $\nu =\mathrm{\Delta }t\mathrm{/}h\ge 0$. At each face $(i+1\mathrm{/}2)$ choose explicit speeds $S_{L,i+1\mathrm{/}2}<0<S_{R,i+1\mathrm{/}2}$ and evaluate the middle branch of the HLL flux HLL two-wave approximate flux on $(U_i,U_{i+1})$. The first-order HLL update is $U_i^{+}=U_i-\nu ({\widehat{F}}_{i+1\mathrm{/}2}-{\widehat{F}}_{i-1\mathrm{/}2}),i(\mathrm{m}\mathrm{o}\mathrm{d}N)\mathrm{.}$ The speeds are supplied data. Neither this definition nor later algebraic results assert that they enclose the wave speeds of an exact nonlinear Riemann solution.

Fix $\gamma >1$ and $\epsilon >0$, and use the state variables and internal energy density from Admissible states and pressure for one-dimensional ideal-gas Euler. Define the positive-margin set $K_{\gamma ,\epsilon }:=\{(\rho ,m,E):\rho \ge \epsilon ,E-\frac{m^2}{2\rho }\ge \epsilon \}\mathrm{.}$ Every member has positive density and positive pressure. Closedness and convexity are proved in Euler positive states are convex and positive-margin sets are closed convex ↗; those properties are not being inferred from this definition alone.

Fix $\gamma >1$. For $U=(\rho ,m,E)\in {\mathcal{G}}_{\gamma }$, let $e=E-m^2\mathrm{/}(2\rho )$, $p=(\gamma -1)e$, $u=m\mathrm{/}\rho$, and $c=\sqrt{\gamma p\mathrm{/}\rho }$. If $a\ge \mathrm{\mid }u\mathrm{\mid }+c$, then both split states $U_{\delta }=U+\delta F(U),\delta \in \{-1\mathrm{/}a,1\mathrm{/}a\},$ belong to ${\mathcal{G}}_{\gamma }$, where $F(U)=(m,m^2\mathrm{/}\rho +p,u(E+p))$. More precisely, with $x=1+\delta u$ their density and internal-energy density are ${\rho }_{\delta }=\rho x,e_{\delta }=ex-\frac{{\delta }^2{p}^2}{2\rho x}>0.$

Let $(\eta ,q)$ be a convex entropy pair on an open domain $\mathcal{A}$, with entropy variable $v$ and potential $\psi$ as in Convex entropy pairs, entropy variables, and entropy potentials for systems. A continuous two-point flux $F^{ec}:\mathcal{A}\times \mathcal{A}\to {\mathbb{R}}^m$ is entropy-conservative in Tadmor's sense if it is consistent and symmetric, $F^{ec}(U,U)=F(U)$, and satisfies the exact jump identity $(v(U_R)-v(U_L){)}^T{F}^{ec}(U_L,U_R)=\psi (U_R)-\psi (U_L)$ for every pair $U_L,U_R\in \mathcal{A}$. This definition asserts no existence for a prescribed entropy pair.

Fix $\gamma >1$ and $\epsilon >0$. The ideal-gas admissible set ${\mathcal{G}}_{\gamma }=\{(\rho ,m,E):\rho >0,E-\frac{m^2}{2\rho }>0\}$ and the positive-margin set $K_{\gamma ,\epsilon }$ from Positive-margin admissible states for ideal-gas Euler are convex, and $K_{\gamma ,\epsilon }$ is closed in ${\mathbb{R}}^3$.

Let $(\eta ,q)$ be an entropy pair with entropy variable $v$ and potential $\psi$, and let $F^{ec}$ be a consistent entropy-conservative flux as in Tadmor entropy-conservative two-point flux. For each pair $(U_L,U_R)\in {\mathcal{A}}^2$, choose a symmetric positive-semidefinite matrix $D(U_L,U_R)\in {\mathbb{R}}^{m\times m}$. The associated entropy-stable dissipative flux is $F^{es}(U_L,U_R):=F^{ec}(U_L,U_R)-\frac{1}{2}D(U_L,U_R)(v(U_R)-v(U_L))\mathrm{.}$ Positive semidefinite means $z^{T}D(U_L,U_R)z\ge 0$ for every $z\in {\mathbb{R}}^m$. No fully discrete time-step property is included.

Use the first-order HLL update of First-order finite-volume update using supplied HLL speeds. At a face with left and right states $U_L,U_R$, supplied speeds $S_L<0<S_R$, and physical flux values $F_L=F(U_L),F_R=F(U_R)$, define $U_{\ast }:=\frac{S_R{U}_R-S_L{U}_L-(F_R-F_L)}{S_R-S_L}\mathrm{.}$ If the one-sided CFL bounds $2\nu (-S_L)\le 1$ and $2\nu S_R\le 1$ hold, the averages of the HLL fan over the adjacent left and right half-cells are, respectively, $A^{-}=(1-2\nu (-S_L))U_L+2\nu (-S_L)U_{\ast },A^{+}=(1-2\nu S_R)U_R+2\nu S_R{U}_{\ast }\mathrm{.}$ For every cell $i$, the updated mean satisfies $U_i^{+}=\frac{1}{2}(A_{i-1\mathrm{/}2}^{+}+A_{i+1\mathrm{/}2}^{-}),$ where each face uses its own supplied speeds and intermediate state.

Let $K$ be a convex set in the state domain, and consider a periodic uniform mesh and one first-order HLL update as defined in First-order finite-volume update using supplied HLL speeds. Assume every cell mean $U_i\in K$. At every face impose the HLL fan-average hypothesis that the two adjacent half-cell averages $A^{-},A^{+}$ from First-order HLL updates are averages of adjacent half-cell fan averages lie in $K$, with that lemma's local CFL bounds. Then every updated mean $U_i^{+}$ belongs to $K$. A sufficient condition for the fan-average hypothesis is that every HLL intermediate state $U_{\ast }\in K$, because the stated CFL makes each half-cell average a convex combination of a cell state and $U_{\ast }$. The result does not assert a bound on exact nonlinear wave speeds.

Let $(\eta ,q)$ be a convex entropy pair on an open set $\mathcal{A}$, with entropy variable $v$ and potential $\psi$ as in Convex entropy pairs, entropy variables, and entropy potentials for systems. On a periodic mesh of $N\ge 2$ cells of width $\mathrm{\Delta }x>0$, let differentiable states $U_i(t)\in \mathcal{A}$ satisfy ${\dot{U}}_i=-\frac{F_{i+1\mathrm{/}2}^{es}-F_{i-1\mathrm{/}2}^{es}}{\mathrm{\Delta }x},$ where every face flux is the entropy-stable dissipative flux of Entropy-stable flux from positive-semidefinite entropy-variable dissipation. Define ${\widehat{q}}_{i+1\mathrm{/}2}:={\left(\frac{v_i+v_{i+1}}{2}\right)}^T{F}_{i+1\mathrm{/}2}^{es}-\frac{{\psi }_i+{\psi }_{i+1}}{2}\mathrm{.}$ Then each cell satisfies $\frac{d}{dt}\eta (U_i)+\frac{{\widehat{q}}_{i+1\mathrm{/}2}-{\widehat{q}}_{i-1\mathrm{/}2}}{\mathrm{\Delta }x}\le 0.$ Summing over the periodic mesh gives nonincrease of total discrete entropy. This is a semidiscrete result; it asserts no entropy inequality for a chosen fully discrete time integrator.

Fix $\gamma >1$, $\epsilon >0$, a finite nonempty node set $J$, positive weights $w_j$ summing to one, and candidate states $V_j\in {\mathbb{R}}^3$ with mean $\bar{V}={\sum }_j{w}_j{V}_j\in K_{\gamma ,\epsilon }$. Apply the maximal convex scaling limiter of Maximal convex scaling toward an admissible cell mean with $K=K_{\gamma ,\epsilon }$. Its output is well-defined, every limited node lies in $K_{\gamma ,\epsilon }$, and the weighted cell mean remains $\bar{V}$. Hence every limited node has positive density and positive pressure. If the unscaled nodes already lie in $K_{\gamma ,\epsilon }$, the maximal parameter is ${\theta }_{\ast }=1$. In particular, if smooth target states stay a distance at least $\delta >0$ from ${\mathbb{R}}^3\setminus K_{\gamma ,\epsilon }$ and each reconstruction node has error less than $\delta$, the limiter is inactive and preserves that reconstruction's accuracy. No positivity between finitely many nodes is claimed.

Fix $\gamma >1$ and a periodic uniform mesh. Suppose every cell state $U_i\in {\mathcal{G}}_{\gamma }$. Apply the first-order HLL update of First-order finite-volume update using supplied HLL speeds. At each face $(i+1\mathrm{/}2)$ choose a symmetric HLL branch $S_L=-a_{i+1\mathrm{/}2}$, $S_R=a_{i+1\mathrm{/}2}$ with $a_{i+1\mathrm{/}2}\ge \max (\mathrm{\mid }{u}_i\mathrm{\mid }+c_i,\mathrm{\mid }{u}_{i+1}\mathrm{\mid }+c_{i+1}),2\nu a_{i+1\mathrm{/}2}\le 1.$ Then one first-order HLL update preserves ${\mathcal{G}}_{\gamma }$; 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.