DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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Explicit entropy-conservative two-point Euler flux

Definition

For two admissible states $U_{L}, U_{R} \in G$ , let $\overset{ˉ}{U} = (U_{L} + U_{R}) / 2$ , $Δ v = v (U_{R}) - v (U_{L})$ , and $Δ ψ_{j} = ψ_{j} (U_{R}) - ψ_{j} (U_{L})$ , where $v$ and $ψ_{j}$ are given by Strict convexity and entropy variables for the ideal-gas Euler flux. Define the symmetric two-point flux

$F_{j}^{e c} (U_{L}, U_{R}) = {\begin{cases} F_{j} (U_{L}), & Δ v = 0, \\ F_{j} (\overset{ˉ}{U}) + \frac{Δ ψ_{j} - Δ v^{T} F_{j} (\overset{ˉ}{U})}{∥ Δ v ∥^{2}} Δ v, & Δ v \neq 0. \end{cases}$

The norm is Euclidean in the $d + 2$ conservative-state coordinates. The convexity result The ideal-gas admissible state set is open and convex ensures $\overset{ˉ}{U} \in G$ .

Depends on

Used by

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Sources

Eitan Tadmor, Entropy Stable Schemes, Chapter 18, Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, 2016, complete chapter (standard reference, not scraped)
F. Ismail and P. L. Roe, Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks, Journal of Computational Physics 228 (2009), 5410–5436, complete paper (standard reference, not scraped)