= Godunov numerical flux
{c}
{title2=$F_G(a,b)$}
= Godunov flux
{c}
{synonym}
For a <scalar conservation law> whose flux is a <convex function>, the <Godunov numerical flux> is the physical flux at the origin of the <entropy solution> of the <Riemann problem> between states $a,b$. It is $\min_{u\in[a,b]}f(u)$ for $a\le b$ and $\max_{u\in[b,a]}f(u)$ for $a>b$. For a <rarefaction wave>, the <characteristic speed> is monotone across its states, so the origin sees either an endpoint or a zero-speed minimizing state. For a <shock wave>, its speed is $(f(a)-f(b))/(a-b)$; the origin sees the left state if this speed is positive and the right state if negative, selecting the larger endpoint flux. Unlike the <Godunov flux>, the <Engquist-Osher flux> at reversed transsonic states may equal $f(a)+f(b)-f(s_*)$.
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