= Sonic-point splitting of a convex Engquist-Osher flux
{title2=$F(a,b)=f(s_*)+f^+(a)+f^-(b)$}
For a flux that is a differentiable <convex function> with unique minimum at $s_*$, define $f^+(u)=f(u)-f(s_*)$ for $u>s_*$ and zero otherwise, and $f^-(u)=f(u)-f(s_*)$ for $u<s_*$ and zero otherwise. The <Engquist-Osher flux> is $F(a,b)=f(s_*)+f^+(a)+f^-(b)$. Thus transsonic <rarefaction wave> states $a\le s_*\le b$ give $F=f(s_*)$, while reversed transsonic states give $F=f(a)+f(b)-f(s_*)$. This follows by integrating the positive and negative parts of $f'$.
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