= Rankine-Hugoniot conditions
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{title2=$\sigma[u]=[f(u)]$}
{wiki=Rankine–Hugoniot_conditions}
= Rankine-Hugoniot condition
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{synonym}
The <Rankine-Hugoniot condition> expresses <conservation laws> across a moving discontinuity of a <weak solution>. Across a discontinuity $x=\sigma t$ in $u_t+f(u)_x=0$, with left and right states $u_-$ and $u_+$, conservation requires
$$
\sigma(u_+-u_-)=f(u_+)-f(u_-).
$$
Writing $[u]=u_+-u_-$ and $[f(u)]=f(u_+)-f(u_-)$ gives $\sigma[u]=[f(u)]$. The same relation holds componentwise for a <system of conservation laws>; in <compressible flow>, conservation of <mass>, <momentum> and <energy> supplies the jump conditions for a <shock wave>. Integrating the <weak solution> identity over a thin space-time region crossing the discontinuity yields this balance between the swept-up conserved quantity and the difference of fluxes.
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