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4.2.2 Rankine-Hugoniot conditions

The Rankine-Hugoniot jump conditions (see Sect. 3.1.5) for the 1D Euler equations (40), describing a shock with speed \bgroup\color{DEFcolor}$s$\egroup become

$\displaystyle s   \left( \rho_\mathrm{r}
\right)$ $\textstyle =$ $\displaystyle \rho v_{_\mathrm{r}}
\rho v_{_\mathrm{l}}$ (200)
$\displaystyle s   \left( \rho v_{_\mathrm{r}}
\rho v_{_\mathrm{l}}
\right)$ $\textstyle =$ $\displaystyle \left( \rho v_{}v_{} + P\right)_\mathrm{r}
\left( \rho v_{}v_{} + P\right)_\mathrm{l}$ (201)
$\displaystyle s   \left( {\rho e_{\rm ik}}_\mathrm{r}
{\rho e_{\rm ik}}_\mathrm{l}
\right)$ $\textstyle =$ $\displaystyle \left( \left[ \rho e_{\rm ik}+ P\right] \; v_{} \right)_\mathrm{r}
\left( \left[ \rho e_{\rm ik}+ P\right] \; v_{} \right)_\mathrm{l}$ (202)
They can only be fulfilled for certain combinations of \bgroup\color{DEFcolor}$q_\mathrm{l}$\egroup and \bgroup\color{DEFcolor}$q_\mathrm{r}$\egroup.

An arbitrary Riemann problem typically causes more than one jump.