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 $s$ become

$$s \left( \rho_\mathrm{r} - \rho_\mathrm{l} \right) = \rho v_{_\mathrm{r}} - \rho v_{_\mathrm{l}}$$ (200)

$$s \left( \rho v_{_\mathrm{r}} - \rho v_{_\mathrm{l}} \right) = \left( \rho v_{}v_{} + P\right)_\mathrm{r} - \left( \rho v_{}v_{} + P\right)_\mathrm{l}$$ (201)

$$s \left( {\rho e_{\rm ik}}_\mathrm{r} - {\rho e_{\rm ik}}_\mathrm{l} \right) = \left( \left[ \rho e_{\rm ik}+ P\right] \; v_{} \right)_\mathrm{r} - \left( \left[ \rho e_{\rm ik}+ P\right] \; v_{} \right)_\mathrm{l}$$ (202)

They can only be fulfilled for certain combinations of $q_\mathrm{l}$ and $q_\mathrm{r}$.

An arbitrary Riemann problem typically causes more than one jump.