3.1.5 Shock speed II

During this time $\bgroup\color{DEFcolor}$\Delta t$\egroup$ the states $\bgroup\color{DEFcolor}$q_\mathrm{l}$\egroup$ , $\bgroup\color{DEFcolor}$q_\mathrm{r}$\egroup$ and fluxes $\bgroup\color{DEFcolor}$f \! \left( q_\mathrm{l} \right)$\egroup$ , $\bgroup\color{DEFcolor}$f \! \left( q_\mathrm{r} \right)$\egroup$ on the left and right do not change (much) and we get

$\begin{displaymath} \Delta x q_\mathrm{l} - \Delta x q_\mathrm{r} + \D... ...thrm{l} \right) = O \! \left( \Delta t^2 \right) \enspace . \end{displaymath}$

(167)

For $\bgroup\color{DEFcolor}$\Delta x=s \Delta t$\egroup$ and $\bgroup\color{DEFcolor}$\Delta t\rightarrow 0$\egroup$ we get the Rankine-Hugoniot jump condition

$\begin{displaymath} s \left( q_\mathrm{r} - q_\mathrm{l} \right) = f \! \left( q_\mathrm{r} \right) - f \! \left( q_\mathrm{l} \right) \end{displaymath}$

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which gives for the shock speed in general

$\begin{displaymath} s = \frac{f \! \left( q_\mathrm{r} \right) - f \! \left( q_\mathrm{l} \right)} {q_\mathrm{r} - q_\mathrm{l}} \end{displaymath}$

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and for Burgers' equation

$\begin{displaymath} s = \frac{\frac{1}{2} v_{\mathrm{r}}^2 - \frac{1}{2} v_... ... \left( v_{\mathrm{r}} + v_{\mathrm{l}} \right) \enspace . \end{displaymath}$

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