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1.1.4 The prototype numerical solution of the Euler equations

The hydrodynamics equations (1) or (2) can be put into the form

\begin{displaymath}
\frac{\partial }{\partial t} \left( \rho, \rho\ensuremath{\...
...box{\boldmath$\scriptscriptstyle v$}}}, \rho e_{\rm ik}\right)
\end{displaymath} (7)
where the function \bgroup\color{DEFcolor}$f$\egroup contains the terms with the spatial derivatives.

Idea:

  1. take initial state $\left( \rho, \rho\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$}}
{\m...
...ldmath$\scriptstyle x$}}
{\mbox{\boldmath$\scriptscriptstyle x$}}}, t_0 \right)$ given on a grid
  2. compute $\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$}}
{\mbox{\boldmath$\te...
...}}
{\mbox{\boldmath$\scriptstyle v$}}
{\mbox{\boldmath$\scriptscriptstyle v$}}}$, $e_{\rm i}$, and $P$
  3. compute the spatial derivatives to get the right-hand side of Eq. (7)
  4. get a small change of $\left( \rho, \rho\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$}}
{\m...
...iptstyle v$}}
{\mbox{\boldmath$\scriptscriptstyle v$}}}, \rho e_{\rm ik}\right)$
  5. update $\left( \rho, \rho\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$}}
{\m...
...iptstyle v$}}
{\mbox{\boldmath$\scriptscriptstyle v$}}}, \rho e_{\rm ik}\right)$
  6. restart at 2

There 1000 ways how this can go wrong...