1.1.4 The prototype numerical solution of the Euler equations

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

$$\frac{\partial }{\partial t} \left( \rho, \rho\ensuremath{\... ...box{\boldmath$\scriptscriptstyle v$}}}, \rho e_{\rm ik}\right)$$ (7)

where the function $f$ 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...