I'm working on the calculation of the [Euler equations](http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)) with the [finite volume method](http://en.wikipedia.org/wiki/Finite_volume_method). Unfortunately I'm not allowed to do a division. So I'm wondering if there's a form which does not need a division.

At the moment the Euler equations look like this:

$$ \frac{\partial}{\partial t}
\begin{pmatrix}
\rho \\ \rho v_1 \\ \rho v_2 \\ \rho v_3 \\ \rho E
\end{pmatrix}
= -\mathrm{div}
\begin{pmatrix} \rho v_1 & \rho v_2 & \rho v_3 \\ 
\rho v_1^2 + p & \rho v_1 v_2 & \rho v_1 v_3 \\ 
\rho v_2 v_1 & \rho v_2^2 + p & \rho v_2 v_3 \\ 
\rho v_3 v_1 & \rho v_3 v_2 & \rho v_3^2 + p \\ 
(\rho E + p) v_1 & (\rho E + p) v_2 & (\rho E + p) v_3
\end{pmatrix} $$

As you can see, I first need to calculate $\frac{\rho v_1}{\rho}$ to get $v_1$ so I can calculate e.g. $\rho v_1^2$