Calculate Euler equations of fluid dynamics without division?

Question

I'm working on the calculation of the Euler equations with the 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. $\rho v_1^2$

Answer 1

An alternative to the conservative formalism of the Euler is the primitive form: $$ \frac{\partial\rho}{\partial t}+\nabla\cdot\rho\mathbf v=0 \\ \rho\frac{\partial\mathbf v}{\partial t}+\rho\mathbf v\cdot\nabla\mathbf v+\left(\gamma-1\right)\nabla\left(\rho e\right)=0 \\ \frac{\partial e}{\partial t}+\mathbf v\cdot\nabla e+\left(\gamma-1\right)e\nabla\cdot\mathbf v=0 $$ You shouldn't have to do any division here, as the primitive variables are what are differenced. I haven't really worked with solving a primitive systems, but I don't believe you can use the conventional Riemann solver, as that expects a conservative system.