# What kind of test cases are convenient to use for testing the code for Euler equations of gas dynamics in polar coordinates?

Consider Euler equation of gas dynamics in polar coordinates as
$$\left( \begin{array}{ccc} \rho \\ \rho u_{r} \\ \rho u_{\theta} \\ E \end{array} \right)_{t} + \frac{1}{r} \left( \begin{array}{ccc} r \rho u_{r} \\ r (\rho u_{r}u_{r} + p)\\ r \rho u_{r} u_{\theta}\\ r u_{r}(E + p)\end{array} \right)_{r} + \frac{1}{r} \left( \begin{array}{ccc} \rho u_{\theta} \\ \rho u_{r}u_{\theta} \\ \rho u_{\theta} u_{\theta} + p\\ u_{\theta}(E + p)\end{array} \right)_{\theta} = \left( \begin{array}{ccc} 0 \\ \frac{p}{r} + \frac{\rho u_{\theta} u_{\theta}}{r} \\ -\frac{\rho u_{r} u_{\theta}}{r}\\ 0\end{array} \right)$$

• Velocity of shock waves are my interest for this problem.
• The finite volume numerical method is picked to obtain approximate solutions for this system.

Before I run my code for my problem ,which does not have an analytical solution so I can't be certain if my code is correct or not, is there any test cases that I can use to test my code ?

For my problem boundary conditions are quite challenging, what kind should I pick and how to apply it to my FVM scheme ?

Matlab is the preferable language for this calculation. Any help is appreciated.

Roughly, CFL condition scales as $h^2$ instead of $h$ if you use artificial diffusion terms. For DG methods, it is typically given by: $$\Delta t \sim \frac{1}{\lambda_{max}\frac{N^2}{h} + ||\nu||_{L^\infty}\frac{N^4}{h^2}}$$ where N is the approximation polynomial degree. For reference:
• It is going to be exactly like cartesian coordinates. The only difference is $x$ and $y$ are now $r$ and $\theta$. Take a smooth soution for your unknowns (in terms of $r$ and $\theta$), substitute them in the governing equations and add the extra terms that you get as source terms. – gk1 Nov 18 '14 at 11:47