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

Question

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.

Answer 1

As a starting point, "method of manufactured solutions" can be used to check the validity of your code. For information regarding this: http://www.innovative-cfd.com/manufactured-solutions.html, or you can just google about manufactured solutions.

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:

• http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-0-387-72065-4
• http://www.researchgate.net/publication/48208658_Viscous_Shock_Capturing_in_a_Time-Explicit_Discontinuous_Galerkin_Method

Boundary conditions question needs to be more concrete I think for the community to help.