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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.

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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:

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

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  • $\begingroup$ MMS method is definitely useful to check convergence rate for the scheme. I used this method for the 2D Euler equations in cartesian coordinates and it works very well without including boundary conditions. For that case I used sinus and cosines functions which depend only on x and y independent variables. To test above problem, is it going to be the similar case? Could you give me some example? $\endgroup$ – Loading... Nov 18 '14 at 10:41
  • $\begingroup$ 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. $\endgroup$ – gk1 Nov 18 '14 at 11:47

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