For the past several weeks I was attempting to adapt Lax-Wendroff or some similar scheme for polar coordinates. The process was complicated due to me being unable to find step-by-step derivations of used schemes, so i had to guess the process for cartesian coordinates and to repeat the same for polar ones.
I managed to do it for MacCormack's scheme, and it exploded even on somewhat smooth, shockwave-less test.
Due to geometry of my system and because of other equations that govern it (I'm trying to add hydrodynamics into 2d plasm discharge model), I was trying to stick to polar coordinates. Hovewer, now I'm quite certain that I won't be able to find ready-for-use scheme for euler equations in polar coordinates. To derive one on my own seems implausible, too.
Thus my questions arise.
Are there ready-to-use robust solvers for inviscid compressible Euler equations with sharp gradients in polar coordinates?
If there are none, how do I attempt to derive one? Which scheme should I try to derive?
Is it even feasible to try to stick to polar coordinates? Or should I solve Euler equations in cartesian coordinates, and then interpolate values for the rest of my model? If so, should I use uniform grid with cropped cells on the border, or should I use some finite-volume method?
If I should move away from polar coordinates, which scheme should I use? There are quite a lot of schemes, there are different types of error correction mechanisms, etc - but can someone with experience suggest something?