I have part of a problem that is described by the momentum conservation equation:
$\frac{\partial \rho}{\partial t} + \frac{1}{\sin\theta} \frac{\partial}{\partial \theta}(\rho u \sin \theta) =0$
Where $u=f(\theta)$ and $ \rho = f(\theta,t) $ (constant velocity).
Naively one could apply one of the solutions listed here. The problem at hand is best described in spherical polar coordinates (thin spherical shell) and those solutions in Cartesian. must I make some kind of coordinate transformation before discretizing this equation or can I discretize it directly?
Secondly is there any reason one should first expand the derivative in $\theta$ and then attempt to discretize?
As a note - I have done several of the above and have obtained solutions that do not seem to be consistent (physically a couple seem to make sense). I am interested in if there is a proper coordinate transformation that should be made or if any of the previously mentioned methods will suffice.
EDIT:
I define flux as:
$\Phi_{i+1/2} = \dfrac{u_{i+1/2}+|u_{i+1/2}|}{2}\rho_{i} \sin{\theta_{i}} + \dfrac{u_{i+1/2}-|u_{i+1/2}|}{2}\rho_{i+1} \sin{\theta_{i+1}} $
$\Phi_{i-1/2} = \dfrac{u_{i-1/2}+|u_{i-1/2}|}{2}\rho_{i-1} \sin{\theta_{i-1}} + \dfrac{u_{i-1/2}-|u_{i-1/2}|}{2}\rho_{i} \sin{\theta_{i}}$
I'm guessing the 'proper' way to define the flux would be to evaluate $\sin{\theta}$ at the $\pm\frac{1}{2}$ 'cell boundaries' rather than at the cell center. This would be more in line with the definition of flux.
One final question - what should I do at the boundaries ($\theta = 0 $ is particularly a problem and I just avoid that point altogether).