I want to solve this set of 2D advection-diffusion equations of this form in spherical coordinates:
$$ \frac{\partial f}{\partial t}=-\mathbf{u}\cdot\nabla f+\eta\nabla^2f+\eta_1(\mathbf{e}_1\cdot\nabla)(\mathbf{e}_1\cdot\nabla)f+\eta_2(\mathbf{e}_2\cdot\nabla)(\mathbf{e}_2\cdot\nabla)f+s_f(g) $$
$$ \frac{\partial g}{\partial t}=-\nabla\cdot(\mathbf{u}g)+\nabla\cdot(\eta\nabla g)+\nabla\cdot(\eta_1\mathbf{e}_1\mathbf{e}_1\cdot\nabla g)+\nabla\cdot(\eta_2\mathbf{e}_2\mathbf{e}_2\cdot\nabla g)+\nabla\cdot h(f)+s_g(g) $$ where the $\mathbf{e}$ are unit vectors not parallel to the unit vectors in spherical coordinates. Most of the terms depend in a nonlinear manner on $f$ and or $g$, so an explicit algorithm is necessary.
I was thinking of finite volumes because most of the terms in the equation for $g$ are in conservative form, but for the equation for $f$, none of them are. I mean, I guess I can do it anyway, so my question is this: is there a way to take maximal advantage of the finite volume method here? If both equations are not in conservative form, I suspect I will loose a lot of the advantages of having the other equation in such a form? I thought that it could be advantageous to get rid of one derivative order for f by writing (which also works for the directional gradients because $\nabla\cdot\mathbf{e}=0$): $$ \eta\nabla^2f=\nabla\cdot(\eta\nabla f)-\nabla\eta\cdot\nabla f. $$ Or is the finite volume even the preferred method here?