What is a central finite difference approximation on a non-uniform 2D grid to the following mixed derivative, in spherical polar coordinates, accurate to 1st/2nd order?
$$\frac{\partial^2 \Phi}{\partial r \partial \theta}\bigg|_{ij} \; = \;\; ?$$
For a uniform grid in Cartesian coordinates, the equivalent would be (e.g. here):
$$\frac{\partial^2 \Phi}{\partial x \partial z}\bigg|_{ij} \;=\; \frac{\Phi_{i+1,j+1} - \Phi_{i+1,j-1} - \Phi_{i-1,j+1} + \Phi_{i-1,j-1}}{4 \Delta x \Delta z } + \mathcal{O}(\Delta x^2, \Delta z^2)$$
Notes:
- I say "1st/2nd order" because such a scheme, although 2nd-order accurate on a uniform grid, can be as bad as 1st-order accurate on a very non-uniform grid. In practice, such schemes on semi-uniform grids perform somewhere between 1st and 2nd order in accuracy.
- I use the convention where $\theta$ is the colatitudinal coordinate and $\phi$ is the azimuthal coordinate.