I am trying to extend my knowledge on numerical solution of PDE, in this particular case I am learning to use splitting techniques as they provide simple approach to paralellization. However in the field of my interest very often transformations of the coordinate systems are used, which result in mixed derivatives in the transformed equations. In none of the books I've read were there considerations on how to approach second oreder mixed derivatives, nor if it is even possible to do so.
So I have the following problems: Consider any PDE with mixed derivative, for example the one given below:
$$\frac{\partial f}{\partial t}=\frac{\partial^2 f}{\partial u^2}+\frac{\partial^2 f}{\partial u\partial v}+\frac{\partial^2 f}{\partial v^2}$$
How to use splitting technique in case of such or similar equation? Is it even possbile? Are there any ready to use schemes (like Lie-Trotter, Strang, etc.) extensions ?
Maybe this question should be asked in Stack Computational Science, however I think that here I might get more comprehensive and theory based explanation.