Can someone provide a brief list of when, and when not to, use each finite difference approximation for derivative?

Question

So far, I've received one possible scenario, and may be the only one, for using the forward difference in approximating the derivative rather than its central difference counterpart. And that is when dealing with cases where the right and left derivative are not one and the same. But wasn't given an explanation why it works, which I verified in my code. So if anyone can provide references or web links listing common scenarios for each that'd be great.

Answer 1

The central difference scheme is more precise approximation of a derivative of function $f$ at point $x$ than the forward of backward difference scheme, that is clear. Two cases appear in my head, when the latter is preferable:

1. On the domain boundary you obviously cannot use the central difference.
2. When solving hyperbolic PDEs, in which at point $x$ information propagates along characteristic in direction $\textbf{v}$, it is preferable to use backward difference scheme, i.e. in direction $-\textbf{v}$, because this is the region which contains the relevant information. The central scheme might be numerically instable.