ABSTRACT
Finite-Difference methods are rather straightforward discretizations of differential operators. The method goes back to the original Newton-Leibniz definition of a differentiation as the limit of a division of two near-by function evaluation at two near-by input variables. Any order derivative may be approximated by a series expansion of finite difference operators. It may be applied to any appearance of an operator. Moreover, it is an exclusive mathematical recipe. This is also the reason that success may be limited. Finite difference methods have been abandoned as a starting point for spatial discretization except when a problem with some other method needs to be analyzed in-depth and there is serious suspicious that there is a flaw in the discretization method itself. The finite-difference scheme is very valuable if parts of the problem characteristics can be addressed by analytic means.
