ok, so what you are saying the function is smooth, meaning I have all kinds of partial derivatives, however, it is magnitude is very large(think 1/x as s approaches 0). And numerically it is hard to capture this type of functions?
Yes, it's similar to boundary layer behaviour and singular perturbation theory, for which classical FD tends to break down.
https://www.nag.co.uk/market/Peter_Duck ... tation.pdf
The exponentially fitted schemes for convection-diffusion PDE (e.g. BS) I developed are based on singular perturbation theory.