When to use forward difference over central difference/three point difference?

Question

I understand that central difference and three point difference namely $\frac{−3 f(x) +4 f(x+h)− f(x+2h)}{2h}$ provide approximations of the first derivative up to a term of order $h^2$. Forward difference only approximates up to a term of order $h$. So for most situations central difference would be preferred over both three point difference (denominator contains 3! rather than 3) and forward difference.

In what situations would forward difference be better than both central or three point difference? In what situations would three point difference be better?

I can really only think of one situation where three point would be better where we want to approximate an x for which we know nothing about $f(x_0), x_0<x$. I cannot think of any situations that forward difference would be better assuming f(x) is smooth

Answer 1

Here's one scenario: You don't need high accuracy, you just need to code up your ode/pde solver as rapidly as possible and get on with life. Lower-order finite-difference schemes have fewer nonzero sub/super diagonals in their matrix representations, making them easier to code up.

Answer 2

In no particular order:

In hyperbolic PDE, higher order schemes tend to have some subtle detrimental effects, such as numerical dissipation or numerical dispersion. An upwind low order scheme avoids these problems, albeit at higher computational cost. Higher order schemes avoiding these problems exist too of course, but they are much more complicated than you would naively expect.

In situations with low regularity, higher order methods whose order proof is based on a Taylor argument are not higher order at all (because the Taylor argument assumes more derivatives than you have). One way this shows up is in the numerical solution of stochastic differential equations. The obvious counterpart to the forward Euler method in SDE is called the Euler-Maruyama method, and it is actually one of the most prominent numerical methods for SDE. Higher order methods in SDE have to resolve very subtle correlation phenomena that can be harder to resolve than it would be to just run Euler-Maruyama in the first place.

In implicit methods, lower order methods are more "banded"; a method of one order might require solving a tridiagonal system and a method of the next order up might require solving a pentadiagonal system. More banded systems are easier to solve, and easier to store in memory to boot.

Stiff integrators tend to be lower order than nonstiff integrators. I don't really have a good intuition for why this is, though.