Five-point stencil

In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors". It is used to write finite difference approximations to derivatives at grid points. It is an example for numerical differentiation.

In one dimension, if the spacing between points in the grid is h, then the five-point stencil of a point x in the grid is

The first derivative of a function f of a real variable at a point x can be approximated using a five-point stencil as:

The center point f(x) itself is not involved, only the four neighboring points.

This formula can be obtained by writing out the four Taylor series of ${\displaystyle f(x\pm h)}$ and ${\displaystyle f(x\pm 2h)}$ at the point ${\displaystyle a}$, up to terms of h³ (or up to terms of h⁵ to get an error estimation as well), evaluating each series at ${\displaystyle a=x\mp h}$ and ${\displaystyle a=x\mp 2h}$ respectively (to get everything in common terms of ${\displaystyle f^{(k)}(x)}$), and solving this system of four equations to get f ′(x). Actually, we have at points x + h and x − h:

Evaluating ${\displaystyle (E_{1+})-(E_{1-})}$ gives us

The residual term O₁(h⁴) should be of the order of h⁵ instead of h⁴ because if the terms of h⁴ had been written out in (E₁₊) and (E₁₋), it can be seen that they would have canceled each other out by f(x + h) − f(x − h). But for this calculation, it is left like that since the order of error estimation is not treated here (cf below).

Similarly, we have