Five point stencil

One dimension

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

First derivative

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

Obtaining the formula

This formula can be obtained by writing out the four Taylor series of ƒ(x ± h) and ƒ(x ± 2h) up to terms of h ³ (or up to terms of h ⁵ to get an error estimation as well) and solving this system of four equations to get ƒ ^′(x). Actually, we have at points x + h and x − h:

Evaluating (E ₁₊) − (E ₁₋) gives us

Note that the residual term O₁(h ⁴) should be of the order of h ⁵ instead of h ⁴ because if the terms of h ⁴ had been written out in (E ₁₊) and (E ₁₋), it can be seen that they would have canceled each other out by ƒ(x + h) − ƒ(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

and ( E 2 + ) − ( E 2 − ) gives us

In order to eliminate the terms of ƒ ⁽³⁾(x), calculate 8 × (E₁) − (E₂)

thus giving the formula as above. Note: the coefficients of f in this formula, (8, -8,-1,1), represent a specific example of the more general Savitzky-Golay filter

Estimated error

The error in this approximation is of order h ⁴. That can be seen from the expansion

which can be obtained by expanding the left-hand side in a Taylor series. Alternatively, apply Richardson extrapolation to the central difference approximation to f ′ ( x ) on grids with spacing 2h and h.

Higher derivatives

The centered difference formulas for five-point stencils approximating second, third, and fourth derivatives are

Estimated errors

The errors in these approximations are O(h ⁴), O(h ²) and O(h ²) respectively.

Relationship to Lagrange interpolating polynomials

As an alternative to deriving the finite difference weights from the Taylor series, they may be obtained by differentiating the Lagrange polynomials

where the interpolation points are

Then, the quartic polynomial p 4 ( x ) interpolating ƒ(x) at these five points is

and its derivative is

So, the finite difference approximation of ƒ ^′(x) at the middle point x = x₂ is

Evaluating the derivatives of the five Lagrange polynomials at x=x₂ gives the same weights as above. This method can be more flexible as the extension to a non-uniform grid is quite straightforward.

Two dimensions

In two dimensions, if for example the size of the squares in the grid is h by h, the five point stencil of a point (x, y) in the grid is

forming a pattern that is also called a quincunx. This stencil is often used to approximate the Laplacian of a function of two variables:

The error in this approximation is O(h ²), which may be explained as follows:

From the 3 point stencils for the second derivative of a function with respect to x and y:

∂ 2 f ∂ x 2 = f ( x + Δ x , y ) + f ( x − Δ x , y ) − 2 f ( x , y ) Δ x 2 − 2 f ( 4 ) ( x , y ) 4 ! Δ x 2 + ⋯

∂ 2 f ∂ y 2 = f ( x , y + Δ y ) + f ( x , y − Δ y ) − 2 f ( x , y ) Δ y 2 − 2 f ( 4 ) ( x , y ) 4 ! Δ y 2 + ⋯

If we assume Δ x = Δ y = h :

∇ 2 f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 = f ( x + h , y ) + f ( x − h , y ) + f ( x , y + h ) + f ( x , y − h ) − 4 f ( x , y ) h 2 − 4 f ( 4 ) ( x , y ) 4 ! h 2 + ⋯ = f ( x + h , y ) + f ( x − h , y ) + f ( x , y + h ) + f ( x , y − h ) − 4 f ( x , y ) h 2 + O ( h 2 )