# Five-point stencil

An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively).

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

## 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

$\ \{x-2h, x-h, x, x+h, x+2h\}.$

### 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

$f'(x) \approx \frac{-f(x+2 h)+8 f(x+h)-8 f(x-h)+f(x-2h)}{12 h}$

#### 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:

$f(x \pm h) = f(x) \pm h f'(x) + \frac{h^2}{2}f''(x) \pm \frac{h^3}{6} f^{(3)}(x) + O_{1\pm}(h^4). \qquad (E_{1\pm}).$

Evaluating (E 1+) − (E 1−) gives us

$f(x+h) - f(x-h) = 2hf'(x) + \frac{h^3}{3}f^{(3)}(x) + O_1(h^4). \qquad (E_1).$

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

$f(x \pm 2h) = f(x) \pm 2h f'(x) + 2h^2 f''(x) \pm \frac{4h^3}{3} f^{(3)}(x) + O_{2\pm}(h^4). \qquad (E_{2\pm})$

and $(E_{2+})-(E_{2-})$ gives us

$f(x+2h) - f(x-2h) = 4hf'(x) + \frac{8h^3}{3}f^{(3)}(x) + O_2(h^4). \qquad (E_2).$

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

$8f(x+h) - 8f(x-h) - f(x+2h) + f(x-2h) = 12h f'(x) + O(h^4) \,$

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

$\frac{-f(x+2 h)+8 f(x+h)-8 f(x-h)+f(x-2h)}{12 h}=f'(x)-\frac{1}{30} f^{(5)}(x) h^4+O(h^5)$ [1]

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

$f''(x) \approx \frac{-f(x+2 h)+16 f(x+h)-30 f(x) + 16 f(x-h) - f(x-2h)}{12 h^2}$
$f^{(3)}(x) \approx \frac{f(x+2 h)-2 f(x+h) + 2 f(x-h) - f(x-2h)}{2 h^3}$
$f^{(4)}(x) \approx \frac{f(x+2 h)-4 f(x+h)+6 f(x) - 4 f(x-h) + f(x-2h)}{h^4}$

#### Estimated errors

The errors in these approximations are O(h 4), O(h 2) and O(h 2) respectively.[1]

## 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

$\ell_j(\xi) = \prod_{i=0,\, i\neq j}^{k} \frac{\xi-x_i}{x_j-x_i},$

where the interpolation points are

$x_0=x-2h,\quad x_1=x-h,\quad x_2=x,\quad x_3=x+h,\quad x_4=x+2h.$

Then, the quartic polynomial $p_4(x)$ interpolating ƒ(x) at these five points is

$p_4(x) = \sum\limits_{j=0}^4 f(x_j) \ell_j(x)$

and its derivative is

$p_4'(x) = \sum\limits_{j=0}^4 f(x_j) \ell'_j(x).$

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

$f'(x_2) = \ell_0'(x_2) f(x_0) + \ell_1'(x_2) f(x_1) + \ell_2'(x_2) f(x_2) + \ell_3'(x_2) f(x_3) + \ell_4'(x_2) f(x_4) + O(h^4)$

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 (xy) in the grid is

$\{(x-h, y), (x, y), (x+h, y), (x, y-h), (x, y+h)\}, \,$

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

$\Delta f(x,y) \approx \frac{f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y)}{h^2}.$

The error in this approximation is O(h 2),[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:

$\begin{array} {l} \frac{\partial ^2 f}{\partial x^2}= \frac{f\left(x + \Delta x,y\right) + f\left(x - \Delta x,y\right) - 2f(x,y)}{\Delta x^2} - 2\frac{f^{(4)}(x,y)}{4!}\Delta x^2 + \cdots \end{array}$

$\begin{array} {l} \frac{\partial ^2 f}{\partial y^2}= \frac{f\left(x, y + \Delta y\right) + f\left(x, y - \Delta y\right) - 2f(x,y)}{\Delta y^2} - 2\frac{f^{(4)}(x,y)}{4!}\Delta y^2 + \cdots \end{array}$

If we assume $\Delta x=\Delta y=h$:

$\begin{array} {ll} \nabla^2 f &= \frac{\partial ^2 f}{\partial x^2}+\frac{\partial ^2 f}{\partial y^2}\\ \nabla^2 f &= \frac{f\left(x + h,y\right) + f\left(x - h,y\right) + f\left(x, y + h\right) + f\left(x, y - h\right) - 4f(x,y)}{h^2} - 4\frac{f^{(4)}(x,y)}{4!}h^2 + \cdots\\ \nabla^2 f &= \frac{f\left(x + h,y\right) + f\left(x - h,y\right) + f\left(x, y + h\right) + f\left(x, y - h\right) - 4f(x,y)}{h^2} + O\left(h^2\right)\\ \end{array}$