Five-point stencil

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

:$$\backslash \{x-2h,\; x-h,\; x,\; x+h,\; x+2h\backslash \}.$$

The first derivative of a function f of a real variable at a point x can be approximated using a five-point stencil as:{{cite book |title=Numerical Analysis |last=Sauer |first=Timothy |publisher=Pearson |year=2012 |isbn=978-0-321-78367-7 |pages=250}}

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

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

:$$f(x\; \backslash pm\; h)\; =\; f(x)\; \backslash pm\; h\; f\text{'}(x)\; +\; \backslash frac\{h^2\}\{2\}f\text{'}\text{'}(x)\; \backslash pm\; \backslash frac\{h^3\}\{6\}\; f^\{(3)\}(x)\; +\; O\_\{1\backslash pm\}(h^4).\; \backslash qquad\; (E\_\{1\backslash pm\}).$$

Evaluating $(E\_\{1+\})-(E\_\{1-\})$ gives us

:$$f(x+h)\; -\; f(x-h)\; =\; 2hf\text{'}(x)\; +\; \backslash frac\{h^3\}\{3\}f^\{(3)\}(x)\; +\; O\_1(h^4).\; \backslash qquad\; (E\_1).$$

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 {{math|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

:$$f(x\; \backslash pm\; 2h)\; =\; f(x)\; \backslash pm\; 2h\; f\text{'}(x)\; +\; \backslash frac\{4h^2\}\{2!\}\; f\text{'}\text{'}(x)\; \backslash pm\; \backslash frac\{8h^3\}\{3!\}\; f^\{(3)\}(x)\; +\; O\_\{2\backslash pm\}(h^4).\; \backslash qquad\; (E\_\{2\backslash pm\})$$

and $(E\_\{2+\})\; -\; (E\_\{2-\})$ gives us

:$$f(x+2h)\; -\; f(x-2h)\; =\; 4hf\text{'}(x)\; +\; \backslash frac\{8h^3\}\{3\}f^\{(3)\}(x)\; +\; O\_2(h^5).\; \backslash qquad\; (E\_2).$$

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

:$$8f(x+h)\; -\; 8f(x-h)\; -\; f(x+2h)\; +\; f(x-2h)\; =\; 12h\; f\text{'}(x)\; +\; O(h^5)$$

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.

The error in this approximation is of order h ⁴. That can be seen from the expansionAbramowitz & Stegun, Table 25.2

:$$\backslash frac\{-f(x+2\; h)+8\; f(x+h)-8\; f(x-h)+f(x-2h)\}\{12\; h\}=f\text{'}(x)-\backslash frac\{1\}\{30\}\; f^\{(5)\}(x)\; h^4+O(h^5)$$

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

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

:$$\backslash begin\{align\}$$

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}

\\[1ex]

f^{(3)}(x) &\approx \frac{f(x+2 h)-2 f(x+h) + 2 f(x-h) - f(x-2h)}{2 h^3}

\\[1ex]

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}

\end{align}

The errors in these approximations are O(h{{i sup|4}}), O(h{{i sup|2}}) and O(h{{i sup|2}}) respectively.

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

:$$\backslash ell\_j(\backslash xi)\; =\; \backslash prod\_\{i=0,\backslash ,\; i\backslash neq\; j\}^\{k\}\; \backslash frac\{\backslash xi-x\_i\}\{x\_j-x\_i\},$$

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 {{math|f(x)}} at these five points is

:$$p\_4(x)\; =\; \backslash sum\_\{j=0\}^4\; f(x\_j)\; \backslash ell\_j(x)$$

and its derivative is

:$$p\_4\text{'}(x)\; =\; \backslash sum\_\{j=0\}^4\; f(x\_j)\; \backslash ell\text{'}\_j(x).$$

So, the finite difference approximation of {{math|f ′(x)}} at the middle point {{math|1=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 {{math|1=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.

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

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

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

:$$\backslash nabla^2\; f(x,y)\; \backslash approx\; \backslash frac\{f(x-h,y)\; +\; f(x+h,y)\; +\; f(x,y-h)\; +\; f(x,y+h)\; -\; 4f(x,y)\}\{h^2\}.$$

The error in this approximation is O(h ²),Abramowitz & Stegun, 25.3.30 which may be explained as follows:

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

:$$\backslash begin\{align\}$$

\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 \\[1ex]

\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{align}

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

:$$\backslash begin\{align\}$$

\nabla^2 f &= \frac{\partial ^2 f}{\partial x^2} + \frac{\partial ^2 f}{\partial y^2} \\[1ex]

&= \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 \\[1ex]

&= \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{align}

• {{annotated link|Finite difference coefficient}}
• {{annotated link|Iterative Stencil Loops}}
• Nine-point stencil
• Stencil (numerical analysis)
• {{annotated link|Stencil jumping}}

{{Reflist}}

• {{citation |first1=Milton |last1=Abramowitz |author1-link=Milton Abramowitz |first2=Irene A. |last2=Stegun |author2-link=Irene Stegun |year=1970 |title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables |publisher=Dover}}. Ninth printing. Table 25.2.

Category:Numerical differential equations