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{{refimprove|date=July 2008}}
[[Image:CompactStencil.svg|right|thumb|150px|A 2D compact stencil using all 8 adjacent nodes, plus the center node (in red).]]

In [[mathematics]], especially in the areas of [[numerical analysis]] called [[numerical partial differential equations]], a '''compact stencil''' is a type of [[stencil (numerical analysis)|stencil]] that uses only nine nodes for its [[discretization]] method in two dimensions. It uses only the center node and the [[Adjacent (graph theory)|adjacent]] nodes. For any [[structured grid]] utilizing a compact stencil in 1, 2, or 3 [[dimension]]s the maximum number of [[Node (graph theory)|nodes]] is 3, 9, or 27 respectively. Compact stencils may be compared to [[non-compact stencil|non-compact stencils]]. Compact stencils are currently implemented in many [[partial differential equation]] solvers, including several in the topics of CFD, FEA, and other mathematical solvers relating to PDE's<ref>{{Cite journal |last=Spotz |first=William F. |date=1996 |title=High-Order Compact Finite Difference Schemes for Computational Mechanics |url=https://www.researchgate.net/publication/2591103 |journal=The University of Texas at Austin |via=ResearchGate}}</ref><ref>Communications in Numerical Methods in Engineering, Copyright © 2008 John Wiley & Sons, Ltd.</ref>

==Two Point Stencil Example==
The two point stencil for the ''first derivative'' of a function is given by:

<math>
f'(x_0)=\frac{f\left(x_0 + h\right) - f\left(x_0 - h\right)}{2h} + O\left(h^2\right)
</math>.

This is obtained from the [[Taylor series]] expansion of the first derivative of the function given by:

<math>\begin{array} {l}
f'(x_0)=\frac{f\left(x_0 + h\right) - f(x_0)}{h} -\frac{f^{(2)}(x_0)}{2!}h - \frac{f^{(3)}(x_0)}{3!}h^2 - \frac{f^{(4)}(x_0)}{4!}h^3 + \cdots
\end{array}</math>.

Replacing <math>h</math> with <math>-h</math>, we have:

<math>\begin{array} {l}
f'(x_0)=-\frac{f\left(x_0 - h\right) - f(x_0)}{h} + \frac{f^{(2)}(x_0)}{2!}h - \frac{f^{(3)}(x_0)}{3!}h^2 + \frac{f^{(4)}(x_0)}{4!}h^3 + \cdots
\end{array}</math>.

Addition of the above two equations together results in the cancellation of the terms in odd powers of <math>h</math>:

<math>\begin{array} {l}
2f'(x_0)=
\frac{f\left(x_0 + h\right) - f(x_0)}{h}
-\frac{f\left(x_0 - h\right) - f(x_0)}{h}
-2\frac{f^{(3)}(x_0)}{3!}h^2 + \cdots
\end{array}</math>.

<math>\begin{array} {l}
f'(x_0)=
\frac{f\left(x_0 + h\right) - f\left(x_0 - h\right)}{2h} - \frac{f^{(3)}(x_0)}{3!}h^2 + \cdots
\end{array}</math>.

<math>\begin{array} {l}
f'(x_0)=
\frac{f\left(x_0 + h\right) - f\left(x_0 - h\right)}{2h} + O\left(h^2\right)
\end{array}</math>.

==Three Point Stencil Example==
For example, the three point stencil for the ''second derivative'' of a function is given by:

<math>\begin{array} {l}
f^{(2)}(x_0)=
\frac{f\left(x_0 + h\right) + f\left(x_0 - h\right) - 2f(x_0)}{h^2} + O\left(h^2\right)
\end{array}</math>.

This is obtained from the [[Taylor series]] expansion of the first derivative of the function given by:

<math>\begin{array} {l}
f'(x_0)=\frac{f\left(x_0 + h\right) - f(x_0)}{h} -\frac{f^{(2)}(x_0)}{2!}h - \frac{f^{(3)}(x_0)}{3!}h^2 - \frac{f^{(4)}(x_0)}{4!}h^3 + \cdots
\end{array}</math>.

Replacing <math>h</math> with <math>-h</math>, we have:

<math>\begin{array} {l}
f'(x_0)=-\frac{f\left(x_0 - h\right) - f(x_0)}{h} + \frac{f^{(2)}(x_0)}{2!}h - \frac{f^{(3)}(x_0)}{3!}h^2 + \frac{f^{(4)}(x_0)}{4!}h^3 + \cdots
\end{array}</math>.

Subtraction of the above two equations results in the cancellation of the terms in even powers of <math>h</math>:
<math>\begin{array} {l}
0=
\frac{f\left(x_0 + h\right) - f(x_0)}{h}
+\frac{f\left(x_0 - h\right) - f(x_0)}{h}
- 2\frac{f^{(2)}(x_0)}{2!}h - 2\frac{f^{(4)}(x_0)}{4!}h^3 + \cdots
\end{array}</math>.

<math>\begin{array} {l}
f^{(2)}(x_0)=
\frac{f\left(x_0 + h\right) + f\left(x_0 - h\right) - 2f(x_0)}{h^2} - 2\frac{f^{(4)}(x_0)}{4!}h^2 + \cdots
\end{array}</math>.

<math>\begin{array} {l}
f^{(2)}(x_0)=
\frac{f\left(x_0 + h\right) + f\left(x_0 - h\right) - 2f(x_0)}{h^2} + O\left(h^2\right)
\end{array}</math>.

==See also==
*[[Stencil (numerical analysis)]]
*[[Non-compact stencil]]
*[[Five-point stencil]]

== References ==
{{reflist}}

[[Category:Numerical differential equations]]

Compact stencil - Revision history

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