Compact stencil
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In mathematics, especially in the areas of numerical analysis called numerical partial differential equations, a compact stencil is a type of stencil that uses only nine nodes for its discretization method in two dimensions. It uses only the center node and the adjacent nodes. For any structured grid utilizing a compact stencil in 1, 2, or 3 dimensions the maximum number of nodes is 3, 9, or 27 respectively. Compact stencils may be compared to 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{{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}}Communications in Numerical Methods in Engineering, Copyright © 2008 John Wiley & Sons, Ltd.
Two Point Stencil Example
The two point stencil for the first derivative of a function is given by:
f'(x_0)=\frac{f\left(x_0 + h\right) - f\left(x_0 - h\right)}{2h} + O\left(h^2\right)
.
This is obtained from the Taylor series expansion of the first derivative of the function given by:
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}.
Replacing with , we have:
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}.
Addition of the above two equations together results in the cancellation of the terms in odd powers of :
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}.
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}.
f'(x_0)=
\frac{f\left(x_0 + h\right) - f\left(x_0 - h\right)}{2h} + O\left(h^2\right)
\end{array}.
Three Point Stencil Example
For example, the three point stencil for the second derivative of a function is given by:
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}.
This is obtained from the Taylor series expansion of the first derivative of the function given by:
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}.
Replacing with , we have:
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}.
Subtraction of the above two equations results in the cancellation of the terms in even powers of :
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}.
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}.
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}.
See also
References
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