Gauss–Hermite quadrature: Difference between revisions

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[[Image:CompactStencil.svg|right|thumb|150px|A 2D compact stencil using all 8 adjacent nodes, plus the center node (in red).]]
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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 (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 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>W. F. Spotz. High-Order Compact Finite Difference Schemes for Computational Mechanics. PhD thesis, University of Texas at Austin, Austin, TX, 1995.</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 ==
{{refimprove|date=July 2008}}
{{reflist}}
 
 
[[Category:Numerical differential equations]]

Revision as of 20:03, 22 February 2014

It involves expertise and knowledge of various tools and technologies used for creating websites. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Should you go with simple HTML or use a platform like Wordpress. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day.

Thus, it is imperative that you must Hire Word - Press Developers who have the expertise and proficiency in delivering theme integration and customization services. If a newbie missed a certain part of the video then they could always rewind. Which is perfect for building a mobile site for business use. They provide many such popular products which you can buy for your baby. That's a total of 180$ for each Wordpress theme if you sell 3 links and keep the designer link for your own website, or 240$ if you sell all links.

Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. Our skilled expertise, skillfulness and excellence have been well known all across the world. Websites using this content based strategy are always given top scores by Google.

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