Talk:Five-point stencil

Richardson extrapolation

I don't know if this is of interest, but the five-point stencil can indeed be obtained using Richardson extrapolation. The idea is to linearly combine two levels of approximation so as to cancel the leading error terms.

The example here is to take the three point stencil on interlacing grids.

$f_{h}'(x)={\frac {1}{2h}}\left[f(x+h)-f(x-h)\right]+O(h^{2})$

$f_{2h}'(x)={\frac {1}{4h}}\left[f(x+2h)-f(x-2h)\right]+O(4h^{2})$

Using these two approximations, we cancel error terms with Richardson extrapolation

$f'_{RE}(x)={\frac {4}{3}}f_{h}'(x)-{\frac {1}{3}}f'_{2h}(x)={\frac {1}{12h}}\left[-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)\right]$

Anyone want to include this in a tidier form?

Gregvw 14:16, 30 April 2007 (UTC)

Thanks for the explanation. I added a sentence to the article. Feel free to improve on it if it isn't what you had in mind. -- Jitse Niesen (talk) 14:50, 30 April 2007 (UTC)

Weights

Not to clutter the page too much, but there is yet another way to get the weights, which is to differentiate the Langrange interpolating polynomials. Does anyone want me to write an explanation of this? I could make a "Alternate methods of computing the weights" section. I could also make a table of the non-centered weights if this seems useful. Gregvw 14:12, 2 May 2007 (UTC)

My guess would be that adding in the Lagrange interpolating polynomials could make the discussion rather technical and lengthy. Perhaps just a sentence could be added saying that the weights could be obtained that way, without going into details. At leas that's what I'd think. Oleg Alexandrov (talk) 15:13, 2 May 2007 (UTC)

Ok, this is probably overkill and might want some clean-up, but I haven't seen this elsewhere in Wikipedia. Maybe I should just start an article on pseudospectral methods? Gregvw 08:10, 3 May 2007 (UTC)

Thanks, this was shorter than what I thought would be. And yeah, an article on pseudospectral methods would be nice. :) Oleg Alexandrov (talk) 15:05, 3 May 2007 (UTC)

Is this just a way to describe the arrangment of the finite difference approximation and coefficients?

Is that all a stencil is? A notational convenience? Sancho 20:40, 9 December 2007 (UTC)

Yes. At least that's how I use the word stencil. -- Jitse Niesen (talk) 12:06, 10 December 2007 (UTC)