Talk:Relaxation method

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[edit] Over-focused

This article over-focuses on the a specific application of the method, needs to be expanded to the entire general concept. —Preceding unsigned comment added by 129.97.58.55 (talk • contribs) 02:24, October 21, 2007 (UTC)

Can you give an example of the relaxation method applied to something else than the numerical solution of an elliptic p.d.e.?  --Lambiam 11:39, 21 October 2007 (UTC)

[edit] Error term on second-order central difference scheme

Shouldn't the error term on the second-order central difference scheme be O(h^2) instead of O(h^4) ? —Preceding unsigned comment added by Runebarnkob (talk • contribs) 10:33, 22 November 2007 (UTC)

I don't think so. In one dimension
h^2\frac{d^2}{{dx}^2}\varphi(x) = \varphi(x{-}h)-2\varphi(x)+\varphi(x{+}h)\,+\,\mathcal{O}(h^4)\,.
If you do this in two orthogonal directions and add to get h^2{\nabla}^2\varphi(x,y)\,, as used in the equation for φ(x, y), you still have an error term of O(h4).  --Lambiam 14:21, 22 November 2007 (UTC)
I agree that the error term should still be the same as you expand the dimension. But I thought that the one dimensional second order three-point central difference scheme was
h^2\frac{d^2}{{dx}^2}\varphi(x) = \varphi(x{-}h)-2\varphi(x)+\varphi(x{+}h)\,+\,\mathcal{O}(h^2)\,.
Maybe I have a lack in my understanding of the BigO-notation. --Runebarnkob (talk) 04:20, 23 November 2007 (UTC)
That formula is equivalent with
\frac{d^2}{{dx}^2}\varphi(x) = \frac{\varphi(x{-}h)-2\varphi(x)+\varphi(x{+}h)}{h^2}\,+\,\mathcal{O}(1)\,,
which is clearly too weak; it does not tell us that
lim_{h \to 0} \frac{\varphi(x{-}h)-2\varphi(x)+\varphi(x{+}h)}{h^2} = \frac{d^2}{{dx}^2}\varphi(x)\,.
Use the Taylor expansion
\varphi(x{+}h) = \varphi(x) + h\varphi'(x) + \mathcal{O}(h^2)\,
and the same with h replaced by −h, substituting it for φ(x±h) in the second-order difference formula, and you're done.  --Lambiam 08:51, 23 November 2007 (UTC)
I agree with you now. I definitely had a problem with my understanding of the O(). Thanks for your patience, Lambiam. --Runebarnkob (talk) 09:55, 23 November 2007 (UTC)