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This article has some nice info missing from finite difference which is more focused on the general theory of finite differences, not necessarily applied to numerial analysis.
This article shows explicit formulas for several finite difference schemes and outlines how one could obtain other finite difference schemes.
The big question, should it be merged to finite difference, or should it stay by itself? I would be inclined towards the second, but then one would need to modify both this and finite difference so that it is clear which material belongs where and for what reason.
Eventually this may grow into a full-blown finite difference method article which we are missing, provided anybody is willing to do the work. Comments? Oleg Alexandrov ( talk) 03:24, 25 July 2006 (UTC)
The thing is that the finite difference methods/schemes dominates the literature on any search for "finite difference". And I suspect the latter Wikipedia article enjoys view-count popularity for the same reason. Alas with the popularity of FDM, the term "finite difference" itself has changed its meaning [in a lot of the FDM literature] to denote finite difference approximation, i.e. a [finite] difference quotient. The Boole/Jordan-style calculus (and definition[s]) don't seem to be of much interest anymore. The Wikipedia articles on this topic generally suck though, regardless of focus. Some1Redirects4You ( talk) 15:02, 27 April 2015 (UTC)
Does anyone have sufficient practical or theoretical experience on the closing statement:
Usually the Crank-Nicolson scheme is the most accurate scheme for small time steps. The explicit scheme is the least accurate and can be unstable, but is also the easiest to implement and the least numerically intensive. The implicit scheme works the best for large time steps.
What constitutes "small" and "large" time steps? Any boundaries on when the Crank-Nicolson scheme is or isn't the most accurate for small time steps? (When is it "usual" and when is it "unusual"?)
-- KnockNrod 16:42, 3 November 2006 (UTC)
In my lecture course (Imperial College, 2007) The errors in the Crank Nicholson method were presented as being both second order. I cannot find any reference to the more precise O(k^4). I suspect the errors I have been given are the global errors, which are of lower order than the local errors. I recommend it being changed.
The CN method is also stable for all R, which is a large advantage.
Darktachyon ( talk) 18:52, 30 December 2007 (UTC)
I did a bit of changing of this article; I only really got to the first half. I shrank the lead section way down and added a couple new sections talking about the derivation of finite difference methods intuitively and with respect to Taylor's polynomials. I also added the beginnings of a discussion of accuracy and error analysis. Some things I feel need to be addressed in the article:
Also, I suspect what I've written could use a couple of other sets of eyes to have a look over it in case I've made some mathy mistakes. I also marked at least one statement I made that ought to be sourced; if any other editors would like to have a look at this page and mark anything else that needs to be sourced, please do, and I can take care of adding references for them. Duplico ( talk) 22:29, 16 June 2008 (UTC)
Where it says "This explicit method is known to be numerically stable and convergent whenever r <= 1/2", does that mean that we simply have to change the units such that r <= 1/2 ? Or, equivalently, scale the axes of the problem? Clearly that shouldn't work to suddenly make the method stable and convergent. So, the statement should be qualified. — Preceding unsigned comment added by 125.253.44.20 ( talk) 11:00, 24 May 2013 (UTC)
Can someone add the unifying scheme for the Forward Euler, Backward Euler, and Crank-Nicolson schemes ? For those that are not aware perhaps, all these three schemes can be unified by using a varying . The author of the following lecture notes : http://hplgit.github.io/INF5620/doc/notes/main_decay.pdf mentions the rule on page 10.
The notion of scheme needs to be introduced before it's nonchalantly used. Some1Redirects4You ( talk) 16:53, 27 April 2015 (UTC)
Hi, in this edit I have just removed a reference added by a user with a conflict of interest. I do not judge of the quality of the reference. If it is a valuable addition to the article, then a more experienced user can perhaps add it back. Thanks, Ariadacapo ( talk) 11:20, 2 September 2015 (UTC)
Hi, section Comparison looks like meant to be complete example. Meanwhile it doesn't have initial condition specified which cause confusion. Thanks... — Preceding unsigned comment added by 38.97.110.5 ( talk) 17:24, 28 December 2016 (UTC)
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Wikipedia is an encyclopedia, not a math textbook. This article should be rewritten eliminate the use of "we". Ira Leviton ( talk) 21:23, 27 June 2020 (UTC)
The graphic "File:Numerical dispersion of a pulse signal in 1D FDTD.ogv" has an incorrectly labeled y axis.
It says, "Electric field magnitude (Arbitrary)" but it cannot be the magnitude as E<0. It should be labeled, "Electric field (Arbitrary units)" or the graphic itself should be changed to show the magnitude.
Cheers. 73.219.235.86 ( talk) 22:51, 13 February 2024 (UTC)