Talk:Finite difference method

What to do about this article?

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)

Or maybe finite difference scheme could be merged into finite difference, but I don't feel comfortable enough to do it. Oleg Alexandrov (talk) 03:26, 25 July 2006 (UTC)

I think a radical reorganization is in order. I propose the following. Finite-difference method can contain the standard explicit 2nd order scheme for heat equation, some analysis (consistency, stability, convergence), other schemes like implicit in time and Crank-Nicholson (up to here this is the content of Finite difference#Example: the heat equation), discussion on how to handle boundary conditions, the centered-difference scheme for the Laplace equation and the same for the wave equation. This is easily enough for a decent article. We might want to add something about FD methods for ODEs.

Finite difference should contain the definitions of the forward, backward and centred difference, how they approximate the derivative, and perhaps higher-order approximations (basically the contents of this article, finite difference schemes). This should perhaps be merged with difference operator, though that is a rather nice article so we should be careful there.

Most redirects should be changed as a result. -- Jitse Niesen (talk) 11:49, 31 July 2006 (UTC)

If $x_{1}$ is my own experience with finite differences, and $x_{2}$ is your experience with the same thing, then the finite difference $x_{2}-x_{1}$ is actually infinite, so if you wish to reorganize things, that's fine with me. :) I also agree that reorganization is in order. Oleg Alexandrov (talk) 18:25, 31 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)

Ambiguous Terms

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)

I don't think this is referenced or verified. Darktachyon (talk) 18:52, 30 December 2007 (UTC)

Possible Error

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)

Changed some things

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:

• Add something about the derivation of finite difference methods using the method of undetermined coefficients
• Mention computational molecules / stencils
• Add some basic treatment of convergence, stability, consistency analysis -- they're terms that are brought up in the article now without any real definition or explanation anywhere
• There might be too many examples now; I added some more, unfortunately.

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)

You made a good start. Thanks, the article was not in a good shape. I'll see whether I can help you a bit. I think the lead is a bit too short now, but I agree that the ODE example probably shouldn't be there. If we need to delete an example, I'd remove Crank-Nicolson because it's the most complicated one. I agree with your list of things that should be added; I'd also like to see something about the treatment of boundary conditions. I'm not too fussed about references, but others are. If you could just add a couple of general-purpose references on finite difference methods, that would be great.

Now the stuff I'm not so keen on; mainly the section "Accuracy and Order" (minor detail: we don't capitalize section titles, so this should be "Accuracy and order"). As you say, local truncation error is discussed in relation to a method, but it's not clear what the method is here. You are computing the error in the approximation (I replaced i by k to avoid confusion with $i={\sqrt {-1}}$):

$f'(x)\approx {\frac {f(x_{0}+kh)-f(x_{0})}{kh}}.$

That's not an approximation that's used very often, except for k = 1 or k = −1. Furthermore, I think the text is a bit too long, given the overlap with the section "Derivation from Taylor's Polynomial" and with the finite difference article; that's a bit personal though, I like a condensed writing style.

What I would do is to postpone the discussion of order/accuracy until after you've given the simple method (forward in time, central in space) for the heat equation. Then discuss order, stability, convergence in that setting, and then give the other examples. I think that's a natural progression for the reader. What do you think? -- Jitse Niesen (talk) 12:17, 17 June 2008 (UTC)

Aha, thanks for the formatting. I agree that the lead is a little short side, but I thought it would probably be better to build it up from being very short than to try to cut it down usefully from the example-laden section it was before.

I agree completely with you about the accuracy and order section; I felt somewhat like I was sort of cramming it in where it shouldn't be, but error analysis is so central to the study of finite difference methods that adding it, even poorly, would make the article better than it would be without it.

Also, wow, it turns out I used a really stupid method for approximating the first derivative in the Accuracy and order section. Meant to do the standard forward-difference method for the first derivative, but it got mangled in a notation change.

Finally, I agree with your suggested reorganization and progression. I'll see if I can drum up some references and massage the Accuracy and order section into something a little more appropriate. Duplico (talk) 16:36, 23 June 2008 (UTC)

Stability criterion - unit dependent?

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)

The $\theta$ (theta)-Rule

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 $\theta$. The author of the following lecture notes : http://hplgit.github.io/INF5620/doc/notes/main_decay.pdf mentions the rule on page 10.

Jargon tag

The notion of scheme needs to be introduced before it's nonchalantly used. Some1Redirects4You (talk) 16:53, 27 April 2015 (UTC)

Conflict-of-interest reference removal

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)

Comparison section missing initial condition

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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Tone of article

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)

Error with "File:Numerical dispersion of a pulse signal in 1D FDTD.ogv"

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)