Talk:Linear multistep method

Dahlquist Barriers

I think that can be useful to put a link to the Dahlquist barrier, since they're strictly related to this argument —Preceding unsigned comment added by 79.21.118.218 (talk) 13:24, 26 June 2008 (UTC)

AB coefficients

I believe the Adams-Bashforth Formula is wrong. I get the correct result for the Moulton example, but I can't get the Bashforth to jive with the error conditions. Or am I just crazy? Sim 01:04, 20 December 2006 (UTC)

It's correct.

The coefficients are correct. I just checked the following paper:

Purser, R. and L. Leslie. "Generalized Adams-Bashforth time integration schemes for a semi-Lagrangian model employing the second-derivative form of the horizontal momentum equations." Journal of the Royal Meteorological Society 122 (1996): 737-763.

Thanks, Wojciech.

Function of one variable or two?

In the examples, wouldn't it be better to be consistent as to how many variables f is a function of? It seems like there are a bunch of missing ts.129.2.134.104 (talk) 12:22, 21 July 2008 (UTC)

Stability and A-stability

It is important to distinguish between stability of a scheme, which shows that the scheme gives the correct result if the time-step is small enough, and A-stability, which gives an estimate of how big the time-step can be in practical implementations. e.g. The explicit Euler method (first order Adams-Bashforth) is strongly stable ($z=1$ is the only root of the characteristic polynomial), but it won't give accurate results for stiff equations unless the step-size is extremely small.

Unfortunately, A-stability doesn't have an article of its own yet, and I don't know when it will have. In the meantime, I've redirected the links to A-stability so that they point to the A-stability section of the article on stiff equations. If someone reads this comment and an A-stability page is up and running, please redirect the links to the A-stability page. --Woodford (talk) 09:30, 29 September 2008 (UTC)

For practice as I am new, and because an article on Zero stability was requested on the "Things to do" page for WP Mathematics, I wrote Zero stability and put a link there to this page; however, apparently "Zero stability" redirects here anyway (and indeed there was a sentence here defining it). My little article is barely more than a stub, but it does have an example. Can someone edit the redirect so that it points there? Or comment on that page? Thanks. Rob.Corless (talk) 21:48, 9 March 2022 (UTC)

AB coefficients

I noticed that the sum of the coefficients is always 1, for both Adams methods. Does anyone know a link to a proof? --Astabada (talk) 16:18, 31 March 2009 (UTC)

It follows from the formula

$\sum _{k=0}^{s}b_{k}=s+\sum _{k=0}^{s-1}ka_{k}$

in the Analysis section (follow the reference provided for a proof, though I guess it's enough to study what happens if you apply the method to the equation y' = 1). The Adams methods all have

$a_{s-1}=1,a_{s-2}=\ldots =a_{0}=0,$

so the formula says that the sum of the coefficients is 1. -- Jitse Niesen (talk) 01:36, 1 April 2009 (UTC)

A coefficients of 4 levels

The coefficient 2774/360 is mistake, this coefficient should be 1387/360. This fact can prove using formula $\sum _{i=0}^{s}b_{i}=1$ —Preceding unsigned comment added by 217.197.4.125 (talk) 10:53, 25 April 2009 (UTC)

You're absolutely right. Many thanks for notifying us of this mistake. -- Jitse Niesen (talk) 11:41, 25 April 2009 (UTC)

Adams-Bashforth-Moulton formatting edits

Putting everything into an align removed the kludge with the bullet points (which where visually misaligned due to the multiline 5th order equation), but lost the wikilinks in the supporting text as a consequence (see http://en.wiki.x.io/w/index.php?title=Linear_multistep_method&diff=556821718&oldid=556820556 ). Was this an acceptable regression? Nelfin (talk) 04:40, 26 May 2013 (UTC)

consistent display for the 5-step Adams--Moulton method

Hi --- the coefficients for the 5-step Adams--Moulton are not reduced as they are for the others.

I understand sometimes the preferred format is to have the same denominator whereas at other times it may be preferred to have simplified expressions.

We should at least be consistent :)

To minimize the amount of editing, my suggestion is to simplify / reduce the fractions in the 5-step Adams--Moulton method to be consistent with the rest of the methods. — Preceding unsigned comment added by 24.244.29.62 (talk) 16:39, 26 February 2016 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Linear multistep method/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Has quite a bit of the basic info, but needs some reorganization. Include other methods (e.g., BDF), Dahlquist barrier, example. Jitse Niesen (talk) 07:15, 10 May 2007 (UTC) Needs references. Geometry guy 23:28, 10 July 2007 (UTC)

Last edited at 12:51, 22 July 2007 (UTC). Substituted at 02:17, 5 May 2016 (UTC)