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942:{\displaystyle {\begin{aligned}{\text{predictor }}&\\r_{n+1}&=r_{n}+hv_{n}+{\frac {h^{2}}{6}}(4a_{n}-a_{n-1})+{\frac {h^{4}}{8}}r_{n}^{(4)};\\{\text{corrector }}&{\text{ (after computation of }}a_{n+1})\\r_{n+1}&=r_{n}+hv_{n}+{\frac {h^{2}}{6}}(a_{n+1}-2a_{n})-{\frac {h^{4}}{12}}r_{n}^{(4)};\\hv_{n+1}&=r_{n+1}-r_{n}+{\frac {h^{2}}{6}}(2a_{n+1}+a_{n})-{\frac {h^{4}}{24}}r_{n}^{(4)}.\end{aligned}}}
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Additionally, methods of order 4 and 5 are given and numerically compared to Verlet, Adams-Multon-Bashford multistep, Rahman and
Nordsiek methods. Verlet is not used in the leapfrog or velocity Verlet variants, in this situation Beeman's third order method appeared more stable for larger steplengths.
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It looks like the equation was made by someone substituting one of the predictor-corrector modifications and removing the predictor-corrector terms. I removed the reference from the page that links to the incorrect source, and added one with a more complete list. I do not know if the error term is
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I've put a citation for the original Beeman paper into the source (hidden in a comment). If someone who has access to J. Comp. Phys. could please have a look at it and confirm that it is an appropriate reference, then that should save a little bit of time putting the details together!
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The errors of derivative are almost universally one order worse than the errors of the main value. So I will be surprised if the order would be dt^4 for both position and velocity. I took the liberty to edit the formula, it would be still worth to find the original paper anyway
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The first source says the error term on velocity is O(dt^4), whereas the other two sources claim O(dt^3). The best step is to probably find Beeman's original paper on the algorithm and be certain one way or another.
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I was unable to find the original paper from Beeman on it. That second equation looked a little screwy to me to, but I didn't know it well enough to second guess it properly. --
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Seems to have the basic info (except for stability), but needs a lot of editing. What equation is solved? What is the point of the predictor/corrector section?
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I think it's O(dt^3) also, however velocity verlet is an example of an algorithm with the same order of error for position and velocity. --
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988:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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correct, but the source I added seems much more reliable, so I suspect the error term from that source is also correct.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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The introduction mentions "Beeman" several times but never includes his or her first name.
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Last edited at 07:29, 10 May 2007 (UTC). Substituted at 01:48, 5 May 2016 (UTC)
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steps are the same as in the article, the corrector steps are at odds.--
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changes in velocity could not be proportional to velocity itself!
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272:At least it makes sense now. Thanks a lot
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