Talk:Runge–Kutta methods

message: "...on the order of h4"
That's the first mention of h in the article.  Perhaps it should be explained somewhere?  --Doradus 03:28, 11 Jul 2004 (UTC)
replies: text: Oops, my mistake.  It's fine as-is.  --Doradus 03:29, 11 Jul 2004 (UTC)

text: Perhaps someone can explain what h is?  —Preceding unsigned comment added by 203.122.193.233 (talk) 17:13, 2 December 2010 (UTC)
replies: text:  changed the section to mention  h = \Delta t  Rcgldr (talk) 01:54, 27 June 2011 (UTC)
replies: text:  mention of h = Δt was removed from the article. Since the article makes many references to t as time, it would make since to at least mention that h = Δt (a time step) somewhere early on. Even in the cases where t is not time, since the article updates t via t_{n+1} = t_n + h, it would still make sense to mention that h = Δt. Rcgldr (talk) 16:46, 1 February 2017 (UTC)
thread_title:  Definition of ''h'' 

message: I removed the last phrase of "The RK4 method is a fourth-order method, meaning that the total accumulated error is on the order of h4, and that it gives the exact integral for polynomials up to the fourth order." because it is not clear for somebody who does not know the subject. To make it clear, it should be explained that methods for solving ODEs can also be used for calculating integrals, that the error in solving ODEs is related to the error in calculating integrals, and that the latter is related to the degree of polynomials that can be integrated exactly. While all this can be done, I think it is not relevant for this article; perhaps it can be added to numerical ordinary differential equations. -- Jitse Niesen 00:03, 8 Jun 2005 (UTC)
thread_title: Local error

message: The Usage section in the article as of now is quite voluminous and I also think it violates WP:NOT#HOWTO. --Berland 18:33, 28 June 2007 (UTC)
replies: text: I agree it's too long, but to me it's more of an example than a how to. More seriously though, it seems to follow a different notation from the rest of the article and so is more likely to confuse than clarify. Is an example really needed here or is the article clear without it?--RDBury (talk) 05:39, 21 November 2007 (UTC)
thread_title:  The Usage-section 

message: How is 'Runge' pronounced? I'm not wild about IPA phonetics, but a "rhymes with" would be awfully useful. :)  —Preceding unsigned comment added by Myself248 (talk • contribs) 23:18, 24 November 2007 (UTC)
replies: text: The 'u' is (approximately) pronounced like the 'oo' in 'look', the 'ng' is as in 'sing', and the 'e' is as in 'the'. I don't know a word which rhymes with 'Runge'. -- Jitse Niesen (talk) 16:45, 25 November 2007 (UTC)
thread_title:  Pronunciation? 

message: I suppose the banner saying that purpose of the wikipedia is only to present facts is put there by some sort of robot, but it is unfortunate.  The wiki disavows being a reference that meets scholarly standards.  So an emphasis on merely "presenting facts" seems rather silly.   Doing so would result in some rather barren pages for mathematics.  
It would be useful to have an example of stability analysis.  From an article on the scholarpedia, I see that this involves an equation with matrices in it.  For persons who are not at the top of their game in matrix notation, an example would clarify whether a notation means the number 1 or the identity matrix, the absolute value of something or a determinant, etc.  
It is interesting that the vector b, which appears at the bottom of the table is written horizontaly and denoted as b_transpose. This suggests that, in another way of looking at things, b is naturally thought about as a column vector. It would be useful to comment upon this.  
In the spirit of knowing that certain approximations are bad, it would be useful to comment on whether pronouncing "Runge" to rhyme with "lunge" , as mamy USA instructions do, is a good one.
Tashiro 15:22, 1 December 2007 (UTC)
thread_title:  Not to teach subject matter ?! 

message: Whatever the Wiki is, it is presumably trying to present HUMAN useful information, and an example is often used in mathematical articles to better communicate what is meant by what was more abstractly said. The author's example should stay, and  the editor's advisory should go.  —Preceding unsigned comment added by 76.191.132.120 (talk) 20:06, 22 January 2008 (UTC)
thread_title:  Advisory excluding examples from Wiki Articles should be removed ==

message: At the time of writing the article might be clear as crystal to someone who already knows what the Runge-Kutta method is trying to achieve, but to someone who is pretty
much a beginner like me I am left wondering: OK the method produces a series of yk's but what do these represent? In other words what is the meaning of the output of the
method? Feraudyh (talk) 14:00, 23 April 2008 (UTC)
thread_title:  Leaves something to be desired 

message: Runge-Kutta can be used for PDEs as well (I believe mostly parabolic) should a section on using Runge-Kutta for PDEs exist on this page? What is everyones thoughts on this? Im just a mere math undergrad with little sophistication when in comes to numerical analysis. --User:MATThematical
RK methods can be applied to well-posed initial boundary value problems (whether parabolic or hyperbolic), however, the bare application to those methods isn't terribly informative and the analysis of the limitations of those applications is quite in depth. There are literally whole bookshelves worth of material devoted to understanding just that. -- Grigjd3  — Preceding unsigned comment added by Grigjd3 (talk • contribs) 13:23, 27 May 2011 (UTC)
thread_title:  PDEs 

message: I see lots of literature that says the error term is O(h^5) for a single step, and O(h^4) globally, but the error term derivation is never given.  It should be pretty straightforward using the Taylor series.  Would be great if someone could add it (I'd do it myself, but the derivation would probably constitute original research, plus I could make a mistake :P  It must be in a text book that someone has somewhere).  --Numsgil (talk) 08:14, 24 May 2008 (UTC)
thread_title:  Error term 

message: The current name of this page (and others that are linked from it) includes an endash, but it should be a hyphen. This page needs to be moved to its hyphenated version. Among other things, it would make it easier for people to link to it.
--TedPavlic | talk 19:48, 6 August 2008 (UTC)
replies: text: Why exactly should it be a hyphen? According to WP:DASH, one of the roles of en dashes is to indicate disjunction, with as one of its main applications being to serve as "a substitute for some uses of and, to or versus for marking a relationship involving independent elements in certain compound expressions". Runge–Kutta methods are methods due to Messrs. Runge and Kutta, being independent individuals. Still according to WP:DASH: "When naming an article, a hyphen is not used as a substitute for an en dash that properly belongs in the title". As Runge-Kutta methods with a hyphen redirects to Runge–Kutta methods with an en dash, linking is easy enough.  --Lambiam 22:37, 12 August 2008 (UTC)
replies: text: To throw more wood into the fire: It should neither be hyphenated nor en-dashed. (Although, you can argue for the hyphenation; the en-dash is entirely wrong.) I cite Brian Garner on this issue: proper nouns used in a phrasal adjective should not be hyphenated.  — Preceding unsigned comment added by 138.246.2.239 (talk) 11:53, 14 October 2014 (UTC)
thread_title:  Hyphenated name, not endashed 

message: The final Butcher tableau in this article (added in this  edit) looks to be incorrect. I believe it should be
:
\begin{array}{c|cc}
0 & 0 &  \\
1 & \frac{1}{2} & \frac{1}{2}\\
\hline
  &  \frac{1}{2}&\frac{1}{2}\\
\end{array}
Correcting my own LaTeX typos.
--92.242.168.176 (talk) 16:43, 15 March 2010 (UTC)
thread_title:  Crank-Nicolson Example 

message: Is there a problem with the defnition of k_i and the corresponding description of k_i? For example k_1 = h*f(t_n,y_n), and f(...) is dy/dt. so h*f(t_n,y_n) is infact "the difference in y over the interval t -> t+h, based on the slope at the beginning of the interval" and not "the slope at the beginning of the interval" as in the current text. The same for definitions/descriptions of k_2, k_3 and k_4.  —Preceding unsigned comment added by 143.117.13.21 (talk) 16:04, 2 June 2010 (UTC)
replies: text: changed the section so that k_i are termed "deltas" Rcgldr (talk) 01:49, 27 June 2011 (UTC)
replies: text:  the section was changed so the k_i are not defined as h f(...), so they're back to being slopes. Rcgldr (talk) 16:35, 1 February 2017 (UTC)
thread_title:  definition of k as a slope 

message: I was surprised to find no mention of Runge Kutta Merson, which, adding a fifth evaluation to RK4, enables a better interval to be computed.  In cases where stability varies over the range of integration, the cost of the extra evaluation is more than balanced by the reduction in the number of evaluations required.  I regret my maths is not up to the high standard set in this article, but I found RKM so useful that I am mentioning it here hoping that someone better qualified will put it in. John Wheater (talk) 11:14, 3 September 2010 (UTC)
thread_title:  Merson 

message: I understand there exists a class of integrators referred to as Runge-Kutta Nystrom integrators designed to allow one to integrate second order ODEs using the method of lines. My personal literature search has only revealed to me some of the most recent findings about such systems at all. If anyone is knowledgeable about such methods, it would be nice to see an explanation of them and perhaps a couple of useful examples (for instance, ERKNs at various orders).  — Preceding unsigned comment added by Grigjd3 (talk • contribs) 13:29, 27 May 2011 (UTC)
replies: text: Yes, it would be nice to see a page on RK-Nystrom methods, and yes, you're correct they're used to integrate 2nd-order ODEs.   Hairer, Nørsett & Wanner (1993) have a section on numerical methods for second-order ODEs, II.14, pages 283--301, that includes a couple examples that I would be willing to describe.  Where would an appropriate place be to add them?  Should I add them to this page or a new page? MathRocks2012 (talk) 21:14, 6 August 2013 (UTC)
thread_title:  Runge-Kutta Nystrom 

message: I have some major doubts on the derivation of Runge-Kutta fourth order method as it is presented now, even if I found this explanation elsewhere (for example, here) and even if, at first, this sounds right, now I am wondering why you are allowed to take the k_i coefficients up to \mathcal{O}(h^2) shouldn't they be expanded at least up to \mathcal{O}(h^4) to obtain a \mathcal{O}(h^5) expression? Some clarifications are in order. -- CristianCantoro (talk) 14:18, 15 March 2013 (UTC)
replies: text: I agree with the previous statements.  Butcher tableaus were invented to aid with the derivation of order conditions using rooted trees.   We could replace this discussion to define order conditions for any fourth-order method, and then simply remark that one such solution is given by the coefficients provided in the classical Runge-Kutta sheme. MathRocks2012 (talk) 19:08, 9 August 2013 (UTC)
thread_title:  Doubts on the derivation of Runge-Kutta fourth order method 

message: There is a parsing error being thrown in Derivation of the Runge–Kutta fourth-order method. Sizeofint (talk) 11:35, 13 December 2015 (UTC)
thread_title:  Parsing error 

message: Substituted at 02:34, 5 May 2016 (UTC)
thread_title: Assessment comment

message: The picture of the classical method is not consistent with the text. It's the same confusion as earlier mentioned, about the definition of the k's with or without h. In the text, the k's are y-values, and in the picture they are slopes. Madyno (talk) 13:49, 2 March 2019 (UTC)
thread_title:  Picture 

message: --2.231.51.224 (talk) 14:07, 2 January 2020 (UTC)
thread_title:  The first image has writing in German 

message: The article gives the following Butcher Tableau for the  trapezoidal rule. 
:
\begin{array}{c|cc}
0 & 0& 0\\
1 & \frac{1}{2}& \frac{1}{2}\\
\hline
  &  \frac{1}{2}&\frac{1}{2}\\
  & 1 & 0 \\
\end{array}
Is that correct? I did't think that the trapezoidal rule is an embedded method, so it shouldn't have the b^* row. -The-erinaceous-one (talk) 21:26, 14 August 2020 (UTC)
thread_title:  Butcher Tableau for the trapezoidal rule? 

message: I have school-level maths but was able to understand RK4.  Thanks to those who wrote the text so carefully.  I am working with autonomous differential equations.  The one sentence I do not understand is about them, currently written as:
In many practical applications the function f {\displaystyle f} f is independent of t {\displaystyle t} t (so called autonomous system, or time-invariant system, especially in physics), and their increments are not computed at all and not passed to function f {\displaystyle f} f, with only the final formula for t n + 1 {\displaystyle t_{n+1}} t_{n+1} used. 
Meaning of "their increments"?
"With only the final formula . . . used"? Is this saying that only k4 is used?
If anyone could clarify this, I'd be grateful.  — Preceding unsigned comment added by KPD674 (talk • contribs) 00:55, 18 February 2021 (UTC)
replies: text: , I think that the point that sentence is making is that, if f() doesn't depend on t explicitly, then it's not necessary to calculate t_n+h/2 when calculating k_2, etc.  This strikes me as a statement of the obvious.  Given that it's unsourced I wonder whether it would be better to remove that sentence entirely.  Best, Wham2001 (talk) 07:54, 18 February 2021 (UTC)
thread_title:  Autonomous (time invariant) equations 

message: Would there be any objections to adding the Kutta-Merson method to the section on adaptive RK methods? See, e.g., https://encyclopediaofmath.org/wiki/Kutta-Merson method. My impression is that it's less well-known than RKF45, and seems to have a tendency to overestimate errors in some circumstances, but it's also relatively straightforward and uses the traditional RK4 for the the fourth-order method.
If I don't hear any objections I'll likely add it later this week. DrPippy (talk) 14:19, 3 November 2021 (UTC)
thread_title:  Kutta-Merson method 

message: Have these methods been overtaken? Are other methods used today by default? Are they just of interest for teaching purposes? Are they still used if computing power is limited? Are they used as part of more modern methods? I think we should be told.  Burraron (talk) Burraron (talk) 10:22, 5 April 2024 (UTC)
thread_title:  Still used today?