Talk:Euler integration

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Shouldn't this article discuss the use of the Euler method for solving general ODEs rather then using trajectories as a specific example? Thydzik 15:12, 9 November 2005 (UTC)

It should indeed. I'd be very grateful if you could amend the article to talk about the general case. -- Jitse Niesen (talk) 15:29, 9 November 2005 (UTC)

(After more than a year): Nothing has changed... just sad (130.113.128.11 19:54, 24 November 2006 (UTC))

[edit] Error discussion

In the "Error" section, someone said

Even if the Δt2 term is removed through a common adjustment to the Euler integrator, the error still contains third-order terms in Δt.

I'm assuming the "common adjustment" is changing:

x(t0 + Δt) = x(t0) + Δtv(t0)

to

x(t_0 + \Delta t) = x(t_0) + \Delta t v(t_0) + \frac{1}{2}\Delta t^2 a(t_0)

Given this, would the error be zero if acceleration were constant (e.g., gravity only)? It seems like this might be worth expanding on - I'm just a little new to the subject matter, so don't want to make bold changes just yet.

Also, what about Verlet Leapfrog and Velocity Verlet? I can't find them on Wikipedia, maybe they should have articles? --- MickWest 20:14, 29 March 2006 (UTC)

I think you're right about what you guess the "common adjustment" means; I'd call it the second-order Taylor method, but I'm sure there are other names for it. It is indeed exact when the acceleration is constant; this might be worthwhile to add.
We do have an article about the Verlet method at Verlet integration. We should probably add some redirects to it though. Where did you look for it? The list of numerical analysis topics might be helpful next time you're looking for an article. We don't have an article on velocity Verlet as far as I'm aware, so please write one (it is mentioned in Beeman's algorithm). -- Jitse Niesen (talk) 00:02, 30 March 2006 (UTC)
I saw the Verlet integration article, but I was looking for "Verlet leapfrog", red-linked as leapfrog method. from Discrete element method, see [1]. I need to do some more research before I edit anything here, but will hopefull get to it soon. Thanks for the response. MickWest 02:22, 30 March 2006 (UTC)
Fair enough. The Verlet method and the leapfrog method are almost the same, so that's why they are sometimes taken together and called the Verlet/leapfrog method. To wit, the website you provide defines the leapfrog method as
x_{k+1} = x_k + \Delta t \, v_{k+1/2}
v_{k+3/2} = v_{k+1/2} + \Delta t \, f(x_{k+1})
From the first equation, we get
x_{k+1} - 2x_k + x_{k-1} = (x_{k+1}-x_k) - (x_k-x_{k-1}) = \Delta t \, v_{k+1/2} - \Delta t \, v_{k-1/2}
and from the second equation, this is equal to (Δt)2f(xk), so we get
x_{k+1} - 2x_k + x_{k-1} = (\Delta t)^2 f(x_k). \,
This is equivalent to
x_{k+1} = 2x_k - x_{k-1} + (\Delta t)^2 f(x_k), \,
which is the formula on Verlet integration.
Yes, this should most definitely be added to that page. -- Jitse Niesen (talk) 02:43, 30 March 2006 (UTC)

[edit] Error Correction

In the part that compares Euler’s integration and Taylor expansion, the difference between both expressions has a sign mistake. The 1/2 part sould be -1/2 instead of the postivie expression shown in the article.

Yep, you're right. Thanks for catching that. -- Jitse Niesen (talk) 12:30, 14 April 2006 (UTC)
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