Talk:Von Neumann stability analysis

Introduction text

during the World War II after having been briefly described in a 1947 article by British researchers Am I misinterpreting this completely, it seems to suggest that WWII happened after 1947. Keith 11:13, 16 July 2009 (UTC) —Preceding unsigned comment added by 131.180.16.252 (talk)

Notation: Discrete vs. Analytical

Salih: Looking at eq. (1), it seems to me that u_i^n (lower case u) is the exact solution of the discretized equation, while U_i^n (upper case U) is the exact solution of the partial differential equation. If not, then either eq. (1) is only approximately true, or epsilon is identically zero? JJL (talk) 17:37, 26 March 2009 (UTC)

Agree that there is some ambiguity regarding the meaning of u_i^n and U_i^n. I will try fixing it. Thanks for pointing it out. Salih (talk) 03:52, 27 March 2009 (UTC)

I clarified my comments above. The notation is inconsistent throughout the literature, but what I typed above matches e.g. G.D. Smith's text and seems to me to be what is intended here. I was trying with my edit to make it better without having to do a full fix of it. JJL (talk) 03:58, 27 March 2009 (UTC)

Hope I have removed the ambiguity. pl see it now. Most of the material I have taken from the J. D. Andersons' book titled "Computational Fluid Dynamics" Salih (talk) 04:23, 27 March 2009 (UTC)

Hmmmmm, I think "Since exact solution $U_{i}^{n}$ must satisfy the discretized equation" is still incorrect, or at least in conflict with eq. (1). From eq. (1), u_i^n (lower case u) is the exact solution of the discretized equation. I don't think that U_i^n (upper case U) is ever defined. Is it as simple as change the U_i^n (upper case U) to u_i^n (upper case u)? I'll try an edit. JJL (talk) 14:07, 27 March 2009 (UTC)

It is the exact solution of the "discretized equation" (not the exact solution of PDE) that has to satisfy the discretized equation exactly. I don't see any problem with your edits, except, I feel the solution of the discretized equation computed in the absence of round-off error may be better denoted by U_i^n. Salih (talk) 14:37, 27 March 2009 (UTC)

But doesn't eq. (1) define it as u_i^n? What's the benefit of also having U_i^n, if they're the same? JJL (talk) 14:47, 27 March 2009 (UTC)

In eq. (1) u's are variables, whereas the solution is a numerical value. Salih (talk) 16:26, 27 March 2009 (UTC)

That's what N is supposed to be, right? JJL (talk) 18:13, 27 March 2009 (UTC)

There is a difference; N denotes the "numerical solution" of the difference equation while U denotes the "exact solution" of the difference equation. Salih (talk) 04:40, 28 March 2009 (UTC)

Equation (2)

Eq. (2) seems plausible, and the basic idea is right, but I do not believe that it's exactly right. An initial data error at time level zero is governed by the same equation, because the round-off error at every level precisely follows the FDE? I've always done von Neumann analysis by expanding u_i^n in a discrete FS. That matches Smith, Morton and Mayers, and Strikwerda, for example. How can one derive eq. (2) from eq. (1) and the defn. of the error? JJL (talk) 14:47, 27 March 2009 (UTC)

Since the numerical solution satisfies eq. (1), we have

$\qquad N_{i}^{n+1}=N_{i}^{n}+r\left(N_{i+1}^{n}-2N_{i}^{n}+N_{i-1}^{n}\right)$

We also have

$N_{i}^{n}=U_{i}^{n}+\epsilon _{i}^{n}$

Thus, we may write

$\left(U_{i}^{n+1}+\epsilon _{i}^{n+1}\right)=\left(U_{i}^{n}+\epsilon _{i}^{n}\right)+r\left[\left(U_{i+1}^{n}+\epsilon _{i+1}^{n}\right)-2\left(U_{i}^{n}+\epsilon _{i+1}^{n}\right)+\left(U_{i-1}^{n}+\epsilon _{i+1}^{n}\right)\right]$

From the above equation it follows that

$\qquad \epsilon _{i}^{n+1}=\epsilon _{i}^{n}+r\left(\epsilon _{i+1}^{n}-2\epsilon _{i}^{n}+\epsilon _{i-1}^{n}\right)$

Hope it's clear. Salih (talk) 16:43, 27 March 2009 (UTC)

But the article says: "$N_{i}^{n}$ is the numerical solution obtained in finite precision arithmetic". By definition, $N_{i}^{n}$ is not the solution of the discretized equation but rather a perturbed version of it. E.g., if we take $r=1/\pi$, the computed $N_{i}^{n+1}$ cannot possibly the desired one in finite precision. So, I disagree with your first statement above. JJL (talk) 17:39, 27 March 2009 (UTC)

If you have access to "Computational Fluid Dynamics: The Basics with Applications" by J. D. Anderson Jr., please refer to it. That is my source. Salih (talk) 18:04, 27 March 2009 (UTC)

I don't have that source to hand, but what's written is at odds with the three numerical PDEs texts I've mentioned above. The N variable doesn't satisfy the same FDE. The theoretical value of u does, but not the computed value of N. Can you expand on why you believe (or what Anderson says about) "Since the numerical solution satisfies eq. (1), we have " and the equation in N that follows it? JJL (talk) 18:13, 27 March 2009 (UTC)

Anderson's book does not elaborate on it. If something is at odds with G. D. Smith's or any other books you have, please edit it to fix the problem and source it accordingly. Salih (talk) 18:32, 27 March 2009 (UTC)

Will do! The basic idea is certainly right, and the initial error does die off as indicated, but epsilon seems to be a cumulative error and that's a bit more involved. I'll rework it this weekend per Smith. JJL (talk) 19:26, 27 March 2009 (UTC)

That's fine! Salih (talk) 04:00, 28 March 2009 (UTC)

I want to add here that the description of this error as that coming from fixed-precision arithmetic is entirely incorrect, whether it is in Anderson's text or not. As JJL said. The accumulated error that comes about from truncating the Taylor series of the PDE completely dwarfs the computational arithmetic error. — Preceding unsigned comment added by 140.180.11.141 (talk) 18:28, 18 May 2012 (UTC)

Seconding this -- it's completely wrong in a way that should be pretty obvious to numerical methods people. Anderson's text getting this wrong doesn't outweigh all the others that got it right. — Preceding unsigned comment added by 68.36.201.166 (talk) 22:26, 12 September 2013 (UTC)

I have the book which is cited (Anderson) and I also though it was wrong in assuming that the numerical solution satisfy the difference equation (hence I came here to see if there might be a better explanation). The book does mention that there is a round off error, but does not include it into the equation. To the above post: don't confuse stability and accuracy, they are connected, but not the same ;) — Preceding unsigned comment added by Serpej (talk • contribs) 10:48, 25 May 2012 (UTC)

Notation confusion in example

I don't have the time to fix this right now, but I just wanted to point out that it appears that in the example the symbol i is used both for the iteration number and for an imaginary number. moink (talk) 13:09, 23 November 2009 (UTC)

You are right, there is a room for confusion. I am going to fix it. Salih (talk) 13:44, 23 November 2009 (UTC)

Courant–Friedrichs–Lewy Condition

Von Neumann stability analysis seems strongly related to the Courant–Friedrichs–Lewy condition (especially considering the example from this article) but I don't think they are the same thing because all the references are different. Could some one explain how the two ideas are related? Njerseyguy (talk) 21:23, 12 May 2010 (UTC)