Talk:Stiff equation

I would like to know what people think about the trapezoidal method for solving first order differential equations with respect to stability and local and global error? 86.20.210.160 12:47, 11 February 2006 (UTC)

The trapezoidal method is second order, A-stable and implicit. Whether this makes it a good method depends on the application. -- Jitse Niesen (talk) 14:45, 11 February 2006 (UTC)

Is it large?

"with the Jacobian of f having at least one eigenvalue m ∈ C and | m | is large."

How mathematically to define if m is large? —Preceding unsigned comment added by 129.175.51.75 (talk) 13:53, 7 August 2009 (UTC)

I believe I have answered your question. See "Stiffness ratio, Characterization of stiffness, and Etymology" below. Anita5192 (talk) 06:49, 7 November 2011 (UTC)

In the same sentence: are we sure we are using the correct C symbol here: this one means "set of continuous functions" but perhaps we meant "set of complex numbers" as I don't see how an eigenvalue could be a function. Thanks. Erwan 79.80.42.47 (talk) 16:00, 7 March 2010 (UTC)

An error maybe?

«Still using the same problem, we reattempt with the trapezoidal method, that is, the two-stage Adams-Bashforth multistep method,»

Maybe here should be Adams-Moulton method (implicit) and not Adams-Bashforth (explicit)?

You're right. Thank you very much for mentioning this. -- Jitse Niesen (talk)

Stability graph for methods

It would be interesting to see a stability graph in C plane here for various methods. I remember one in my college book (look, ma, I've been on college, hoooo), but I believe it is copyrighted. Plain formula does not give that sense of increased stability in trapezoid method over Euler. My $0.02. -- Mtodorov 69 12:57, 5 April 2007 (UTC)

Quite right. If anyone can be bothered to draw and upload figures of the stability regions for Euler, trapezoidal and A-B methods, please do. Perhaps I will get round to it someday. Paul Matthews 11:55, 26 April 2007 (UTC)

problem with picture

I don't know who created the algorithm for the approximations but if you use euler's method on the stated differential, the approximations would not oscillate wildly as stated in the article. the approximations would smooth out as the slope of the tangents to the function approach zero. Perhaps a "stiffer" function should be chosen.--Cronholm144 05:43, 20 May 2007 (UTC)

I beg to differ. Take Euler's method with step size h = 1/4 for y' = − 15y,y(0) = 1:

$y_1 = y_0 + hf(y_0) = 1 + \frac14 \cdot (-15) \cdot 1 = -\frac{11}4 = -2.75$

$y_2 = y_1 + hf(y_1) = -2.75 + \frac14 \cdot (-15) \cdot \left( -\frac{11}4 \right) = -\frac{121}{16} = 7.5625$

etc. -- Jitse Niesen (talk) 07:52, 20 May 2007 (UTC)

Blast! Negative slope and cursory observations/comments will be my undoing. Sorry about my foolishness.--Cronholm144 08:22, 20 May 2007 (UTC)

Beware of negative slopes -- Jitse Niesen (talk) 08:48, 20 May 2007 (UTC)

Also, red/green is not good for red/green colorblindness.--Lingwitt (talk) 04:01, 13 March 2008 (UTC)

implicit rk-method

hwo about an implicit rk-method in the plot? the plot isn't very friendly for rk-methods ;) —Preceding unsigned comment added by 91.130.182.31 (talk) 15:19, 22 April 2009 (UTC)

Useful new section?

Personally I would find it very useful if there was a section that stated which methods are commonly used with stiff equations/systems. It is my understanding from the article that one of the more stable methods is the Trapezoid Method though I could be wrong as the article isn't very clear for people with a mid range math knowledge. —Preceding unsigned comment added by 161.65.16.253 (talk) 04:55, 5 August 2009 (UTC)

Stability for RK4 Methods

The table of stability polynomials for the various RK4 methods (different B coefficients) appears to be erroneous. To quote Ascher and Petzold, 1998, SIAM, p88:

"Thus we note that all p-stage explicit Runge-Kutta methods of order p [which the RK4 method in question is] have the same regions of absolute stability"

Namely, this region is 1 + z + z^2/2 + z^3/6 + z^4/24.

Moreover, Hairer & Wanner, 1996, Springer, p17 equation (2.12) arrives at the very same conclusion.

--EvilGuru (talk) 23:51, 10 November 2010 (UTC)

Nonlinear case.

I just rephrased a bunch of the introduction. I was left with this, though, which I think needs more details:

Generalizing to the nonlinear case, the equation

$\mathbf{y}' = \mathbf{f}(\mathbf{y}, t),$

is stiff if the Jacobian of f has at least one eigenvalue, $m \in \mathbb{C}$, for which | m | is large.

But "large" needs to be with respect to something. In the linear case, y' = Ky + f(t), the Jacobian of f(y,t) is really a matrix with K as a left block and the identity as the right block (that is, $y'=[K I] [y^\top f(t)^\top]^\top$). With just that, then we would be talking about m as an eigenvalue of the left block of J rather than of all of J. It seems like what really matters is if any eigenvalues, m, of the left block of J (the block with $\partial y'_i/\partial y_j$ terms) to the eigenvalue, λ of the $\partial y'_i/\partial t$ column... Maybe I just answered my question. Of course, what is "large" when comparing different units... hmmm... —Ben FrantzDale (talk) 18:28, 12 November 2010 (UTC)

I believe I have answered your question. See "Stiffness ratio, Characterization of stiffness, and Etymology" below. Anita5192 (talk) 06:51, 7 November 2011 (UTC)

CFL and Neuman

Don't CFL and Neuman deserve some kind of mention with regards to the stability analysis? Or am I missing something terribly? 192.16.184.140 (talk) 13:44, 20 December 2010 (UTC)

This is about ODEs and not about PDEs. --P. Birken (talk) 15:48, 13 August 2011 (UTC)

Additions of User:Ramiamro

A while ago, Ramiamro and an IP (probably him) added a couple of sections, which I just removed. The first one was concerned with general definitions for methods for IVPs. The second one with an obscure example taken from a very specific article that has not really anything to do with stiffness, but just with the stability domain, for which already the instructive example of the explicit Euler and the trapezoidal rule is given. Finally, a section about plotting stability domains, which was first of all wrong and second again is not about stiffness in the first place, but again about stability domains. Best, --P. Birken (talk) 15:53, 13 August 2011 (UTC)

Stiffness ratio, Characterization of stiffness, and Etymology

I just added some new material to clarify the stiffness concept to readers and to answer some questions of the other editors.

A second paragraph has been added to the Introduction, introducing some additional concepts.

A new section, Stiffness ratio introduces and defines the stiffness ratio and how it encapsulates the behavior of the transient components of the solution to a class of differential systems.

The existing section, Characterization of stiffness previously claimed that an equation is stiff if, for an eigenvalue m, | m | is large. I have never seen this in the literature, so I replaced it with what I have seen in several places in the literature: that, among other things, an eigenvalue has negative real part. What was correct in this section I absorbed into the new section, Stiffness ratio. This section now describes several empirical statements about the phenomenon of stiffness and explains somewhat the vagueness of these statements.

A new section, Etymology explains possible sources of the name stiffness, and builds upon material in the earlier sections by using that material to construct an example which demonstrates a correlation between the physical stiffness of a physical system and the mathematical stiffness of its associated differential system. The notion of large is briefly explained.

Several new References have been added and some footnotes refering to them to support claims made in the text. Anita5192 (talk) 06:46, 7 November 2011 (UTC)