Talk:Stiff equation/Archive 1

Archive 1

Trapezoidal method

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)

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)? — Preceding unsigned comment added by 213.130.27.39 (talk) 06:44, 13 March 2006 (UTC)

You're right. Thank you very much for mentioning this. -- Jitse Niesen (talk) — Preceding unsigned comment added by Jitse Niesen (talk • contribs) 06:54, 13 March 2006 (UTC)

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 $y'=-15y,y(0)=1$:

$y_{1}=y_{0}+hf(y_{0})=1+{\frac {1}{4}}\cdot (-15)\cdot 1=-{\frac {11}{4}}=-2.75$

$y_{2}=y_{1}+hf(y_{1})=-2.75+{\frac {1}{4}}\cdot (-15)\cdot \left(-{\frac {11}{4}}\right)=-{\frac {121}{16}}=7.5625$

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)