RE: Re: odd behavior of NDSolve
- To: mathgroup@smc.vnet.net
- Subject: [mg11747] RE: [mg11702] Re: odd behavior of NDSolve
- From: R Finley <trfin@umsmed.edu>
- Date: Thu, 26 Mar 1998 03:09:14 -0500
Hi Selwyn,
I agree that it is probably more correct to think of the problem in this
case as one of wrong step size instead of accumulated error, although
the two are not independent. It appears that in this case the wrong
step is taken as the first step and I note that there is a warning to
this effect in the documentation to NDSolve...where they give one
solution to this problem as specifying the StartingStepSize. Indeed
that works in your case (StartingStepSize->1)and is probably a better
solution than changing the MaxStepSize although I guess it would depend
on the interval you were interested in. One can also get the correct
result, however, by changing AccuracyGoal->Infinity.
RF
-----Original Message-----
From: Selwyn Hollis [SMTP:shollis@peachnet.campus.mci.net] To:
mathgroup@smc.vnet.net
Sent: Saturday, March 21, 1998 5:36 PM To: mathgroup@smc.vnet.net
Subject: [mg11702] Re: odd behavior of NDSolve
Richard Finley wrote:
> Hi Selwyn,
>
> Actually, I don't believe there is any mystery behind the behavior.
> Since it is a numerical integration routine, it is subject to
> accumulated error depending on step size, maximum number of steps,
> etc...and it just so happens in this case that it becomes unstable
> around 12.03.
Accumulated error is something I'm quite familiar with, but this
behavior is not about accumulated error. NDSolve is taking ONE giant
step, right past the interesting part of the solution when tmax >
12.03. As Paul Abbott correctly pointed out, the real cause is the
default value of Infinity for MaxStepSize. (Seems to me that a more
sensible default for MaxStepSize might be something like tmax/10.)
Selwyn
-----Original Message-----
> From: Selwyn Hollis [SMTP:shollis@peachnet.campus.mci.net] To:
To: mathgroup@smc.vnet.net
> mathgroup@smc.vnet.net
> Sent: Tuesday, March 17, 1998 9:43 AM To: mathgroup@smc.vnet.net
> Subject: [mg11747] [mg11702] [mg11580] odd behavior of NDSolve
>
> I've discovered a very disturbing problem with NDSolve. I'd like to
> know
> if others can reproduce this.
>
> Here's an example:
>
> f[x_, y_] :
Exp[-(x^2 + y^2)]
>
> fx[x_, y_] ¿[x, y], x]; fy[x_, y_]
[f[x, y], y];
>
> Clear[x0, y0, u0, v0];
> diffeqs {x''[t] Ð-fx[x[t], y[t]],
> y''[t] ßy[x[t], y[t]],
> x[0] , x'[0] Ðu0, y[0] Ðy0, y'[0] Ðv0}
>
> endTime .03;
> {x0, y0, u0, v0} L 3, -0.7, -0.17};
> soln õatten[NDSolve[diffeqs, {x, y}, {t, 0, endTime}]];
> r[t_] t], y[t]} /. soln;
> path rametricPlot[r[t], {t, 0, endTime},
> PlotRange -> {{-4, 4}, {-4, 4}}];
>
> You should see a curve with a slight bend to it. Now change endTime to
> 12.04. The bend is gone. Shouldn't we get the same curve for 0 < t <
> 12.03? Either I'm missing something or NDSolve has a serious problem.
>
--
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dr. Selwyn Hollis
Associate Professor of Mathematics
Armstrong Atlantic State University
Savannah, GA 31419 USA
<http://www.math.armstrong.edu/faculty/hollis/>
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~