From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Determining stiff DEs
Date: 26 Jan 2001 12:19:33 GMT
Newsgroups: sci.math.num-analysis
Summary: When is a differential equation "stiff"?

In article <94rkmi$d5g$1@nnrp1.deja.com>,
 Josh <josh_downer@my-deja.com> writes:
|> Is there a method for determining if a differential equation is 'stiff'?
|> I know that the idea is that there should be multiple time-scales in the
|> equations, but is there a measure of the 'stiffness'?
|> 
this is a qualitative characterization as is "illconditioned" for a matrix.
roughly one would qualify an ode as "stiff" if there are time-scales
which are large compared to the length of the timeinterval where the
solution is sought, that is 
   |lambda|/(t_end - t_beg) >> 1
so you need a small stepsize if you use explicit integrators.
the best indicator of stiffness is running an explicit integrator with
a good error estimation/stepsize control. this will integrate with
stepsizes near the boundary of the stability region, i.e. stepsizes
like
     1/|lambda|

a crude first information is given by the eigenvalues of the jacobian
of f(t,y) with respect to y. But this may be misleading, since the behaviour 
of a nonlinear ode is not correctly described by its linearization.

a better characterization may be with respect to a specific initial value
problem:
an initial value problem is stiff if the vicinity of its true solution 
contains trajectories of the differential equation whose time derivatives are
much larger than that of the true solution itself.
typical example: (from Gear)

y' = lambda*(y-exp(-t))-exp(-t), y(0)=1, 
true solution 
y(t)=exp(-t)
solution manifold  y(t)=c*exp(lambda*t)+exp(-t) for arbitrary c. 
take lambda=-10000, then the time derivatives of the true solution 
are all +(-)exp(-t) but those of the solution manifold are 
c*(10000^k)exp(-10000*t)+(-)exp(-t), (c depends on the initial value)
so near any (t0,exp(-t0)) there are solutions of the ode with hugh 
derivatives thereas the true solution of the ivp is smooth.
hope this helps
peter
==============================================================================

From: "Leifa" <leif-a-somewhere@upnorth.no>
Subject: Re: Determining stiff DEs
Date: Sat, 27 Jan 2001 00:30:50 +0100
Newsgroups: sci.math.num-analysis


Peter Spellucci <spellucci@mathematik.tu-darmstadt.de> wrote in message
news:94rq0l$nhp$1@sun27.hrz.tu-darmstadt.de...
[quote of previous article deleted --djr]

Best definition I ever heard was from Prof. Nørsett in Norway.

"A system is stiff when stepsize  is controlled by stability rather than
accuracy."

Happy computing

Leif-Arne

Leif-A-Stopjunk@online.no

topjunk is not a part of my e-mail address