Re: Can I solve that with Runge-Kutta?

From: Gerald F. Thomas (gfthomas_at_sympatico.ca)
Date: 03/09/05 

Date: Wed, 9 Mar 2005 11:50:28 -0500

"Donald R. Fredkin" <drfredkin@sbcglobal.net> wrote in message
news:Xns9613E9067DFB2dlnet@207.115.63.158...
> "J DUDLEY" <sbdpumbaa@blueyonder.co.uk> wrote in
> news:x1oXd.19219$y25.6413@fe3.news.blueyonder.co.uk:
>
> > What is in the Langevin equation that inhibits a Runge-Kutta code?
> > Typically all I have found that prevents 'success' is stiffness, with
> > stiffness increasing (approximately) as the square of the number of
> > elements.
>
> The Langevin equation is a stochastic differential equation (SDE). None
of
> the standard techniques for solution of ordinary differential equations
are
> applicable. You must, to start, understand what it means to solve an
> equation containing terms which are given only in a statistical sense.
And
> then you must recognize the extreme lack of smoothness in the random
terms.
> Stiffness is not an issue, and the BDF methods appropriate to stiff
> equations are guaranteed to fail for SDEs.
>

Indeed.

> There is an extensive literature on the subject. A reasonably undemanding
> starting point might be "Numerical solution of stochastic differential
> equations" by Peter E. Kloeden and Eckhard Platen, Springer-Verlag, 1992.
>
> I do not know of any canned codes.

DESMOND J. HIGHAM, AN ALGORITHMIC INTRODUCTION TO NUMERICAL SIMULATION OF
STOCHASTIC DIFFERENTIAL EQUATIONS, in SIAM Review, Education Section, Vol
43, 2001, 525--546, gives a link to Matlab .m scripts that illustrate this
tutorial on SDE's. Also www-math.bgsu.edu/~zirbel/sde/langevin.m might be
of interest to the OP. 

-- 
You're Welcome,
Gerry T.
______
"Things are not what they seem; or, to be more accurate, they are not only
what they seem, but very much else besides." -- Aldous Huxley.

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