[ODE] Euler vs. Runge-Kutta and adaptive step sizes

[Mike Cline]

I have not seen this time-stepping scheme extended to higher-order 
integrators, but I have read a paper by Mihai Anitescu (who I think has 
done work with Stewart and Trinkle) which extends this time-stepping scheme 
to an implicit integrator (a linearized version of backward-Euler 
method).  This is useful for systems with stiff forces -- e.g. stiff spring 
forces.   With explicit methods, these systems tend to "blow up" much easier.

For what it is worth, I have also seen Anitescu claim in one of his papers 
that extending the time-stepping scheme to higher order integrators is 
"non-trivial".

Mike



>I believe the difficult part of non-Euler integration is the fact that
>ODE uses Stewart/Trinkle's time-stepping algorithm, where the numerical
>integration scheme which is used forms part of the system of DAE's to be
>solved. The integrator isn't separate; it's tightly bound into the
>entire solution scheme, since a "time stepping" algorithm solves not for
>instantaneous forces, but rather for their integrals over finite time
>periods (thus requiring a numerical integration scheme as part of the solution
>process).
>
>I have only seen one paper which deals with extending Stewart/Trinkle's
>time-stepping altorithm to higher-order integrators. See some previous
>messages on this topic,
>
>http://q12.org/pipermail/ode/2002-March/000976.html
>http://q12.org/pipermail/ode/2002-March/000978.html
>
>Regarding the last message, the author of the dissertation did not want
>to release his source code at this time.
>
>I'd be interested in hearing any progress on higher-order integrators.
>Reading Stewart/Trinkle's description of the time-stepping algorithm for
>the case of an Euler integrator sounds logical enough, but extending this
>to a higher-order integrator seems like a tedious problem - certainly more
>so than with non-time-stepping algorithms.
>
>-Norman
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