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

[nlin@nlin.net]

"Martin C. Martin" wrote:
> Subject: [ODE] Euler vs. Runge-Kutta and adaptive step sizes
> 
> Russ, I take it you use Euler integration rather than, say, fourth order
> Runge-Kutta?  If so, why?  Would a fourth order Runge-Kutta be a lot
> more work?

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