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Re: [Phys-L] [Phys-l] accurate numerical solution of equations of motion




On 2009, Oct 22, , at 12:18, John Denker <jsd@av8n.com> wrote:

OK, here's the situation, as I understand it at the moment:

Symplectic integrators, such as the Euler-Cromer method,
are well suited to Hamiltonian dynamical systems.

It turns out that the system I was dealing with last month --
a model of the aeronautical "turns on pylons" maneuver -- is
*not* a Hamiltonian dynamical system. The model does not
conserve energy; not even close. That's because the model
does not account for engine thrust, aerodynamic drag, or
the component of gravity along the aircraft trajectory
during climbs and descents. I know from experience that
these contributions, taken together, have little effect on
the maneuver, so I am justified in leaving them out of the
model ... but it makes the model ineligible to use the
Euler-Cromer method.

It would be nice to have a name for the method I described
this morning, using stride=2 for one integration and second-
order Euler for the other ... but I don't think it would be
appropriate to name it after me. I can't cite any literature
that says to do it exactly that way, but I would be quite
astonished if it was original. I suspect any competent
"old school" numerical analyst would do something like it,
or something better, as a matter of routine. For now I
will just call it the "old school" method.



AND:


On 2014, Apr 05, , at 11:04, John Denker <jsd@av8n.com> wrote:


I recommend sticking with symplectic methods. A fourth-order
symplectic method won't be any worse than a non-symplectic
method of the same order. For a Hamiltonian system it will be
much better. For a /slightly/ dissipative system it should
retain some of its advantages.



In searching AAPT pubs. for further understanding, tho I’d decided to settle on the LPA (Euler-Cromer) I found:


"Numerical integration routines designed for introductory physics courses, such as the last‐point approximation and the second Taylor approximation, are incompatible with velocity‐dependent forces. A general purpose routine which handles resistive, Coriolis, and magnetic forces, as well as conservative forces, is obtained by combining the fundamental Euler method with Richardson extrapolation. Further, this Euler–Richardson method is almost as efficient as the last‐point approximation and the second Taylor approximation for simple central force problems and is more efficient for difficult problems, such as Earth–Moon orbits."
‎scitation.aip.org/docserver/fulltext/aapt/journal/ajp/62/3/1.17610.pdf?expires=1398106582&id=id&accname=394397&checksum=4128D05954F8021E38B465DDEDFA2D4A


AJP 62 259 (1994)

I also found numerous other articles on numerical algorithms including reviews of computational physics texts.

Finally. Is a valid measure of the accuracy of an algorithm to compare a fit using the analytic solution (when available) with the numerical data. For example: use k1*cos(k2*t+k3)*exp(-k4*t) +k5 with F = a1*x + a2*xdot solved numerically, instead of (for example) a phase plot (xdot and x)?

When I’ve used this method with no dissipation, i.e. just a simple harmonic oscillator and the the LPA (Euler -Cromer), the error (in position) increases with time. The period does appear to remain the same.



bc found an old (ca 1971) article comparing various numericals with their efficiencies. (time and accuracy criteria)


Choosing Algorithms for Numeric Integration

http://scitation.aip.org/content/aapt/journal/ajp/39/11/10.1119/1.1976678