Deformation of a phase space subset by interaction picture dynamics II. UM 2009-08-03

Leapfrog method

Asynchronous leapfrog method

Runge-Kutta second order

Direct midpoint method (Störmer-Verlet)

The first part of the computation shown in movie3 covered the first 10 revolutions of our 'Kepler particle'; the present, shorter animation covers revolutions 20.25 - 22.5 und shows much more deformation of the phase cell (which was a circle in the initia state). To reasonably represent its shape, the contour is now defined by 1000 phase points instead of 200. All other data of the computation remained unchanged. Recall that what here is referred to as 'interaction picture dynamics' visualized the error induced by the time-discrete computation of the trajecories: if such errors would not exist, the phase cell would neither move nor deform. (It should be mentioned that the overall motion of the cell is not shown in the movies - we 'center' the cell computationally.) The main intention of this investigation was to learn whether my brain child 'asynchronos leapfrog integrator' behaves much like the normal leapfrog integrator. Taking into account that the asnchronous method originates from the normal method by only a minor reorganization of the information flow, one could in fact expect that there would be no larger difference in behavior. However, as the present animations show, there are substantial differences. The asynchronous method causes much less local agitation (of errors), than the original method. That the asynchronous method allows the loop to cross over itself came as a surprise to me; it is allowed to happen since states of the asynchronous leapfrog method hold two further parameters in addition to position and momentum. Since the movie frames only show the later, we can't see why one and the same visible point may in one dynamical situation go one way and in another situation an other way. These additional state descriptors are also central to understanding the reversible trajecories of irreversible equations reported in the section Work in progress.