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Comparing motion of polyspherical particles computed with high and with low numerical precision,
UM 2009-03-03

Left image: computation done with 32 digital places numerical precision. Right image: 10 digital places.
See article8 for an explanation of
polyspherical particles and a robust and efficient computational implementation
of their dynamics.
Towards the end of the simulation the left and the right image begin to
look different and finally in the right image (the one which was computed with the lower precision)
the mirror symmetry between the red and green particles gets lost.
The system, which is here shown twice for comparison of two computations,
consists of 12 particles which which make a free fall in a cylindrical drum
with a closing lid at both ends. Each particle consists of three spherical
components which strongly overlap. Six of the particles are generated by a
a procedure which involves random variations of the parametes (such as radii
of the spherical componets). The other six particles are defined as
mirror images of the original particles. Also the initial placement of
the particles shows this mirror symmetry, as does the containing drum,
and the homogeneous gravitation field which causes the motion of the
particles, which are initially at rest.
Under these conditions the exact solution of the equation of motion (we use
simple elastic forces between the spherical components of the polyspherical
particles and between particles and drum) remains mirror symmetric for all times.
Numerical errors destroy this symmetry slowly but effective.
The rotational degree of freedom lets the
effect of a collision depend much more sensitively on the collision geometry
than it would be the case for simply spherical particles.
The images shown are parallel projections along the drum's
axis so that the mirror symmetry under consideration reflects itself in the
left to right mirror symmetry of the green area (original particles) and red area
(particles with mirror symmetric initial condition).
See also the final section of the rational for my
multiple precision Ruby class R .
Total energy is very well conserved in both computations, with only tiny
oscillations and no detectable trend. This is a beneficial property of the
integrator which is used here. Less robust integrators distroy the mirror
symmetry much earlier.
Remark: When, as a student, I had an awesome look into the 10 digit logarithmic tables
by Peters and Bauschinger, I did not anticipate that I would ever consider 10 digits
'low numerical pecision'.