The simulations above were finished in around 1 minute for `Pelegant` and 10
minutes for `elegant` on the AMD Athlon nodes of the weed cluster in APS.
To convince ourselves the results of the two
versions of `elegant` are same, one can compare the output files with the
sddsdiff command, which should be available in the SDDS toolkit.
For example, to examine the particle coordinates at interior points, we can type:

sddsdiff manyParticles_p.w2 manyParticles_s.w2

which should return that two results are identical.

Validating a parallel program against a uniprocessor program with the
requirement of bitwise identical result is notoriously difficult [2], as we
may meet some new problems raised from parallel computing, such as different
ordering of summations, non-scalable random number generator. Although the
simulation results with the discrepancies should conform to IEEE 754 within
some tolerance, more consistent results can be expected with more accurate
numerical algorithm, such as Kahan's summation formula [3], which has been
employed in both serial and parallel versions of `elegant`.

We ran a regression test of 92 cases and validated the results of `Pelegant`
with `elegant`. As the random number sequences generated by one CPU
and multiple CPUs usually are not same, some test examples can't be validated
by comparing the results of `elegant` and `Pelegant`. Those examples
have been validated either by mathematical formulae or their physical meaning.