  We present extensive Monte-Carlo spin dynamics simulations of the classical
XY model in three dimensions on a simple cubic lattice with periodic boundary
conditions. A recently developed efficient integration algorithm for the
equations of motion is used, which allows a substantial improvement of
statistics and large integration times. We find spin wave peaks in a wide range
around the critical point and spin diffusion for all temperatures. At the
critical point we find evidence for a violation of dynamic scaling in the sense
that independent components of the dynamic structure factor S(q,w) require
different dynamic exponents in order to obtain scaling. Below the critical
point we investigate the dispersion relation of the spin waves and the
linewidths of S(q,w) and find agreement with mode coupling theory. Apart from
strong spin wave peaks we observe additional peaks in S(q,w) which can be
attributed to two-spin wave interactions. The overall lineshapes are also
discussed and compared to mode coupling predictions. Finally, we present first
results for the transport coefficient D(q,w) of the out-of-plane magnetization
component at the critical point, which is related to the thermal conductivity
of He4 near the superfluid-normal transition.
