  Spin-dynamics techniques have been used to perform large-scale simulations of
the dynamic behavior of the classical Heisenberg antiferromagnet in simple
cubic lattices with linear sizes $L\leq 60$. This system is widely recognized
as an appropriate model for the magnetic properties of RbMnF$_3$.
Time-evolutions of spin configurations were determined numerically from coupled
equations of motion for individual spins using a new algorithm implemented by
Krech {\it etal}, which is based on fourth-order Suzuki-Trotter decompositions
of exponential operators. The dynamic structure factor was calculated from the
space- and time-displaced spin-spin correlation function. The crossover from
hydrodynamic to critical behavior of the dispersion curve and spin-wave
half-width was studied as the temperature was increased towards the critical
temperature. The dynamic critical exponent was estimated to be $z=(1.43\pm
0.03)$, which is slightly lower than the dynamic scaling prediction, but in
good agreement with a recent experimental value. Direct, quantitative
comparisons of both the dispersion curve and the lineshapes obtained from our
simulations with very recent experimental results for RbMnF$_3$ are presented.
