  We apply the spectral-Lagrangian method of Gamba and Tharkabhushanam for
solving the homogeneous Boltzmann equation to compute the low probability tails
of the velocity distribution function, $f$, of a particle species. This method
is based on a truncation, $Q^{\operatorname{tr}}(f,f)$, of the Boltzmann
collision operator, $Q(f,f)$, whose Fourier transform is given by a weighted
convolution. The truncated collision operator models the situation in which two
colliding particles ignore each other if their relative speed exceeds a
threshold, $g_{\text{tr}}$. We demonstrate that the choice of truncation
parameter plays a critical role in the accuracy of the numerical computation of
$Q$. Significantly, if $g_{\text{tr}}$ is too large, then accurate numerical
computation of the weighted convolution integral is not feasible, since the
decay rate and degree of oscillation of the convolution weighting function both
increase as $g_{\text{tr}}$ increases. We derive an upper bound on the
pointwise error between $Q$ and $Q^{\text{tr}}$, assuming that both operators
are computed exactly. This bound provides some additional theoretical
justification for the spectral-Lagrangian method, and can be used to guide the
choice of $g_{\text{tr}}$ in numerical computations. We then demonstrate how to
choose $g_{\text{tr}}$ and the numerical discretization parameters so that the
computation of the truncated collision operator is a good approximation to $Q$
in the low probability tails. Finally, for several different initial
conditions, we demonstrate the feasibility of accurately computing the time
evolution of the velocity pdf down to probability density levels ranging from
$10^{-5}$ to $10^{-9}$.
