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Spectral computation of low probability tails for the homogeneous Boltzmann equation

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 $10^{-9}$.