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, $Q^{\mathrm{tr}}(f,f)$, 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 degree of oscillation of the convolution weighting function increases 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 ${10}^{-9}$.

我们使用Gamba和Tharkabhushanam的拉格朗日谱方法来求解齐次Boltzmann方程，以计算粒子物种的速度分布函数f的低概率尾部。此方法基于截断，${问}^{\mathrm{TR}}（F，F）$是玻尔兹曼碰撞算子的 $问（F，F）$，其傅里叶变换由加权卷积给出。截断的碰撞算子模拟了两个碰撞粒子的相对速度超过阈值时彼此忽略的情况，$G_{\text{TR}}$。我们证明截断参数的选择在Q数值计算的准确性中起着至关重要的作用。重要的是，如果$G_{\text{TR}}$ 太大，则无法正确计算加权卷积积分的数值，因为卷积加权函数的振荡程度会随着 $G_{\text{TR}}$增加。我们得出Q和之间的逐点误差的上限${问}^{\text{TR}}$，假设两个运算符都经过精确计算。该界限为光谱-拉格朗日方法提供了其他理论依据，可用于指导选择$G_{\text{TR}}$在数值计算中。然后，我们演示如何选择$G_{\text{TR}}$以及离散化的数值参数，因此截断的碰撞算子的计算可以很好地逼近低概率尾部中的Q。最后，对于几种不同的初始条件，我们证明了精确计算速度pdf的时间演化到概率密度水平（从${10}^{-5}$ 至 ${10}^{-9}$。