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 $10^{-9}$.
翻译:我们应用Gamba 和 Tharkabhushanam 的光谱- Lagrangian 方法解决同质的 Boltzmann 方程式, 来计算粒子物种速度分布函数的低概率尾数($f$) 。 这个方法基于Boltzmann 碰撞操作员的 truncation name{tr ⁇ {(f,f)$, 美元(f,f) 美元, 其Freier 变换是由一个加权变换提供的。 快速碰撞操作操作器模拟两个相交粒子如果其相对速度超过一个阈值, 则彼此忽略的参数。 $g> text{ text{ tr} 的低概率尾数尾数尾数尾数尾数尾数尾数尾数尾数尾数尾数尾数($美元) 。 我们从此直径直的直径直的直径数数数轴值计算器, 直径直到直径直的直径直径直径直的直值 。