Abstract

In this study, a spatio-temporal approach for the solution of the time-dependent Boltzmann (transport) equation is derived. Finding the exact solution using the Boltzmann equation for the general case is generally an open problem and approximate methods are usually used. One of the most common methods is the spherical harmonics method (the $P_N$ approximation), when the exact transport equation is replaced with a closed set of equations for the moments of the density with some closure assumption. Unfortunately, the classic $P_N$ closure yields poor results with low-order N in highly anisotropic problems. Specifically, the tails of the particles’ positional distribution as attained by the $P_N$ approximation are inaccurate compared to the true behavior. In this work, we present a derivation of a linear closure that even for low-order approximation yields a solution that is superior to the classical $P_N$ approximation. This closure is based on an asymptotic derivation both for space and time of the exact Boltzmann equation in infinite homogeneous media. We test this approximation with respect to the one-dimensional benchmark of the full Green function in infinite media. The convergence of the proposed approximation is also faster when compared to (classic or modified) $P_N$ approximation.

摘要

在这项研究中，导出了时空相关的玻尔兹曼（运输）方程解的时空方法。对于一般情况，使用Boltzmann方程寻找精确解通常是一个开放性问题，通常使用近似方法。最常见的方法之一是球谐函数法（$P_{ñ}$近似）），当在某些封闭假设的情况下，针对密度矩用一组封闭的方程式代替精确的输运方程式时。不幸的是，经典$P_{ñ}$在高度各向异性的问题中，使用低阶N进行封闭会产生较差的结果。具体来说，通过$P_{ñ}$与真实行为相比，近似值不准确。在这项工作中，我们提出了线性闭包的推导，即使对于低阶逼近，它也能产生优于经典闭包的解。$P_{ñ}$近似。这种闭合是基于无限均质介质中精确Boltzmann方程的空间和时间的渐近推导。我们针对无限媒体中完整Green函数的一维基准测试了这种近似值。与（经典或修改）相比，建议近似值的收敛速度也更快。$P_{ñ}$ 近似。