In this paper we propose a novel numerical approach for the Boltzmann equation with uncertainties. The method combines the efficiency of classical direct simulation Monte Carlo (DSMC) schemes in the phase space together with the accuracy of stochastic Galerkin (sG) methods in the random space. This hybrid formulation makes it possible to construct methods that preserve the main physical properties of the solution along with spectral accuracy in the random space. The schemes are developed and analyzed in the case of space homogeneous problems as these contain the main numerical difficulties. Several test cases are reported, both in the Maxwell and in the variable hard sphere (VHS) framework, and confirm the properties and performance of the new methods.

在本文中，我们为具有不确定性的玻尔兹曼方程提出了一种新颖的数值方法。该方法将经典直接模拟蒙特卡罗（DSMC）方案在相空间中的效率与随机Galerkin（sG）方法在随机空间中的准确性结合在一起。这种混合配方使得可以构建保留溶液主要物理性质以及随机空间中光谱准确性的方法。该方案是在空间均质问题的情况下开发和分析的，因为这些问题包含主要的数值困难。在Maxwell和可变硬球（VHS）框架中都报告了几个测试用例，它们证实了新方法的特性和性能。