We consider the method-of-moments approach to solve the Boltzmann equation of rarefied gas dynamics, which results in the following moment-closure problem. Given a set of moments, find the underlying probability density function. The moment-closure problem has infinitely many solutions and requires an additional optimality criterion to single-out a unique solution. Motivated from a discontinuous Galerkin velocity discretization, we consider an optimality criterion based upon L2-minimization. To ensure a positive solution to the moment-closure problem, we enforce positivity constraints on L2-minimization. This results in a quadratic optimization problem with moments and positivity constraints. We show that a (Courant-Friedrichs-Lewy) CFL-type condition ensures both the feasibility of the optimization problem and the L2-stability of the moment approximation. Numerical experiments showcase the accuracy of our moment method.

中文翻译：

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气体动力学玻尔兹曼方程的一种正稳定的基于L2最小化的矩方法

我们考虑使用矩量法来求解稀薄气体动力学的 Boltzmann 方程，这导致了以下矩量闭合问题。给定一组矩，找到潜在的概率密度函数。矩闭合问题有无限多个解，需要额外的最优性标准来挑出唯一的解。受不连续 Galerkin 速度离散化的启发，我们考虑了基于 L2 最小化的最优性标准。为了确保对矩闭合问题的积极解决方案，我们对 L2 最小化强制执行正约束。这导致具有矩和正约束的二次优化问题。我们表明（Courant-Friedrichs-Lewy）CFL 类型的条件确保了优化问题的可行性和矩近似的 L2 稳定性。数值实验展示了我们的矩方法的准确性。