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 $L^2$-minimization. To ensure a positive solution to the moment-closure problem, we enforce positivity constraints on $L^2$-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 $L^2$-stability of the space-time discrete moment approximation. We provide an extension of our method to multi-dimensional space-velocity domains and perform several numerical experiments to showcase its accuracy.

我们考虑采用矩量法来求解稀有气体动力学的玻尔兹曼方程，从而导致以下矩闭合问题。给定一组矩，找到潜在的概率密度函数。矩闭合问题具有无限多的解决方案，并且需要附加的最优性准则才能找出唯一的解决方案。基于不连续的Galerkin速度离散化，我们考虑基于${\mathrm{大号}}^{\mathrm{2个}}$-最小化。为确保矩闭合问题的正解，我们对${\mathrm{大号}}^{\mathrm{2个}}$-最小化。这导致具有力矩和正约束的二次优化问题。我们证明（Courant-Friedrichs-Lewy）CFL型条件确保了优化问题的可行性和${\mathrm{大号}}^{\mathrm{2个}}$时空离散矩逼近的稳定性。我们提供了将方法扩展到多维空间速度域的方法，并进行了一些数值实验以证明其准确性。