Previous works have developed boundary conditions that lead to the $L^2$-boundedness of solutions to the linearised moment equations. Here we present a spatial discretization that preserves the $L^2$-stability by recovering integration-by-parts over the discretized domain and by imposing boundary conditions using a simultaneous-approximation-term (SAT). We develop three different forms of the SAT using: (i) characteristic splitting of moment equation's boundary conditions; (ii) decoupling of moments in moment equations; and (iii) characteristic splitting of Boltzmann equation's boundary conditions. We discuss how the first two forms differ in terms of their usage and implementation. We show that the third form is equivalent to using an upwind kinetic numerical flux along the boundary, and we argue that even though it provides stability, it prescribes the incorrect number of boundary conditions. Using benchmark problems, we compare the accuracy of moment solutions computed using different SATs. Our numerical experiments also provide new insights into the convergence of moment approximations to the Boltzmann equation's solution.

以前的作品已经开发出导致边界条件的 ${\mathrm{大号}}^2$线性矩方程解的有界性。在这里，我们提出了一个空间离散化，它保留了 ${\mathrm{大号}}^2$通过在离散域上恢复逐部分积分并使用同时逼近项（SAT）施加边界条件来提高稳定性。我们使用以下三种形式开发S​​AT：（i）矩方程边界条件的特征分解；（ii）力矩方程中的力矩解耦；（iii）玻尔兹曼方程边界条件的特征分裂。我们讨论前两种形式在用法和实现方面的不同。我们表明，第三种形式等效于沿边界使用逆风动力学数值通量，并且我们认为，即使它提供了稳定性，也规定了不正确数量的边界条件。使用基准问题，我们比较使用不同SAT计算的矩解的准确性。我们的数值实验还为矩近似逼近Boltzmann方程的解提供了新的见解。