We have developed the Boltzmann moment equation approach for the dynamics of stars (BEADS-2D), which is a finite-difference Eulerian numerical code designed for the modelling of anisotropic and non-axisymmetric flat stellar discs. The BEADS-2D code solves the Boltzmann moment equations up to second order in the thin-disc approximation. This allows us to obtain the anisotropy of the velocity ellipsoid and the vertex deviation in the plane of the disc. We apply the BEADS-2D code to study numerically the properties of the stellar velocity distribution in stellar discs which have developed a saturated, two-armed spiral structure. We follow the growth of the spiral structure deeply into the non-linear regime by solving the Boltzmann moment equations up to second order. By adopting the thin-disc approximation, we restrict our study of the stellar velocity distribution to the plane of the stellar disc. We find that the outer (convex) edges of stellar spiral arms are characterized by peculiar properties of the stellar velocity ellipsoids, which make them distinct from most other galactic regions. In particular, the ratio of the smallest versus largest principal axes of the stellar velocity ellipsoid can become abnormally small (as compared to the rest of the disc) near the outer edges of spiral arms. Moreover, the epicycle approximation fails to reproduce the ratio of the tangential versus radial velocity dispersions in these regions. These peculiar properties of the stellar velocity distribution are caused by large-scale non-circular motions of stars, which in turn are triggered by the non-axisymmetric gravitational field of stellar spiral arms. The magnitude of the vertex deviation appears to correlate globally with the amplitude of the spiral stellar density perturbations. However, locally there is no simple correlation between the vertex deviation and the density perturbations. The results have been published in Monthly Notices of the Royal Astronomical Society. This research was done in collaboration with Dr. Christian Theis.