Boltzmann moment equation approach for the numerical study of   anisotropic stellar disks
  We present 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 disks. The BEADS-2D code solves the Boltzmann moment equations up to second order in the thin-disk approximation. This allows us to obtain the anisotropy of the velocity ellipsoid and the vertex deviation in the plane of the disk. We study the time-dependent evolution of exponential stellar disks in the linear regime and beyond. The disks are initially characterized by different values of the Toomre parameter Q_s and are embedded in a dark matter halo, yielding a rotation curve composed of a rigid central part and a flat outer region. Starting from a near equilibrium state, several unstable modes develop in the disk. In the early linear phase, the very centre and the large scales are characterized by growing one-armed and bisymmetric positive density perturbations, respectively. This is in agreement with expectations from the swing amplification mechanism of short-wavelength trailing disturbances, propagating through the disk centre. In the late linear phase, the overall appearance is dominated by a two-armed spiral structure localized within the outer Lindblad resonance (OLR). During the non-linear evolutionary phase, radial mass redistribution due to the gravitational torques of spiral arms produces an outflow of mass, which forms a ring at the OLR, and an inflow of mass, which forms a transient central bar. This process of mass redistribution is self-regulatory and it terminates when spiral arms diminish due to a shortage of matter. Finally, a compact central disk and a diffuse ring at the OLR are formed (the abstract is abridged). 
E. I. Vorobyov (1), Ch. Theis (2) ((1) The University of Western   Ontario, Canada, (2) Institute for Astronomy, Vienna, Aus...