  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).
