  A numerical method presented by Imshennik et al. (2002) is used to solve the
two dimensional axisymmetric hydrodynamic problem on the formation of a
toroidal atmosphere during the collapse of an iron stellar core and outer
stellar layers. An evolutionary model from Boyes et al. (1999) with a total
mass of $25M_{\odot}$ is used as the initial data for the distribution of
thermodynamic quantities in the outer shells of a high-mass star. We analyze in
detail the results of three calculations in which the difference mesh and the
location of the inner boundary of the computational region are varied. In the
initial data, we roughly specify an angular velocity distribution that is
actually justified by the final result - the formation of a hydrostatic
equilibrium toroidal atmosphere with reasonable total mass, $M^{tot} = (0.117
\div 0.122)M_{\odot}$, and total angular momentum, $J^{tot} = (0.445 \div
0.472) x 10^{50} erg \cdot s$, for the two main calculations. We compare the
numerical solution with our previous analytical solution in the form of
toroidal atmospheres (Imshennik and Manukovskii 2000). This comparison
indicates that they are identical if we take into account the more general and
complex equation of state with a nonzero temperature and self-gravitation
effects in the atmosphere. Our numerical calculations, first, prove the
stability of toroidal atmospheres on characteristic hydrodynamic time scales
and, second, show the possibility of sporadic fragmentation of these
atmospheres even after a hydrodynamic equilibrium is established. The
calculations were carried out under the assumption of equatorial symmetry of
the problem and up to relatively long time scales $(\approx 10s)$.
