Circumstellar disks are commonly associated with pre-main-sequence sta
rs. Under the action of viscous stress, angular momentum is transferre
d outward and matter diffuses inward. With the aid of a fully two-dime
nsional, axisymmetric hydrodynamical scheme we investigate the accreti
on of matter from protostellar disks onto their central stars. In orde
r to obtain an asymptotic steady state solution we follow the evolutio
n of the flow for over 10(3) dynamical timescales. For typical ranges
of accretion rates we confirm the existence of two quasi-equilibrium s
tates of the disk, separated by thermal instability. For relatively lo
w accretion rates (<10(-6) M. yr(-1)), typical for classical T Tauri s
tars, the disk is geometrically thin and fully convective. In the regi
on adjacent to the stellar surface, i.e., in the boundary layer, the d
isk is optically thin. For relatively high accretion rates (more than
a few 10(-5) M. yr(-1)), typical for FU Orionis systems, the disk is g
eometrically thick and radiative. The incoming matter spreads rapidly
around the protostellar envelope to form a dense, hot shell. Within th
e stellar envelope, and in the disk for a low mass-inflow rate, we wer
e able to resolve directly several convection cells, which are driven
by the ionization of hydrogen. These eddies provide an effective visco
us stress such that the accretion of rapidly rotating disk material ge
nerally induces an increase in the spin of the outer convective envelo
pe of the star.