  (abridged) When preplanetary bodies reach proportions of ~1 km or larger in
size, their accretion rate is enhanced due to gravitational focusing (GF). We
have developed a new numerical model to calculate the collisional evolution of
the gravitationally-enhanced growth stage. We validate our approach against
existing N-body and statistical codes. Using the numerical model, we explore
the characteristics of the runaway growth and the oligarchic growth accretion
phases starting from an initial population of single planetesimal radius R_0.
In models where the initial random velocity dispersion (as derived from their
eccentricity) starts out below the escape speed of the planetesimal bodies, the
system experiences runaway growth. We find that during the runaway growth phase
the size distribution remains continuous but evolves into a power-law at the
high mass end, consistent with previous studies. Furthermore, we find that the
largest body accretes from all mass bins; a simple two component approximation
is inapplicable during this stage. However, with growth the runaway body stirs
up the random motions of the planetesimal population from which it is
accreting. Ultimately, this feedback stops the fast growth and the system
passes into oligarchy, where competitor bodies from neighboring zones catch up
in terms of mass. Compared to previous estimates, we find that the system
leaves the runaway growth phase at a somewhat larger radius. Furthermore, we
assess the relevance of small, single-size fragments on the growth process. In
classical models, where the initial velocity dispersion of bodies is small,
these do not play a critical role during the runaway growth; however, in models
that are characterized by large initial relative velocities due to external
stirring of their random motions, a situation can emerge where fragments
dominate the accretion.
