  We have developed a cosmic ray (CR) shock code in one dimensional spherical
geometry with which the particle distribution, the gas flow and their nonlinear
interaction can be followed numerically in a frame comoving with an expanding
shock. In order to accommodate a very wide dynamic range of diffusion length
scales in the CR shock problem, we have incorporated subzone shock tracking and
adaptive mesh refinement techniques. We find the spatial grid resolution
required for numerical convergence is less stringent in this code compared to
typical, fixed-grid Eulerian codes. The improved convergence behavior derives
from maintaining the shock discontinuity inside the same grid zone in the
comoving code. That feature improves numerical estimates of the compression
rate experienced by CRs crossing the subshock compared to codes that allow the
subshock to drift on the grid. Using this code with a Bohm-like diffusion model
we have calculated the CR acceleration and the nonlinear feedback at supernova
remnant shocks during the Sedov-Taylor stage. Similarly to plane-parallel
shocks, with an adopted thermal leakage injection model, about 10^{-3} of the
particles that pass through the shock and up to 60 % of the explosion energy
are transferred to the CR component. These results are in good agreement with
previous nonlinear spherical CR shock calculations of Berezhko and
collaborators.
