Numerical Studies of Diffusive Shock Acceleration at Spherical Shocks
  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. 
Hyesung Kang and T. W. Jones