## Comoving spherical grid

In order to ensure that the shock remains near the middle of the computational domain at all levels of refined grids, for a spherically expanding shock, it is necessary define a comoving frame which expands with the instantaneous shock speed. Following the conventional cosmological approach (Ryu et al., 1993), the comoving radial coordinate, x = r/a, is adopted, where a is the expansion factor and a = 1 at the start of simulations. The expansion rate, a = (us - vs )xs , is found from the condition that the shock speed is zero at the comoving frame. Here us and vs are the shock radial velocities in the Eulerian frame and in the comoving frame, respectively. Then the comoving density and pressures are defined as p = pa3, Pg = Pga3, Pc = Pca3 . (4.27.1)

The gas-dynamic equation with CR pressure terms in the spherical comoving frame can be written as follows:

dt a ox ax

^ +1A pv2 + Pg + Pc )= - ax P2 - - - axp , (4.27.3)

dt a ox

dt a dxy g

V dPc 2-{pegv + Pgv)- — peg - axpv - Z(x,t). (4.26.4)

The injection energy loss term, L(r, t), accounts for the energy of the supra-thermal particles injected to the CR component at the sub-shock. The deceleration rate is calculated numerically by a = (an — an-i )/Atn . The diffusion-convection equation for the function f = p4 f, where f (p, r, t/ is the comoving CR distribution function, is given by

where y = ln(p/ and k(x,p/ is the diffusion coefficient.