upstream gas pressure with f (p) ^ p

Fig. 4.23.1. Evolution of a CR modified Mach 10 plane shock using two different schemes for solving the diffusion convection equation. The simulations involve evolution of an initial Mach 10 shock discontinuity with an initially uniform CR pressure equal to twice the upstream gas pressure with f (p)^ p~5. The time is in diffusion time units, td = k(p = 1)/u2 , and momentum p in units mpc. The shock structure and immediate post-

Fig. 4.23.1. Evolution of a CR modified Mach 10 plane shock using two different schemes for solving the diffusion convection equation. The simulations involve evolution of an initial Mach 10 shock discontinuity with an initially uniform CR pressure equal to twice the upstream gas pressure with f (p)^ p~5. The time is in diffusion time units, td = k(p = 1)/u2 , and momentum p in units mpc. The shock structure and immediate post-

shock particle distribution functions are shown at t = 0, 2.5, 5, 7.5, and 10. The solid lines (dynamical variables) and the large stars (distribution function) represent solutions based on 8 momentum bins and the new scheme described in the text. The dotted lines and small triangles come from a conventional finite difference scheme using 96 momentum points. The new code took about 80% less execution time. According to Jones and Kang (2003).

In Fig. 4.23.1 it was used the diffusion coefficient k(pp05. In diffusion time units, = k(p = 1)/uS , the shock structure and immediate-post-shock particle distribution function are shown at t = 0, 2.5, 5, 7.5, 10. The conventional diffusion convection equation solution used 96 momentum points spanning the momentum range of ln(p) from -1 to 8. The coarse-grained solution used 9 logarithmically expanding momentum bins. It gave a factor 5 reduction in computation time. The code has been tested with simple CR injection models by currently implementing the thermal injection scheme of Gieseler et al. (2000) and porting the routine to CRASH of CR AMR code.

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