An Efficient Numerical Scheme for Simulating Particle Acceleration in   Evolving Cosmic-Ray Modified Shocks
  We have developed a new, very efficient numerical scheme to solve the CR diffusion convection equation that can be applied to the study of the nonlinear time evolution of CR modified shocks for arbitrary spatial diffusion properties. The efficiency of the scheme derives from its use of coarse-grained finite momentum volumes. This approach has enabled us, using $\sim 10 - 20$ momentum bins spanning nine orders of magnitude in momentum, to carry out simulations that agree well with results from simulations of modified shocks carried out with our conventional finite difference scheme requiring more than an order of magnitude more momentum points. The coarse-grained, CGMV scheme reduces execution times by a factor approximately half the ratio of momentum bins used in the two methods. Depending on the momentum dependence of the diffusion, additional economies in required spatial and time resolution can be utilized in the CGMV scheme, as well. These allow a computational speed-up of at least an order of magnitude in some cases. 
T. W. Jones and H. Kang