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We present an efficient algorithm to follow spectral evolution of Cosmic Rays(CR) coupled with an MHD system on Eulerian grids. The algorithm is designedfor studies of CR energy spectrum evolution in MHD simulations of a galacticinterstellar medium. The base algorithm for CR transport relies on thetwo-moment piece-wise power-law method, known also as Coarse Grained MomentumFinal Volume (CGMV), for solving the Fokker-Planck CR transport equation, witha low number of momentum-bins extending over several decades of the momentumcoordinate. We propose an extension of the CGMV with a novel feature whichallows momentum boundaries to change in response to CR momentum gains or lossesnear the extremes of the population distribution. Our extension involves aspecial treatment of momentum bins containing spectral cutoff. Contrary to theregular bins of fixed width, those bins have variable-width, and their outeredges coincide with spectral cutoffs. The cutoff positions are estimated fromthe particle number density and energy density in the outer bins for an assumedsmall value of an additional parameter representing the smallest physicallysignificant level of CR spectral energy density. We performed a series ofelementary tests to validate the algorithm and demonstrated, whenever possible,that results of the test simulations correspond, with a reasonable accuracy, tothe results of analogous analytical solutions. In a more complex test ofgalactic CR-driven wind problem we obtained results consistent withexpectations regarding the effects of advection, diffusion, adiabatic, andsynchrotron cooling of a CR population.

Implementation of CR Energy SPectrum (CRESP) algorithm in PIERNIK MHD code. I. Spectrally resolved propagation of CR electrons on Eulerian grids