We analyzed the properties of shock waves in galaxy clusters, by using the data of simulations for the large-scale structure of the universe with the spatial resolution of up to 25 kpc/h. In a substantial fraction of clusters, we found that strong shocks with Mach number of several or larger exist in outskirts within the virial radius. They are produced by the accretion of warm gas flowing from filaments to clusters, and generate large cosmic-ray fluxes. The cosmic rays advect into cluster cores, but may temporally induce the population inversion, that is, larger population at larger radius, suggested by recent radio and ${gamma}$-ray observations.