Shock waves form during the hierarchical formation of the large-scale structure (LSS) of the universe. Particularly, the gas in the intracluster medium (ICM) is in the form of hot tenuous plasma, and thus shocks in clusters of galaxies are collisionless as in other astrophysical environments. The shocks induced as a consequence of major mergers in merging clusters are energetic enough to be observed in radio as well as in X-ray; diffuse radio sources known as radio relics are explained by synchrotron-emitting electrons accelerated at the shocks. Yet, there remain unresolved puzzles, including a discrepancy that for some radio relics the shock Mach number inferred from the radio spectral index is substantially larger than that estimated from X-ray observations, and the lack of observation of pion0 decay gamma-ray from inelastic collisions of shock-accelerated protons with ICM protons. Toward exploring these puzzles, we present recent works on the nature of intracluster shock waves studied with simulations for the LSS formation of the universe and also with PIC (particle-in-cell) simulations for plasma processes around the shocks.