Cosmological shock waves result from supersonic flow motions induced by
hierarchical clustering during the large-scale structure formation in
the Universe. Suprathermal particles are known to be produced via plasma
interactions at collisionless shocks in tenuous plasmas and they can be
further accelerated to become cosmic rays (CRs) via diffusive shock
acceleration (DSA). The presence of CR electrons has been inferred from
observations of diffuse radio halos and relics in some merging galaxy
clusters. We have calculated the emissions from CR electrons accelerated
at weak planar shocks, using time-dependent DSA simulations that include
energy losses via synchrotron emission and Inverse Compton scattering.
The simulated nonthermal emission are used to model the synchrotron
emission from several observed radio relics.