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.