  Recent radio observations have identified a class of structures, so-called
radio relics, in clusters of galaxies. The radio emission from these sources is
interpreted as synchrotron radiation from GeV electrons gyrating in
microG-level magnetic fields. Radio relics, located mostly in the outskirts of
clusters, seem to associate with shock waves, especially those developed during
mergers. In fact, they seem to be good structures to identify and probe such
shocks in intracluster media (ICMs), provided we understand the electron
acceleration and re-acceleration at those shocks. In this paper, we describe
time-dependent simulations for diffusive shock acceleration at weak shocks that
are expected to be found in ICMs. Freshly injected as well as pre-existing
populations of cosmic-ray (CR) electrons are considered, and energy losses via
synchrotron and inverse Compton are included. We then compare the synchrotron
flux and spectral distributions estimated from the simulations with those in
two well-observed radio relics in CIZA J2242.8+5301 and ZwCl0008.8+5215.
Considering that the CR electron injection is rather inefficient at weak shocks
with Mach number M <~ a few, the existence of radio relics could indicate the
pre-existing population of low-energy CR electrons in ICMs. The implication of
our results on the merger shock scenario of radio relics is discussed.
