  Cosmic ray electron (CRE) acceleration and cooling are important physical
processes in astrophysics. We develop an approximative framework to treat CRE
physics in the parallel smoothed particle hydrodynamics code Gadget-3. In our
methodology, the CRE spectrum of each fluid element is approximated by a single
power-law distribution with spatially varying amplitude, upper cut-off, lower
cut-off, and spectral index. We consider diffusive shock acceleration to be the
source of injection, and oppositely the sinking processes is attributed to
synchrotron radiation, inverse Compton scatters, and Coulomb scatters. The
adiabatic gains and losses are also included. We show that our formalism
produces the energy and pressure with an accuracy of $ > 90\%$ for a free
cooling CRE spectrum. Both slope and intensity of the radio emission computed
from the CRE population given by our method in cosmological hydro-simulation
coincide well with observations, and our results also show that relaxed
clusters have lower fluxes. Finally, we investigate several impacts of the CRE
processes on the cosmological hydro-simulation, we find that: (1) the pressure
of the CRE spectrum is very small and can be ignored in hydro-simulation, (2)
the impacts of the CRE processes on the gas phase-space state of
hydro-simulation is up to $3\%$, (3) the CRE processes induce a $5\%$ influence
on the mass function in the mass range $10^{12} -10^{13} h^{-1} M_{\odot}$, (4)
The gas temperature of massive galaxy cluster is influenced by the CRE
processes up to $\sim 10\%$.
