  Cosmic ray (CR) electrons reveal key insights into the non-thermal physics of
the interstellar medium, galaxies, galaxy clusters, and active galactic nuclei
by means of their inverse Compton {\gamma}-ray emission and synchrotron
emission in magnetic fields. While magnetohydrodynamical (MHD) simulations with
CR protons capture their dynamical impact on these systems, only few
computational studies include CR electron physics because of the short cooling
time-scales and complex hysteresis effects, which require a numerically
expensive, high-resolution spectral treatment. Since CR electrons produce
important non-thermal observational signatures, such a spectral CR electron
treatment is important to link MHD simulations to observations. We present an
efficient post-processing code for Cosmic Ray Electron Spectra that are evolved
in Time (CREST) on Lagrangian tracer particles. The CR electron spectra are
very accurately evolved on comparably large MHD time steps owing to an
innovative hybrid numerical-analytical scheme. CREST is coupled to the
cosmological MHD code AREPO and treats all important aspects of spectral CR
electron evolution such as adiabatic expansion and compression, Coulomb losses,
radiative losses in form of inverse Compton, bremsstrahlung and synchrotron
processes, diffusive shock acceleration and reacceleration, and Fermi-II
reacceleration. After showing various code validations of idealized one-zone
simulations, we study the coupling of CREST to MHD simulations. We demonstrate
that the CR electron spectra are efficiently and accurately evolved in
shock-tube and Sedov-Taylor blast wave simulations. This opens up the
possibility to produce self-consistent synthetic observables of non-thermal
emission processes in various astrophysical environments.
