  Relativistic jets are intrinsic phenomena of active galactic nuclei (AGN) and
quasars. They have been observed to also emanate from systems containing
compact objects, such as white dwarfs, neutron stars and black hole candidates.
The corresponding Lorentz factors, $\Gamma$, were found to correlate with the
compactness of the central objects. In the case of quasars and AGNs, plasmas
with $\Gamma$-factors larger than $8$ were detected. However, numerically
consistent modelling of propagating shock-fronts with $\Gamma \geq 4$ is a
difficult issue, as the non-linearities underlying the transport operators
increase dramatically with $\Gamma$, thereby giving rise to a numerical
stagnation of the time-advancement procedure or alternatively they may diverge
completely. In this paper, we present a unified numerical solver for modelling
the propagation of one-dimensional shock fronts with high Lorentz factors. The
numerical scheme is based on the finite-volume formulation with adaptive mesh
refinement (AMR) and domain decomposition for parallel computation. It unifies
both time-explicit and time-implicit numerical schemes within the framework of
the pre-conditioned defect-correction iteration solution procedure. We find
that time-implicit solution procedures are remarkably superior over their
time-explicit counterparts in the very high $\Gamma$-regime and therefore most
suitable for consistent modelling of relativistic outflows in AGNs and
micro-quasars.
