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Relativistic jets are intrinsic phenomena of active galactic nuclei (AGN) andquasars. They have been observed to also emanate from systems containingcompact objects, such as white dwarfs, neutron stars and black hole candidates.The corresponding Lorentz factors, $\Gamma$, were found to correlate with thecompactness of the central objects. In the case of quasars and AGNs, plasmaswith $\Gamma$-factors larger than $8$ were detected. However, numericallyconsistent modelling of propagating shock-fronts with $\Gamma \geq 4$ is adifficult issue, as the non-linearities underlying the transport operatorsincrease dramatically with $\Gamma$, thereby giving rise to a numericalstagnation of the time-advancement procedure or alternatively they may divergecompletely. In this paper, we present a unified numerical solver for modellingthe propagation of one-dimensional shock fronts with high Lorentz factors. Thenumerical scheme is based on the finite-volume formulation with adaptive meshrefinement (AMR) and domain decomposition for parallel computation. It unifiesboth time-explicit and time-implicit numerical schemes within the framework ofthe pre-conditioned defect-correction iteration solution procedure. We findthat time-implicit solution procedures are remarkably superior over theirtime-explicit counterparts in the very high $\Gamma$-regime and therefore mostsuitable for consistent modelling of relativistic outflows in AGNs andmicro-quasars.

Time-implicit schemes in fluid dynamics? -- Their advantage in the regime of ultra-relativistic shock fronts