An implicit advection scheme for modeling relativistic shocks with high Lorentz factors

Jet-plasmas emanating from the vicinity of relativistic objects and in gamma-ray bursts have been observed to propagate with Lorentz factors laying in the range between one and several hundreds. On the other hand, the numerical studies of such flows have been focussed so far mainly on the lowest possible range of Lorentz factors $\Gamma,$ specifically, on the regime $1\leq \Gamma \leq 5.$ Therefore, relativistic flows with high $\Gamma-$factors have poorly studied, as most numerical methods are found to encounter severe numerical difficulties or even become numerically unstable for $\Gamma \gg 1. In this paper we present an implicit numerical advection scheme for modeling the propagation of relativistic plasmas with shocks, discuss its consistency with respect to both the internal and total energy formulation in general relativity. Using the total energy formulation, the scheme is found to be viable for modeling moving shocks with moderate Lorentz factors, though with relatively small Courant numbers. In the limit of high Lorentz factors, the internal energy formulation in combination with a fine-tuned artificial viscosity is much more robust and efficient. We confirm our conclusions by performing test calculations and compare the results with analytical solutions of the relativistic shock tube problem. The aim of the present modification is to enhance the robustness of the general relativistic implicit radiative MHD solver: GR-I-RMHD (http://www1.iwr.uni-heidelberg.de/groups/compastro/home/gr-i-mhd-solver) and extend its range of applications into the high$\Gamma-$regime.}