Compact objects like black holes and neutron stars interacting with the relativistic plasma in the sorrounding region are believed to be responsible for many high-energy phenomena in astrophysics. Active galactic nuclei, relativistic jets from compact objects, gamma-ray bursts, extraction of rotational energy from the ergosphere of a Kerr black hole represent physical examples where the interplay between **strong magnetic and strong gravitational fields** is the crucial factor for understanding their dynamics.

The theoretical framework in which these mechanisms are treated is that of general relativistic magnetohydrodynamics (GRMHD). Since relativistic magnetized flows are often associated with the formation of strong shocks at high Lorentz factors, the numerical solution of the GRMHD equations is very efficiently performed through shock-capturing Godunov-type methods, that are based on the conservative formulation of the equations.

# Numerical relativistic magnetohydrodynamics

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