The magnetically polarized matter in astrophysical systems may be relevant in some magnetically dominated regions .
For instance , the funnel that is generated in some highly magnetized disks configurations whereby relativistic jets are thought to spread , or in pulsars where the fluids are subject to very intense magnetic fields .
With the aim of dealing with magnetic media in the astrophysical context , we present for the first time the conservative form of the ideal general relativistic magnetohydrodynamics ( GRMHD ) equations with a non-zero magnetic polarization vector m ^ { \mu } .
Then , we follow the Anile method to compute the eigenvalue structure in the case where the magnetic polarization is parallel to the magnetic field , and it is parametrized by the magnetic susceptibility \chi _ { m } .
This approximation allows us to describe diamagnetic fluids , for which \chi _ { m } < 0 , and paramagnetic fluids where \chi _ { m } > 0 .
The theoretical results were implemented in the CAFE code to study the role of the magnetic polarization in some 1D Riemann problems .
We found that independently of the initial condition , the first waves that appear in the numerical solutions are faster in diamagnetic materials than in paramagnetic ones .
Moreover , the constant states between the waves change notably for different magnetic susceptibilities .
All these effects are more appreciable if the magnetic pressure is much bigger than the fluid pressure .
Additionally , with the aim of analysing a magnetic media in a strong gravitational field , we carry out for the first time the magnetized Michel accretion of a magnetically polarized fluid .
With this test , we found that the numerical solution is effectively maintained over time ( t > 4000 ) , and that the global convergence of the code is \gtrsim 2 for \chi _ { m } \lesssim 0.005 , for all the magnetic field strength \beta we considered .
Finally , when \chi _ { m } = 0.008 and \beta \geq 10 , the global convergence of the code is reduced to a value between first and second order .