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High-energy phenomena in astrophysics involve quite generally a combinationof relativistic motions and strong gravity. The simultaneous solution ofEinstein equations and General Relativistic MHD equations is thus necessary tomodel with accuracy such phenomena. The so-called Conformally Flat Condition(CFC) allows a simplified treatment of Einstein equations, that can beparticularly efficient in those contexts where gravitational wave emission isnegligible, like core-collapse, or the formation/evolution of neutron stars. Wehave developed a set of codes to model axisymmetric MHD flows, in GeneralRelativity, where the solution of Einstein equations is achieved with asemi-spectral scheme. Here, we will show how this framework is particularlywell suited to investigate neutron star equilibrium models in the presence ofstrong magnetic fields and we will present the XNS code, that has been recentlydeveloped and here updated to treat poloidal and mixed configurations.

Solving the 3+1 GRMHD equations in the eXtended Conformally Flat Condition: the XNS code for magnetized neutron stars