General relativistic magnetohydrodynamics in axisymmetric dynamical spacetimes: the X-ECHO code

Abstract

We present a new numerical code, X-ECHO, for general relativistic magnetohydrodynamics (GRMHD) in dynamical spacetimes. This aims at studying astrophysical situations where strong gravity and magnetic fields are both supposed to play an important role, such as in the evolution of magnetized neutron stars or in the gravitational collapse of the magnetized rotating cores of massive stars, which is the astrophysical scenario believed to eventually lead to (long) GRB events. The code extends the Eulerian conservative high-order (ECHO) scheme (Del Zanna et al. 2007, A&A, 473, 11) for GRMHD, here coupled to a novel solver of the Einstein equations in the extended conformally flat condition (XCFC). We solve the equations in the 3 + 1 formalism, assuming axisymmetry and adopting spherical coordinates for the conformal background metric. The GRMHD conservation laws are solved by means of shock-capturing methods within a finite-difference discretization, whereas, on the same numerical grid, the Einstein elliptic equations are treated by resorting to spherical harmonics decomposition and are solved, for each harmonic, by inverting band diagonal matrices. As a side product, we built and make available to the community a code to produce GRMHD axisymmetric equilibria for polytropic relativistic stars in the presence of differential rotation and a purely toroidal magnetic field. This uses the same XCFC metric solver of the main code and has been named XNS. Both XNS and the full X-ECHO codes are validated through several tests of astrophysical interest.

General relativistic models for rotating magnetized neutron stars in conformally flat space–time

“…The electromagnetic tensor can be defined either in terms of the comoving magnetic and electric field b µ and e µ (whenever a flow velocity can be defined), or in terms of the so called Eulerian electric and magnetic field E µ and B µ (purely spatial vectors), following the same 3+1 splitting of the metric (Del Zanna et al 2007;Bucciantini & Del Zanna 2011). In the region of space occupied by matter, the ideal MHD condition is e µ = u ν F µν = 0, and only the comoving magnetic field does not vanish.…”

“…When this condition is relaxed, non-ideal effects such as dynamo or reconnection processes may take place in both the interior and the magnetosphere of magnetars. For applications to the numerical modeling of relativistic plasmas, see Bucciantini & Del Zanna (2013) and Del Zanna et al (2016).…”

“…Our approach to the numerical solution of Einstein equations together with the equilibrium conditions for the matter distribution and the electromagnetic field is based on the so called 3+1 splitting of the space-time metric and fluid quantities (see for example Gourgoulhon 2007, Gourgoulhon 2010, Alcubierre 2008, Baumgarte & Shapiro 2010, Bucciantini & Del Zanna 2011, to which the reader is referred for a detailed description). In such formalism the generic line element can be written in the ADM form (Arnowitt, Deser & Misner 1962) …”

Three-dimensional evolution of differentially rotating magnetized neutron stars

“…This amplified magnetic field could have an impact on the dynamics of the mergers because it contributes to the angular momentum transport and the magnetic pressure may modify the structure of the objects formed after the merger. Motivated by this expectation, several groups have implemented the magnetohydrodynamics (MHD) code in the framework of numerical relativity [23][24][25][26][27][28]. These numerical codes developed have been applied to collapse of magnetized hypermassive neutron stars (HMNS) [29][30][31], magnetized neutron starblack hole binary merger [32,33], evolution of magnetized neutron stars [34][35][36], and magnetorotational collapse of massive stellar cores [24,37].…”

Gravitational waves in dynamical spacetimes with matter content in the fully constrained formulation

“…Recent works [61] overcome some pathological problems related with non-local uniqueness in the elliptic equations in CFC, and also in FCF. The equations were rewritten in such a way that these problems were solved, and the new scheme has been used successfully in some applications [47,62,63].…”