Magnetohydrodynamics in full general relativity: Formulation and tests

The University of Tokyo, Tōkyō, Japan
Physical review D: Particles and fields (Impact Factor: 4.86). 07/2005; 72(4). DOI: 10.1103/PhysRevD.72.044014
Source: arXiv

ABSTRACT A new implementation for magnetohydrodynamics (MHD) simulations in full general relativity (involving dynamical spacetimes) is presented. In our implementation, Einstein's evolution equations are evolved by a BSSN formalism, MHD equations by a high-resolution central scheme, and induction equation by a constraint transport method. We perform numerical simulations for standard test problems in relativistic MHD, including special relativistic magnetized shocks, general relativistic magnetized Bondi flow in stationary spacetime, and a longterm evolution for self-gravitating system composed of a neutron star and a magnetized disk in full general relativity. In the final test, we illustrate that our implementation can follow winding-up of the magnetic field lines of magnetized and differentially rotating accretion disks around a compact object until saturation, after which magnetically driven wind and angular momentum transport inside the disk turn on. Comment: 28 pages, to be published in Phys. Rev. D

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    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.
    Astronomy and Astrophysics 04/2011; 528. DOI:10.1051/0004-6361/201015945 · 4.48 Impact Factor
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    ABSTRACT: We obtain the general forms for the current density and the vorticity from the integrability conditions of the basic equations which govern the stationary states of axisymmetric magnetized self-gravitating barotropic objects with meridional flows under the ideal magnetohydrodynamics (MHD) approximation. As seen from the stationary condition equations for such bodies, the presence of the meridional flows and that of the poloidal magnetic fields act oppositely on the internal structures. The different actions of these two physical quantities, the meridional flows and the poloidal magnetic fields, could be clearly seen through stationary structures of the toroidal gaseous configurations around central point masses in the framework of Newtonian gravity because the effects of the two physical quantities can be seen in an amplified way for toroidal systems compared to those for spheroidal stars. The meridional flows make the structures more compact, i.e. the widths of toroids thinner, while the poloidal magnetic fields are apt to elongate the density contours in a certain direction depending on the situation. Therefore, the simultaneous presence of the internal flows and the magnetic fields would work as if there were no such different actions within and around the stationary gaseous objects such as axisymmetric magnetized toroids with internal motions around central compact objects under the ideal MHD approximation, although these two quantities might exist in real systems.
    Monthly Notices of the Royal Astronomical Society 05/2013; 431(2):1453-1469. DOI:10.1093/mnras/stt275 · 5.23 Impact Factor
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    ABSTRACT: The monograph originating from lectures is devoted to the 3+1 formalism in general relativity. It starts with three chapters on basic differential geometry, the geometry of single hypersurfaces embedded in space-time, and the foliation of space-time by a family of spacelike hypersurfaces. Based on this, the decomposition of the Einstein equations relative to the decomposition is presented, formulating the Cauchy problem with constraints and giving rise to the ADM formalism. This is followed by the 3+1 treatment of a perfect fluid and the electromagnetic field as possible sources of gravitation. After these fundamentals, a technical chapter occurs that prepares the following more special aspects of the 3+1 formalism. It motivates and performs a conformal decomposition of the 3-metric on each hypersurface of a 3+1 foliation and introduces some conformal transformation of the 3-metric and the corresponding rewriting of the Einstein equations. In the following chapter, after providing a definition of asymptotic flatness, the global quantities are introduced that one may associate to the space-time or to each slice of the 3+1 foliation: the ADM mass, the ADM linear momentum, the total angular momentum, the Komar mass and the Komar angular momentum. The subsequent chapter shows how the conformal decomposition can be used to solve the constraint equations to get valid initial data for the time evolution. Afterwards, the choice of the foliation and the spatial coordinates in modern numerical relativity is reviewed. The final chapter presents various schemes for the time integration of the 3+1 Einstein equations, putting some emphasis on the successful BSSN scheme. Two appendices on basic tools of the 3+1 formalism and some computer algebra codes based on the Sage system complete the book. With the attempt to make the text self-consistent and complete, the calculations are rather detailed such that the book is well suitable for undergraduate and graduate students.
    Lecture Notes in Physics 01/2012; DOI:10.1007/978-3-642-24525-1


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