Article
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
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 "Consider for example the shallow water equations on the sphere [34] as a model for the global air and water flow and the " shallow water " magnetohydrodynamic equations [14,27] as a model for the global dynamics in the solar tachocline. Further examples include surface acoustic waves [33], (magneto)hydrodynamics in general relativity [11,12,28], the flow of oil on a moving water surface, the transport processes on cell surfaces [1,26], surfactants on the interfacial hypersurface between two phases in multiphase flow [6], and the flow of a fluid in fractured porous media whose fractions are considered as a lower dimensional mani fold [15]. Scalar conservation laws have been established as a good model problem for studying the nonlinear effects in such systems. "
[Show abstract] [Hide abstract] ABSTRACT: In this paper we establish wellposedness for scalar conservation laws on closed manifolds M endowed with a constant or a timedependent Riemannian metric for initial values in L^\infty(M). In particular we show the existence and uniqueness of entropy solutions as well as the L^1 contraction property and a comparison principle for these solutions. Throughout the paper the flux function is allowed to depend on time and to have nonvanishing divergence. Furthermore, we derive estimates of the total variation of the solution for initial values in BV(M), and we give, in the case of a timeindependent metric, a simple geometric characterisation of flux functions that give rise to total variation diminishing estimates. 
 "One such approach is known as the constraint transport technique [3] [4] which adopts a particular algorithm that staggers the variables appropriately to ensure the satisfaction of the constraint at roundoff level within Finite Difference and Finite Elements techniques. This approach has been quite successful in a number of applications across different disciplines and particularly relevant in astrophysics applications [5] [6] [7] [8] [9] [10] [11] [12]. However, by design it imposes limits on the algorithmic options available to an implementation. "
[Show abstract] [Hide abstract] ABSTRACT: We study and develop constraint preserving boundary conditions for the Newtonian magnetohydrodynamic equations and analyze the behavior of the numerical solution upon considering different possible options. We concentrate on both the standard ideal MHD system and the one augmented by a “pseudo potential” to control the divergence free constraint. We show how the boundary conditions developed significantly reduce the violations generated at the boundaries at the numerical level and how lessen their influence in the interior of the computational domain by making use of the available freedom in the equations. 
 "The code is axisymmetric and, as in the hydrodynamic case, the " cartoon " method is employed [6] for evolving the BSSN equations, which makes possible the use of Cartesian coordinates in axisymmetric simulations, and a cylindrical grid is used for the GRMHD equations. It has recently been applied in the study of the collapse of magnetized hypermassive neutron stars to black holes in the context of gammaray burst mechanisms, as we discuss in Section 5. Duez et al. [108]: The development of this code has proceeded simultaneously to that of [366], to the point that they share some important features and both have been used to study similar astrophysical problems, whose results have been presented in joint publications [105, 106]. This code solves the GRMHD equations in both two dimensions (axisymmetry) and three dimensions, using a Cartesian grid for the latter case and the " cartoon " method in the former (although a cylindrical grid is used for the induction and MHD evolution equations). "
[Show abstract] [Hide abstract] ABSTRACT: This article presents a comprehensive overview of numerical hydrodynamics and magnetohydrodynamics (MHD) in general relativity. Some significant additions have been incorporated with respect to the previous two versions of this review (2000, 2003), most notably the coverage of generalrelativistic MHD, a field in which remarkable activity and progress has occurred in the last few years. Correspondingly, the discussion of astrophysical simulations in generalrelativistic hydrodynamics is enlarged to account for recent relevant advances, while those dealing with generalrelativistic MHD are amply covered in this review for the first time. The basic outline of this article is nevertheless similar to its earlier versions, save for the addition of MHDrelated issues throughout. Hence, different formulations of both the hydrodynamics and MHD equations are presented, with special mention of conservative and hyperbolic formulations well adapted to advanced numerical methods. A large sample of numerical approaches for solving such hyperbolic systems of equations is discussed, paying particular attention to solution procedures based on schemes exploiting the characteristic structure of the equations through linearized Riemann solvers. As previously stated, a comprehensive summary of astrophysical simulations in strong gravitational fields is also presented. These are detailed in three basic sections, namely gravitational collapse, blackhole accretion, and neutronstar evolutions; despite the boundaries, these sections may (and in fact do) overlap throughout the discussion. The material contained in these sections highlights the numerical challenges of various representative simulations. It also follows, to some extent, the chronological development of the field, concerning advances in the formulation of the gravitational field, hydrodynamics and MHD equations and the numerical methodology designed to solve them. To keep the length of this article reasonable, an effort has been made to focus on multidimensional studies, directing the interested reader to earlier versions of the review for discussions on onedimensional works.