Symmetry Without Symmetry: Numerical Simulation of Axisymmetric Systems Using Cartesian Grids

Alcubierre, Miguel; Brandt, Steven R.; Brügmann, Bernd; Holz, D. E.; Seidel, Edward; Takahashi, Ryōji; Thornburg, Jonathan

doi:10.1142/s0218271801000834

Cited by 151 publications

(188 citation statements)

References 27 publications

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“…We evolve only the x-z plane [a (2+1) dimensional problem]. We adopt the cartoon method [39] for evolving the BSSN equations, and use cylindrical coordinates for evolving the induction and MHD equations. In this scheme, the coordinate x is identified with the cylindrical radius ̟, and the y-direction corresponds to the azimuthal direction.…”

Section: Methodsmentioning

confidence: 99%

Evolution of magnetized, differentially rotating neutron stars: Simulations in full general relativity

et al. 2006

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We study the effects of magnetic fields on the evolution of differentially rotating neutron stars, which can be formed in stellar core collapse or binary neutron star coalescence. Magnetic braking and the magnetorotational instability (MRI) both act on differentially rotating stars to redistribute angular momentum. Simulations of these stars are carried out in axisymmetry using our recently developed codes which integrate the coupled Einstein-Maxwell-MHD equations. We consider stars with two different equations of state (EOS), a gamma-law EOS with Γ = 2, and a more realistic hybrid EOS, and we evolve them adiabatically. Our simulations show that the fate of the star depends on its mass and spin. For initial data, we consider three categories of differentially rotating, equilibrium configurations, which we label normal, hypermassive and ultraspinning. Normal configurations have rest masses below the maximum achievable with uniform rotation, and angular momentum below the maximum for uniform rotation at the same rest mass. Hypermassive stars have rest masses exceeding the mass limit for uniform rotation. Ultraspinning stars are not hypermassive, but have angular momentum exceeding the maximum for uniform rotation at the same rest mass. We show that a normal star will evolve to a uniformly rotating equilibrium configuration. An ultraspinning star evolves to an equilibrium state consisting of a nearly uniformly rotating central core, surrounded by a differentially rotating torus with constant angular velocity along magnetic field lines, so that differential rotation ceases to wind the magnetic field. In addition, the final state is stable against the MRI, although it has differential rotation. For a hypermassive neutron star, the MHD-driven angular momentum transport leads to catastrophic collapse of the core. The resulting rotating black hole is surrounded by a hot, massive, magnetized torus undergoing quasistationary accretion, and a magnetic field collimated along the spin axis-a promising candidate for the central engine of a short gamma-ray burst.

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Section: Methodsmentioning

confidence: 99%

Evolution of magnetized, differentially rotating neutron stars: Simulations in full general relativity

et al. 2006

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“…In the following of this subsection, we assume that Einstein's equations are solved in the Cartesian coordinates (x, y, z) for simplicity. Although we apply the implementation described here to axisymmetric issues as well as nonaxisymmetric ones, this causes no problem since Einstein's equations in axial symmetry can be solved using the so-called Cartoon method in which an axisymmetric boundary condition is appropriately imposed in the Cartesian coordinates [32][33][34]: In the Cartoon method, the field equations are solved only in the y = 0 plane, and grid points of y = ±∆x (∆x denotes the grid spacing in the uniform grid) are used for imposing the axisymmetric boundary conditions. We solve Einstein's evolution equations in our latest BSSN formalism [35,6].…”

Section: B Einstein's Equationmentioning

confidence: 99%

“…In the axisymmetric case, the equations for (ρ * , S i , S 0 ) should be written in the cylindrical coordinates (̟, ϕ, z) when we adopt the Cartoon method for solving Einstein's evolution equations [32][33][34]. On the other hand, in the standard Cartoon method, Einstein's equations are solved in the y = 0 plane for which x = ̟, S ̟ = S x , S ϕ = xS y , and other similar relations hold for vector and tensor quantities.…”

Section: Grmhd Equationsmentioning

confidence: 99%

Magnetohydrodynamics in full general relativity: Formulation and tests

2005

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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.04.25. Dm, 04.40.Nr, 47.75.+f, 95.30.Qd

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“…Additionally boundary conditions in both these classes may be either local or global. For example periodic boundary conditions are global symmetry conditions, the Cartoon [26] boundary condition is a symmetry condition but is local, and radiative boundary conditions are physical and local.…”

Section: Boundary Conditionsmentioning

confidence: 99%

The Cactus Framework and Toolkit: Design and Applications

Goodale

Allen

Lanfermann

et al. 2003

Lecture Notes in Computer Science

Self Cite

245

249

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Abstract. We describe Cactus, a framework for building a variety of computing applications in science and engineering, including astrophysics, relativity and chemical engineering. We first motivate by example the need for such frameworks to support multi-platform, high performance applications across diverse communities. We then describe the design of the latest release of Cactus (Version 4.0) a complete rewrite of earlier versions, which enables highly modular, multi-language, parallel applications to be developed by single researchers and large collaborations alike. Making extensive use of abstractions, we detail how we are able to provide the latest advances in computational science, such as interchangeable parallel data distribution and high performance IO layers, while hiding most details of the underlying computational libraries from the application developer. We survey how Cactus 4.0 is being used by various application communities, and describe how it will also enable these applications to run on the computational Grids of the near future. Application Frameworks in Scientific ComputingVirtually all areas of science and engineering, as well as an increasing number of other fields, are turning to computational science to provide crucial tools to further their disciplines. The increasing power of computers offers unprecedented ability to solve complex equations, simulate natural and man-made complex processes, and visualise data, as well as providing novel possibilities such as new forms of art and entertainment. As computational power advances rapidly, computational tools, libraries, and computing paradigms themselves also advance. In such an environment, even experienced computational scientists and engineers can easily find themselves falling behind the pace of change, while they redesign and rework their codes to support the next computer architecture. This

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Symmetry Without Symmetry: Numerical Simulation of Axisymmetric Systems Using Cartesian Grids

Cited by 151 publications

References 27 publications

Evolution of magnetized, differentially rotating neutron stars: Simulations in full general relativity

Evolution of magnetized, differentially rotating neutron stars: Simulations in full general relativity

Magnetohydrodynamics in full general relativity: Formulation and tests

The Cactus Framework and Toolkit: Design and Applications

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