22,631 research outputs found
Solving the Einstein constraints in periodic spaces with a multigrid approach
Novel applications of Numerical Relativity demand for more flexible
algorithms and tools. In this paper, I develop and test a multigrid solver,
based on the infrastructure provided by the Einstein Toolkit, for elliptic
partial differential equations on spaces with periodic boundary conditions.
This type of boundary often characterizes the numerical representation of
cosmological models, where space is assumed to be made up of identical copies
of a single fiducial domain, so that only a finite volume (with periodic
boundary conditions at its edges) needs to be simulated. After a few tests and
comparisons with existing codes, I use the solver to generate initial data for
an infinite, periodic, cubic black-hole lattice.Comment: 25 pages, 15 figures. Fixed typos, added references and software
release informatio
Relativistic MHD and black hole excision: Formulation and initial tests
A new algorithm for solving the general relativistic MHD equations is
described in this paper. We design our scheme to incorporate black hole
excision with smooth boundaries, and to simplify solving the combined Einstein
and MHD equations with AMR. The fluid equations are solved using a finite
difference Convex ENO method. Excision is implemented using overlapping grids.
Elliptic and hyperbolic divergence cleaning techniques allow for maximum
flexibility in choosing coordinate systems, and we compare both methods for a
standard problem. Numerical results of standard test problems are presented in
two-dimensional flat space using excision, overlapping grids, and elliptic and
hyperbolic divergence cleaning.Comment: 22 pages, 8 figure
Multigrid elliptic equation solver with adaptive mesh refinement
In this paper we describe in detail the computational algorithm used by our
parallel multigrid elliptic equation solver with adaptive mesh refinement. Our
code uses truncation error estimates to adaptively refine the grid as part of
the solution process. The presentation includes a discussion of the orders of
accuracy that we use for prolongation and restriction operators to ensure
second order accurate results and to minimize computational work. Code tests
are presented that confirm the overall second order accuracy and demonstrate
the savings in computational resources provided by adaptive mesh refinement.Comment: 12 pages, 9 figures, Modified in response to reviewer suggestions,
added figure, added references. Accepted for publication in J. Comp. Phy
Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems
We present a modification to the Berger and Oliger adaptive mesh refinement
algorithm designed to solve systems of coupled, non-linear, hyperbolic and
elliptic partial differential equations. Such systems typically arise during
constrained evolution of the field equations of general relativity. The novel
aspect of this algorithm is a technique of "extrapolation and delayed solution"
used to deal with the non-local nature of the solution of the elliptic
equations, driven by dynamical sources, within the usual Berger and Oliger
time-stepping framework. We show empirical results demonstrating the
effectiveness of this technique in axisymmetric gravitational collapse
simulations. We also describe several other details of the code, including
truncation error estimation using a self-shadow hierarchy, and the
refinement-boundary interpolation operators that are used to help suppress
spurious high-frequency solution components ("noise").Comment: 31 pages, 15 figures; replaced with published versio
hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity
A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other discontinuities. This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity. We present a novel hp-adaptive numerical scheme for curvilinear and non-conforming meshes. It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains. The code employs compactification to move the outer boundary near spatial infinity. We explore the properties of the code on some test problems, including one mimicking Neutron stars with phase transitions. We also apply it to construct initial data for two or three black holes
Adaptive Mesh Refinement for Characteristic Codes
The use of adaptive mesh refinement (AMR) techniques is crucial for accurate
and efficient simulation of higher dimensional spacetimes. In this work we
develop an adaptive algorithm tailored to the integration of finite difference
discretizations of wave-like equations using characteristic coordinates. We
demonstrate the algorithm by constructing a code implementing the
Einstein-Klein-Gordon system of equations in spherical symmetry. We discuss how
the algorithm can trivially be generalized to higher dimensional systems, and
suggest a method that can be used to parallelize a characteristic code.Comment: 36 pages, 17 figures; updated to coincide with journal versio
A sparse representation of gravitational waves from precessing compact binaries
Many relevant applications in gravitational wave physics share a significant
common problem: the seven-dimensional parameter space of gravitational
waveforms from precessing compact binary inspirals and coalescences is large
enough to prohibit covering the space of waveforms with sufficient density. We
find that by using the reduced basis method together with a parametrization of
waveforms based on their phase and precession, we can construct ultra-compact
yet high-accuracy representations of this large space. As a demonstration, we
show that less than judiciously chosen precessing inspiral waveforms are
needed for cycles, mass ratios from to and spin magnitudes . In fact, using only the first reduced basis waveforms yields a
maximum mismatch of over the whole range of considered parameters. We
test whether the parameters selected from the inspiral regime result in an
accurate reduced basis when including merger and ringdown; we find that this is
indeed the case in the context of a non-precessing effective-one-body model.
This evidence suggests that as few as numerical simulations of
binary black hole coalescences may accurately represent the seven-dimensional
parameter space of precession waveforms for the considered ranges.Comment: 5 pages, 3 figures. The parameters selected for the basis of
precessing waveforms can be found in the source file
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