251,910 research outputs found
Numerical Relativity in Spherical Polar Coordinates: Off-center Simulations
We have recently presented a new approach for numerical relativity
simulations in spherical polar coordinates, both for vacuum and for
relativistic hydrodynamics. Our approach is based on a reference-metric
formulation of the BSSN equations, a factoring of all tensor components, as
well as a partially implicit Runge-Kutta method, and does not rely on a
regularization of the equations, nor does it make any assumptions about the
symmetry across the origin. In order to demonstrate this feature we present
here several off-centered simulations, including simulations of single black
holes and neutron stars whose center is placed away from the origin of the
coordinate system, as well as the asymmetric head-on collision of two black
holes. We also revisit our implementation of relativistic hydrodynamics and
demonstrate that a reference-metric formulation of hydrodynamics together with
a factoring of all tensor components avoids problems related to the coordinate
singularities at the origin and on the axes. As a particularly demanding test
we present results for a shock wave propagating through the origin of the
spherical polar coordinate system.Comment: 13 pages, 11 figures; matches version published in PR
An interior point algorithm for minimum sum-of-squares clustering
Copyright @ 2000 SIAM PublicationsAn exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the rst time for several fairly large data sets from the literature, including Fisher's 150 iris.This research was supported by the Fonds
National de la Recherche Scientifique Suisse, NSERC-Canada, and FCAR-Quebec
Initial data for black hole-neutron star binaries: a flexible, high-accuracy spectral method
We present a new numerical scheme to solve the initial value problem for
black hole-neutron star binaries. This method takes advantage of the
flexibility and fast convergence of a multidomain spectral representation of
the initial data to construct high-accuracy solutions at a relatively low
computational cost. We provide convergence tests of the method for both
isolated neutron stars and irrotational binaries. In the second case, we show
that we can resolve the small inconsistencies that are part of the
quasi-equilibrium formulation, and that these inconsistencies are significantly
smaller than observed in previous works. The possibility of generating a wide
variety of initial data is also demonstrated through two new configurations
inspired by results from binary black holes. First, we show that choosing a
modified Kerr-Schild conformal metric instead of a flat conformal metric allows
for the construction of quasi-equilibrium binaries with a spinning black hole.
Second, we construct binaries in low-eccentricity orbits, which are a better
approximation to astrophysical binaries than quasi-equilibrium systems.Comment: 19 pages, 11 figures, Modified to match final PRD versio
TESS: A Relativistic Hydrodynamics Code on a Moving Voronoi Mesh
We have generalized a method for the numerical solution of hyperbolic systems
of equations using a dynamic Voronoi tessellation of the computational domain.
The Voronoi tessellation is used to generate moving computational meshes for
the solution of multi-dimensional systems of conservation laws in finite-volume
form. The mesh generating points are free to move with arbitrary velocity, with
the choice of zero velocity resulting in an Eulerian formulation. Moving the
points at the local fluid velocity makes the formulation effectively
Lagrangian. We have written the TESS code to solve the equations of
compressible hydrodynamics and magnetohydrodynamics for both relativistic and
non-relativistic fluids on a dynamic Voronoi mesh. When run in Lagrangian mode,
TESS is significantly less diffusive than fixed mesh codes and thus preserves
contact discontinuities to high precision while also accurately capturing
strong shock waves. TESS is written for Cartesian, spherical and cylindrical
coordinates and is modular so that auxilliary physics solvers are readily
integrated into the TESS framework and so that the TESS framework can be
readily adapted to solve general systems of equations. We present results from
a series of test problems to demonstrate the performance of TESS and to
highlight some of the advantages of the dynamic tessellation method for solving
challenging problems in astrophysical fluid dynamics.Comment: ApJS, 197, 1
Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes
We have developed the formalism necessary to employ the
discontinuous-Galerkin approach in general-relativistic hydrodynamics. The
formalism is firstly presented in a general 4-dimensional setting and then
specialized to the case of spherical symmetry within a 3+1 splitting of
spacetime. As a direct application, we have constructed a one-dimensional code,
EDGES, which has been used to asses the viability of these methods via a series
of tests involving highly relativistic flows in strong gravity. Our results
show that discontinuous Galerkin methods are able not only to handle strong
relativistic shock waves but, at the same time, to attain very high orders of
accuracy and exponential convergence rates in smooth regions of the flow. Given
these promising prospects and their affinity with a pseudospectral solution of
the Einstein equations, discontinuous Galerkin methods could represent a new
paradigm for the accurate numerical modelling in relativistic astrophysics.Comment: 24 pages, 19 figures. Small changes; matches version to appear in PR
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