95 research outputs found
Local characteristic algorithms for relativistic hydrodynamics
Numerical schemes for the general relativistic hydrodynamic equations are
discussed. The use of conservative algorithms based upon the characteristic
structure of those equations, developed during the last decade building on
ideas first applied in Newtonian hydrodynamics, provides a robust methodology
to obtain stable and accurate solutions even in the presence of
discontinuities. The knowledge of the wave structure of the above system is
essential in the construction of the so-called linearized Riemann solvers, a
class of numerical schemes specifically designed to solve nonlinear hyperbolic
systems of conservation laws. In the last part of the review some astrophysical
applications of such schemes, using the coupled system of the
(characteristic) Einstein and hydrodynamic equations, are also briefly
presented.Comment: 20 pages, 4 figures, To appear in the proceedings of the workshop
"The conformal structure of space-time", J. Frauendiener, H. Friedrich, eds,
Springer Lecture Notes in Physic
Gravitational waves in dynamical spacetimes with matter content in the Fully Constrained Formulation
The Fully Constrained Formulation (FCF) of General Relativity is a novel
framework introduced as an alternative to the hyperbolic formulations
traditionally used in numerical relativity. The FCF equations form a hybrid
elliptic-hyperbolic system of equations including explicitly the constraints.
We present an implicit-explicit numerical algorithm to solve the hyperbolic
part, whereas the elliptic sector shares the form and properties with the well
known Conformally Flat Condition (CFC) approximation. We show the stability
andconvergence properties of the numerical scheme with numerical simulations of
vacuum solutions. We have performed the first numerical evolutions of the
coupled system of hydrodynamics and Einstein equations within FCF. As a proof
of principle of the viability of the formalism, we present 2D axisymmetric
simulations of an oscillating neutron star. In order to simplify the analysis
we have neglected the back-reaction of the gravitational waves into the
dynamics, which is small (<2 %) for the system considered in this work. We use
spherical coordinates grids which are well adapted for simulations of stars and
allow for extended grids that marginally reach the wave zone. We have extracted
the gravitational wave signature and compared to the Newtonian quadrupole and
hexadecapole formulae. Both extraction methods show agreement within the
numerical errors and the approximations used (~30 %).Comment: 17 pages, 9 figures, 2 tables, accepted for publication in PR
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
Numerical hydrodynamics in general relativity
The current status of numerical solutions for the equations of ideal general
relativistic hydrodynamics is reviewed. With respect to an earlier version of
the article the present update provides additional information on numerical
schemes and extends the discussion of astrophysical simulations in general
relativistic hydrodynamics. Different formulations of the equations are
presented, with special mention of conservative and hyperbolic formulations
well-adapted to advanced numerical methods. A large sample of available
numerical schemes is discussed, paying particular attention to solution
procedures based on schemes exploiting the characteristic structure of the
equations through linearized Riemann solvers. A comprehensive summary of
astrophysical simulations in strong gravitational fields is presented. These
include gravitational collapse, accretion onto black holes and hydrodynamical
evolutions of neutron stars. 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 on the formulation of the gravitational field and
hydrodynamic equations and the numerical methodology designed to solve them.Comment: 105 pages, 12 figures. The full online-readable version of this
article, including several animations, will be published in Living Reviews in
Relativity at http://www.livingreviews.or
Relativistic hydrodynamics on spacelike and null surfaces: Formalism and computations of spherically symmetric spacetimes
We introduce a formulation of Eulerian general relativistic hydrodynamics
which is applicable for (perfect) fluid data prescribed on either spacelike or
null hypersurfaces. Simple explicit expressions for the characteristic speeds
and fields are derived in the general case. A complete implementation of the
formalism is developed in the case of spherical symmetry. The algorithm is
tested in a number of different situations, predisposing for a range of
possible applications. We consider the Riemann problem for a polytropic gas,
with initial data given on a retarded/advanced time slice of Minkowski
spacetime. We compute perfect fluid accretion onto a Schwarzschild black hole
spacetime using ingoing null Eddington-Finkelstein coordinates. Tests of fluid
evolution on dynamic background include constant density and TOV stars sliced
along the radial null cones. Finally, we consider the accretion of
self-gravitating matter onto a central black hole and the ensuing increase in
the mass of the black hole horizon.Comment: 23 pages, 13 figures, submitted to Phys. Rev.
Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)
This workshop brought together
leading experts, as well as the most
promising young researchers, working on nonlinear
hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in
modeling, analysis, and numerics. Particular topics included ill-/well-posedness,
randomness and multiscale modeling, flows in a moving domain, free boundary problems,
games and control
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