274 research outputs found

### Spectral Methods for Numerical Relativity. The Initial Data Problem

Numerical relativity has traditionally been pursued via finite differencing.
Here we explore pseudospectral collocation (PSC) as an alternative to finite
differencing, focusing particularly on the solution of the Hamiltonian
constraint (an elliptic partial differential equation) for a black hole
spacetime with angular momentum and for a black hole spacetime superposed with
gravitational radiation. In PSC, an approximate solution, generally expressed
as a sum over a set of orthogonal basis functions (e.g., Chebyshev
polynomials), is substituted into the exact system of equations and the
residual minimized. For systems with analytic solutions the approximate
solutions converge upon the exact solution exponentially as the number of basis
functions is increased. Consequently, PSC has a high computational efficiency:
for solutions of even modest accuracy we find that PSC is substantially more
efficient, as measured by either execution time or memory required, than finite
differencing; furthermore, these savings increase rapidly with increasing
accuracy. The solution provided by PSC is an analytic function given
everywhere; consequently, no interpolation operators need to be defined to
determine the function values at intermediate points and no special
arrangements need to be made to evaluate the solution or its derivatives on the
boundaries. Since the practice of numerical relativity by finite differencing
has been, and continues to be, hampered by both high computational resource
demands and the difficulty of formulating acceptable finite difference
alternatives to the analytic boundary conditions, PSC should be further pursued
as an alternative way of formulating the computational problem of finding
numerical solutions to the field equations of general relativity.Comment: 15 pages, 5 figures, revtex, submitted to PR

### The Federal Administrative Court Proposal: An Examination of General Principals

Simulations of relativistic hydrodynamics often need both high accuracy and robust shock-handling properties. The discontinuous Galerkin method combines these featuresâ€”a high order of convergence in regions where the solution is smooth and shock-capturing properties for regions where it is notâ€”with geometric flexibility and is therefore well suited to solve the partial differential equations describing astrophysical scenarios. We present here evolutions of a general-relativistic neutron star with the discontinuous Galerkin method. In these simulations, we simultaneously evolve the spacetime geometry and the matter on the same computational grid, which we conform to the spherical geometry of the problem. To verify the correctness of our implementation, we perform standard convergence and shock tests. We then show results for evolving, in three dimensions, a Kerr black hole; a neutron star in the Cowling approximation (holding the spacetime metric fixed); and, finally, a neutron star where the spacetime and matter are both dynamical. The evolutions show long-term stability, good accuracy, and an improved rate of convergence versus a comparable-resolution finite-volume method

### Estimating the final spin of a binary black hole coalescence

We present a straightforward approach for estimating the final black hole
spin of a binary black hole coalescence with arbitrary initial masses and
spins. Making some simple assumptions, we estimate the final angular momentum
to be the sum of the individual spins plus the orbital angular momentum of a
test particle orbiting at the last stable orbit around a Kerr black hole with a
spin parameter of the final black hole. The formula we obtain is able to
reproduce with reasonable accuracy the results from available numerical
simulations, but, more importantly, it can be used to investigate what
configurations might give rise to interesting dynamics. In particular, we
discuss scenarios which might give rise to a ``flip'' in the direction of the
total angular momentum of the system. By studying the dependence of the final
spin upon the mass ratio and initial spins we find that our simple approach
suggests that it is not possible to spin-up a black hole to extremal values
through merger scenarios irrespective of the mass ratio of the objects
involved.Comment: 9 pages, 8 figure

### Evolving relativistic fluid spacetimes using pseudospectral methods and finite differencing

We present a new code for solving the coupled Einstein-hydrodynamics
equations to evolve relativistic, self-gravitating fluids. The Einstein field
equations are solved on one grid using pseudospectral methods, while the fluids
are evolved on another grid by finite differencing. We discuss implementation
details, such as the communication between the grids and the treatment of
stellar surfaces, and present code tests.Comment: To appear in the Proceedings of the Eleventh Marcel Grossmann Meetin

### A New Generalized Harmonic Evolution System

A new representation of the Einstein evolution equations is presented that is
first order, linearly degenerate, and symmetric hyperbolic. This new system
uses the generalized harmonic method to specify the coordinates, and
exponentially suppresses all small short-wavelength constraint violations.
Physical and constraint-preserving boundary conditions are derived for this
system, and numerical tests that demonstrate the effectiveness of the
constraint suppression properties and the constraint-preserving boundary
conditions are presented.Comment: Updated to agree with published versio

### Orbiting binary black hole evolutions with a multipatch high order finite-difference approach

We present numerical simulations of orbiting black holes for around twelve
cycles, using a high-order multipatch approach. Unlike some other approaches,
the computational speed scales almost perfectly for thousands of processors.
Multipatch methods are an alternative to AMR (adaptive mesh refinement), with
benefits of simplicity and better scaling for improving the resolution in the
wave zone. The results presented here pave the way for multipatch evolutions of
black hole-neutron star and neutron star-neutron star binaries, where high
resolution grids are needed to resolve details of the matter flow

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