7,477 research outputs found
A new dissipation term for finite-difference simulations in Relativity
We present a new numerical dissipation algorithm, which can be efficiently
used in combination with centered finite-difference methods. We start from a
formulation of centered finite-volume methods for Numerical Relativity, in
which third-order space accuracy can be obtained by employing just
piecewise-linear reconstruction. We obtain a simplified version of the
algorithm, which can be viewed as a centered finite-difference method plus some
'adaptive dissipation'. The performance of this algorithm is confirmed by
numerical results obtained from 3D black hole simulations.Comment: Talk presented at the Spanish Relativity Meeting (Tenerife 2007
Constraint-preserving boundary conditions in the 3+1 first-order approach
A set of energy-momentum constraint-preserving boundary conditions is
proposed for the first-order Z4 case. The stability of a simple numerical
implementation is tested in the linear regime (robust stability test), both
with the standard corner and vertex treatment and with a modified
finite-differences stencil for boundary points which avoids corners and
vertices even in cartesian-like grids. Moreover, the proposed boundary
conditions are tested in a strong field scenario, the Gowdy waves metric,
showing the expected rate of convergence. The accumulated amount of
energy-momentum constraint violations is similar or even smaller than the one
generated by either periodic or reflection conditions, which are exact in the
Gowdy waves case. As a side theoretical result, a new symmetrizer is explicitly
given, which extends the parametric domain of symmetric hyperbolicity for the
Z4 formalism. The application of these results to first-order BSSN-like
formalisms is also considered.Comment: Revised version, with conclusive numerical evidence. 23 pages, 12
figure
New Formalism for Numerical Relativity
We present a new formulation of the Einstein equations that casts them in an
explicitly first order, flux-conservative, hyperbolic form. We show that this
now can be done for a wide class of time slicing conditions, including maximal
slicing, making it potentially very useful for numerical relativity. This
development permits the application to the Einstein equations of advanced
numerical methods developed to solve the fluid dynamic equations, {\em without}
overly restricting the time slicing, for the first time. The full set of
characteristic fields and speeds is explicitly given.Comment: uucompresed PS file. 4 pages including 1 figure. Revised version adds
a figure showing a comparison between the standard ADM approach and the new
formulation. Also available at http://jean-luc.ncsa.uiuc.edu/Papers/ Appeared
in Physical Review Letters 75, 600 (1995
Flux Limiter Methods in 3D Numerical Relativity
New numerical methods have been applied in relativity to obtain a numerical
evolution of Einstein equations much more robust and stable. Starting from 3+1
formalism and with the evolution equations written as a FOFCH (first-order flux
conservative hyperbolic) system, advanced numerical methods from CFD
(Computational Fluid Dynamics) have been successfully applied. A flux limiter
mechanism has been implemented in order to deal with steep gradients like the
ones usually associated with black hole spacetimes. As a test bed, the method
has been applied to 3D metrics describing propagation of nonlinear gauge waves.
Results are compared with the ones obtained with standard methods, showing a
great increase in both robustness and stability of the numerical algorithm.Comment: 9 pages, 5 figures. to be published in the Procedings of ERE0
Elliptic solutions and solitary waves of a higher order KdV--BBM long wave equation
We provide conditions for existence of hyperbolic, unbounded periodic and
elliptic solutions in terms of Weierstrass functions of both third and
fifth-order KdV--BBM (Korteweg-de Vries--Benjamin, Bona \& Mahony) regularized
long wave equation. An analysis for the initial value problem is developed
together with a local and global well-posedness theory for the third-order
KdV--BBM equation. Traveling wave reduction is used together with zero boundary
conditions to yield solitons and periodic unbounded solutions, while for
nonzero boundary conditions we find solutions in terms of Weierstrass elliptic
functions. For the fifth-order KdV--BBM equation we show that a parameter
, for which the equation has a Hamiltonian, represents a
restriction for which there are constraint curves that never intersect a region
of unbounded solitary waves, which in turn shows that only dark or bright
solitons and no unbounded solutions exist. Motivated by the lack of a
Hamiltonian structure for we develop bounds, and
we show for the non Hamiltonian system that dark and bright solitons coexist
together with unbounded periodic solutions. For nonzero boundary conditions,
due to the complexity of the nonlinear algebraic system of coefficients of the
elliptic equation we construct Weierstrass solutions for a particular set of
parameters only.Comment: 13 pages, 6 figure
Gauge and constraint degrees of freedom: from analytical to numerical approximations in General Relativity
The harmonic formulation of Einstein's field equations is considered, where
the gauge conditions are introduced as dynamical constraints. The difference
between the fully constrained approach (used in analytical approximations) and
the free evolution one (used in most numerical approximations) is pointed out.
As a generalization, quasi-stationary gauge conditions are also discussed,
including numerical experiments with the gauge-waves testbed. The complementary
3+1 approach is also considered, where constraints are related instead with
energy and momentum first integrals and the gauge must be provided separately.
The relationship between the two formalisms is discussed in a more general
framework (Z4 formalism). Different strategies in black hole simulations follow
when introducing singularity avoidance as a requirement. More flexible
quasi-stationary gauge conditions are proposed in this context, which can be
seen as generalizations of the current 'freezing shift' prescriptions.Comment: Talk given at the Spanish Relativity Meeting, Tenerife, September
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