6,714 research outputs found
The remarkable effectiveness of time-dependent damping terms for second order evolution equations
We consider a second order linear evolution equation with a dissipative term
multiplied by a time-dependent coefficient. Our aim is to design the
coefficient in such a way that all solutions decay in time as fast as possible.
We discover that constant coefficients do not achieve the goal, as well as
time-dependent coefficients that are too big. On the contrary, pulsating
coefficients which alternate big and small values in a suitable way prove to be
more effective.
Our theory applies to ordinary differential equations, systems of ordinary
differential equations, and partial differential equations of hyperbolic type.Comment: 32 pages, 5 figure
Constraint Damping in First-Order Evolution Systems for Numerical Relativity
A new constraint suppressing formulation of the Einstein evolution equations
is presented, generalizing the five-parameter first-order system due to Kidder,
Scheel and Teukolsky (KST). The auxiliary fields, introduced to make the KST
system first-order, are given modified evolution equations designed to drive
constraint violations toward zero. The algebraic structure of the new system is
investigated, showing that the modifications preserve the hyperbolicity of the
fundamental and constraint evolution equations. The evolution of the
constraints for pertubations of flat spacetime is completely analyzed, and all
finite-wavelength constraint modes are shown to decay exponentially when
certain adjustable parameters satisfy appropriate inequalities. Numerical
simulations of a single Schwarzschild black hole are presented, demonstrating
the effectiveness of the new constraint-damping modifications.Comment: 11 pages, 5 figure
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
Simulation of Binary Black Hole Spacetimes with a Harmonic Evolution Scheme
A numerical solution scheme for the Einstein field equations based on
generalized harmonic coordinates is described, focusing on details not provided
before in the literature and that are of particular relevance to the binary
black hole problem. This includes demonstrations of the effectiveness of
constraint damping, and how the time slicing can be controlled through the use
of a source function evolution equation. In addition, some results from an
ongoing study of binary black hole coalescence, where the black holes are
formed via scalar field collapse, are shown. Scalar fields offer a convenient
route to exploring certain aspects of black hole interactions, and one
interesting, though tentative suggestion from this early study is that behavior
reminiscent of "zoom-whirl" orbits in particle trajectories is also present in
the merger of equal mass, non-spinning binaries, with appropriately fine-tuned
initial conditions.Comment: 16 pages, 14 figures; replaced with published versio
Gap-Townes solitons and localized excitations in low dimensional Bose Einstein condensates in optical lattices
We discuss localized ground states of Bose-Einstein condensates in optical
lattices with attractive and repulsive three-body interactions in the framework
of a quintic nonlinear Schr\"odinger equation which extends the
Gross-Pitaevskii equation to the one dimensional case. We use both a
variational method and a self-consistent approach to show the existence of
unstable localized excitations which are similar to Townes solitons of the
cubic nonlinear Schr\"odinger equation in two dimensions. These solutions are
shown to be located in the forbidden zones of the band structure, very close to
the band edges, separating decaying states from stable localized ones
(gap-solitons) fully characterizing their delocalizing transition. In this
context usual gap solitons appear as a mechanism for arresting collapse in low
dimensional BEC in optical lattices with attractive real three-body
interaction. The influence of the imaginary part of the three-body interaction,
leading to dissipative effects on gap solitons and the effect of atoms feeding
from the thermal cloud are also discussed. These results may be of interest for
both BEC in atomic chip and Tonks-Girardeau gas in optical lattices
Stability of solutions to nonlinear wave equations with switching time-delay
In this paper we study well-posedness and asymptotic stability for a class of
nonlinear second-order evolution equations with intermittent delay damping.
More precisely, a delay feedback and an undelayed one act alternately in time.
We show that, under suitable conditions on the feedback operators, asymptotic
stability results are available. Concrete examples included in our setting are
illustrated. We give also stability results for an abstract model with
alternate positive-negative damping, without delay
Mixed Hyperbolic - Second-Order Parabolic Formulations of General Relativity
Two new formulations of general relativity are introduced. The first one is a
parabolization of the Arnowitt, Deser, Misner (ADM) formulation and is derived
by addition of combinations of the constraints and their derivatives to the
right-hand-side of the ADM evolution equations. The desirable property of this
modification is that it turns the surface of constraints into a local attractor
because the constraint propagation equations become second-order parabolic
independently of the gauge conditions employed. This system may be classified
as mixed hyperbolic - second-order parabolic. The second formulation is a
parabolization of the Kidder, Scheel, Teukolsky formulation and is a manifestly
mixed strongly hyperbolic - second-order parabolic set of equations, bearing
thus resemblance to the compressible Navier-Stokes equations. As a first test,
a stability analysis of flat space is carried out and it is shown that the
first modification exponentially damps and smoothes all constraint violating
modes. These systems provide a new basis for constructing schemes for long-term
and stable numerical integration of the Einstein field equations.Comment: 19 pages, two column, references added, two proofs of well-posedness
added, content changed to agree with submitted version to PR
Constraint damping of the conformal and covariant formulation of the Z4 system in simulations of binary neutron stars
Following previous work in vacuum spacetimes, we investigate the
constraint-damping properties in the presence of matter of the recently
developed traceless, conformal and covariant Z4 (CCZ4) formulation of the
Einstein equations. First, we evolve an isolated neutron star with an ideal gas
equation of state and subject to a constraint-violating perturbation. We
compare the evolution of the constraints using the CCZ4 and
Baumgarte-Shibata-Shapiro-Nakamura-Oohara-Kojima (BSSNOK) systems. Second, we
study the collapse of an unstable spherical star to a black hole. Finally, we
evolve binary neutron star systems over several orbits until the merger, the
formation of a black hole, and up to the ringdown. We show that the CCZ4
formulation is stable in the presence of matter and that the constraint
violations are one or more orders of magnitude smaller than for the BSSNOK
formulation. Furthermore, by comparing the CCZ4 and the BSSNOK formulations
also for neutron star binaries with large initial constraint violations, we
investigate their influence on the errors on physical quantities. We also give
a new, simple and robust prescription for the damping parameter that removes
the instabilities found when using the fully covariant version of CCZ4 in the
evolution of black holes. Overall, we find that at essentially the same
computational costs the CCZ4 formulation provides solutions that are stable and
with a considerably smaller violation of the Hamiltonian constraint than the
BSSNOK formulation. We also find that the performance of the CCZ4 formulation
is very similar to another conformal and traceless, but noncovariant
formulation of the Z4 system, i.e. the Z4c formulation.Comment: 15 pages, 11 figures; accepted for publication in Phys. Rev.
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