19,276 research outputs found
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
Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems
We present a modification to the Berger and Oliger adaptive mesh refinement
algorithm designed to solve systems of coupled, non-linear, hyperbolic and
elliptic partial differential equations. Such systems typically arise during
constrained evolution of the field equations of general relativity. The novel
aspect of this algorithm is a technique of "extrapolation and delayed solution"
used to deal with the non-local nature of the solution of the elliptic
equations, driven by dynamical sources, within the usual Berger and Oliger
time-stepping framework. We show empirical results demonstrating the
effectiveness of this technique in axisymmetric gravitational collapse
simulations. We also describe several other details of the code, including
truncation error estimation using a self-shadow hierarchy, and the
refinement-boundary interpolation operators that are used to help suppress
spurious high-frequency solution components ("noise").Comment: 31 pages, 15 figures; replaced with published versio
Unique continuation property with partial information for two-dimensional anisotropic elasticity systems
In this paper, we establish a novel unique continuation property for
two-dimensional anisotropic elasticity systems with partial information. More
precisely, given a homogeneous elasticity system in a domain, we investigate
the unique continuation by assuming only the vanishing of one component of the
solution in a subdomain. Using the corresponding Riemann function, we prove
that the solution vanishes in the whole domain provided that the other
component vanishes at one point up to its second derivatives. Further, we
construct several examples showing the possibility of further reducing the
additional information of the other component. This result possesses remarkable
significance in both theoretical and practical aspects because the required
data is almost halved for the unique determination of the whole solution.Comment: 14 pages, 1 figur
Qualitative properties of solutions to mixed-diffusion bistable equations
We consider a fourth-order extension of the Allen-Cahn model with
mixed-diffusion and Navier boundary conditions. Using variational and
bifurcation methods, we prove results on existence, uniqueness, positivity,
stability, a priori estimates, and symmetry of solutions. As an application, we
construct a nontrivial bounded saddle solution in the plane.Comment: New version with minor change
On the calculation of finite-gap solutions of the KdV equation
A simple and general approach for calculating the elliptic finite-gap
solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is
based on the use of the finite-gap equations and the general representation of
these solutions in the form of rational functions of the elliptic Weierstrass
function. The calculation of initial elliptic finite-gap solutions is reduced
to the solution of the finite-band equations with respect to the parameters of
the representation. The time evolution of these solutions is described via the
dynamic equations of their poles, integrated with the help of the finite-gap
equations. The proposed approach is applied by calculating the elliptic 1-, 2-
and 3-gap solutions of the KdV equations
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