93,016 research outputs found
An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations
We give an example of a well posed, finite energy, 2D incompressible active
scalar equation with the same scaling as the surface quasi-geostrophic equation
and prove that it can produce finite time singularities. In spite of its
simplicity, this seems to be the first such example. Further, we construct
explicit solutions of the 2D Boussinesq equations whose gradients grow
exponentially in time for all time. In addition, we introduce a variant of the
2D Boussinesq equations which is perhaps a more faithful companion of the 3D
axisymmetric Euler equations than the usual 2D Boussinesq equations.Comment: 9 pages; simplified a solution formula in section 4 and added a
sentence on the time growth rate in the solutio
Centrifugal deformations of the gravitational kink
The Kaluza-Klein reduction of 4d conformally flat spacetimes is reconsidered.
The corresponding 3d equations are shown to be equivalent to 2d gravitational
kink equations augmented by a centrifugal term. For space-like gauge fields and
non-trivial values of the centrifugal term the gravitational kink solutions
describe a spacetime that is divided in two disconnected regions.Comment: 8 pages, no figure
2D and 3D reconstructions in acousto-electric tomography
We propose and test stable algorithms for the reconstruction of the internal
conductivity of a biological object using acousto-electric measurements.
Namely, the conventional impedance tomography scheme is supplemented by
scanning the object with acoustic waves that slightly perturb the conductivity
and cause the change in the electric potential measured on the boundary of the
object. These perturbations of the potential are then used as the data for the
reconstruction of the conductivity. The present method does not rely on
"perfectly focused" acoustic beams. Instead, more realistic propagating
spherical fronts are utilized, and then the measurements that would correspond
to perfect focusing are synthesized. In other words, we use \emph{synthetic
focusing}. Numerical experiments with simulated data show that our techniques
produce high quality images, both in 2D and 3D, and that they remain accurate
in the presence of high-level noise in the data. Local uniqueness and stability
for the problem also hold
Self-consistent calculation of the coupling constant in the Gross-Pitaevskii equation
A method is proposed for a self-consistent evaluation of the coupling
constant in the Gross-Pitaevskii equation without involving a pseudopotential
replacement. A renormalization of the coupling constant occurs due to medium
effects and the trapping potential, e.g. in quasi-1D or quasi-2D systems. It is
shown that a simplified version of the Hartree-Fock-Bogoliubov approximation
leads to a variational problem for both the condensate and a two-body wave
function describing the behaviour of a pair of bosons in the Bose-Einstein
condensate. The resulting coupled equations are free of unphysical divergences.
Particular cases of this scheme that admit analytical estimations are
considered and compared to the literature. In addition to the well-known cases
of low-dimensional trapping, cross-over regimes can be studied. The values of
the kinetic, interaction, external, and release energies in low dimensions are
also evaluated and contributions due to short-range correlations are found to
be substantial.Comment: 15 pages, ReVTEX, no figure
Mitigation of numerical Cerenkov radiation and instability using a hybrid finite difference-FFT Maxwell solver and a local charge conserving current deposit
A hybrid Maxwell solver for fully relativistic and electromagnetic (EM)
particle-in-cell (PIC) codes is described. In this solver, the EM fields are
solved in space by performing an FFT in one direction, while using finite
difference operators in the other direction(s). This solver eliminates the
numerical Cerenkov radiation for particles moving in the preferred direction.
Moreover, the numerical Cerenkov instability (NCI) induced by the
relativistically drifting plasma and beam can be eliminated using this hybrid
solver by applying strategies that are similar to those recently developed for
pure FFT solvers. A current correction is applied for the charge conserving
current deposit to correctly account for the EM calculation in hybrid Yee-FFT
solver. A theoretical analysis of the dispersion properties in vacuum and in a
drifting plasma for the hybrid solver is presented, and compared with PIC
simulations with good agreement obtained. This hybrid solver is applied to both
2D and 3D Cartesian and quasi-3D (in which the fields and current are
decomposed into azimuthal harmonics) geometries. Illustrative results for laser
wakefield accelerator simulation in a Lorentz boosted frame using the hybrid
solver in the 2D Cartesian geometry are presented, and compared against results
from 2D UPIC-EMMA simulation which uses a pure spectral Maxwell solver, and
from OSIRIS 2D lab frame simulation using the standard Yee solver. Very good
agreement is obtained which demonstrates the feasibility of using the hybrid
solver for high fidelity simulation of relativistically drifting plasma with no
evidence of the numerical Cerenkov instability
Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers
This paper introduces a new sweeping preconditioner for the iterative
solution of the variable coefficient Helmholtz equation in two and three
dimensions. The algorithms follow the general structure of constructing an
approximate factorization by eliminating the unknowns layer by layer
starting from an absorbing layer or boundary condition. The central idea of
this paper is to approximate the Schur complement matrices of the factorization
using moving perfectly matched layers (PMLs) introduced in the interior of the
domain. Applying each Schur complement matrix is equivalent to solving a
quasi-1D problem with a banded LU factorization in the 2D case and to solving a
quasi-2D problem with a multifrontal method in the 3D case. The resulting
preconditioner has linear application cost and the preconditioned iterative
solver converges in a number of iterations that is essentially indefinite of
the number of unknowns or the frequency. Numerical results are presented in
both two and three dimensions to demonstrate the efficiency of this new
preconditioner.Comment: 25 page
A method for reconstructing the PDF of a 3D turbulent density field from 2D observations
We introduce a method for calculating the probability density function (PDF)
of a turbulent density field in three dimensions using only information
contained in the projected two-dimensional column density field. We test the
method by applying it to numerical simulations of hydrodynamic and
magnetohydrodynamic turbulence in molecular clouds. To a good approximation,
the PDF of log(normalised column density) is a compressed, shifted version of
the PDF of log(normalised density). The degree of compression can be determined
observationally from the column density power spectrum, under the assumption of
statistical isotropy of the turbulence.Comment: 5 pages, 2 figures, accepted for publication in MNRAS Letter
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