14,068 research outputs found
Canonical Formulation of Gravitational Teleparallelism in 2+1 Dimensions in Schwinger's Time Gauge
We consider the most general class of teleparallel gravitational {}{}theories
quadratic in the torsion tensor, in three space-time dimensions, and carry out
a detailed investigation of its Hamiltonian formulation in Schwinger's time
gauge. This general class is given by a family of three-parameter theories. A
consistent implementation of the Legendre transform reduces the original theory
to a one-parameter family of theories. By calculating Poisson brackets we show
explicitly that the constraints of the theory constitute a first-class set.
Therefore the resulting theory is well defined with regard to time evolution.
The structure of the Hamiltonian theory rules out the existence of the
Newtonian limit.Comment: 17 pages, Latex file, no figures; a numerical coefficient has been
corrected and a different result is achieve
Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
Kolmogorov -widths and low-rank approximations are studied for families of
elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay
of the -widths can be controlled by that of the error achieved by best
-term approximations using polynomials in the parametric variable. However,
we prove that in certain relevant instances where the diffusion coefficients
are piecewise constant over a partition of the physical domain, the -widths
exhibit significantly faster decay. This, in turn, yields a theoretical
justification of the fast convergence of reduced basis or POD methods when
treating such parametric PDEs. Our results are confirmed by numerical
experiments, which also reveal the influence of the partition geometry on the
decay of the -widths.Comment: 27 pages, 6 figure
Shenfun -- automating the spectral Galerkin method
With the shenfun Python module (github.com/spectralDNS/shenfun) an effort is
made towards automating the implementation of the spectral Galerkin method for
simple tensor product domains, consisting of (currently) one non-periodic and
any number of periodic directions. The user interface to shenfun is
intentionally made very similar to FEniCS (fenicsproject.org). Partial
Differential Equations are represented through weak variational forms and
solved using efficient direct solvers where available. MPI decomposition is
achieved through the {mpi4py-fft} module (bitbucket.org/mpi4py/mpi4py-fft), and
all developed solver may, with no additional effort, be run on supercomputers
using thousands of processors. Complete solvers are shown for the linear
Poisson and biharmonic problems, as well as the nonlinear and time-dependent
Ginzburg-Landau equation.Comment: Presented at MekIT'17, the 9th National Conference on Computational
Mechanic
A pseudospectral matrix method for time-dependent tensor fields on a spherical shell
We construct a pseudospectral method for the solution of time-dependent,
non-linear partial differential equations on a three-dimensional spherical
shell. The problem we address is the treatment of tensor fields on the sphere.
As a test case we consider the evolution of a single black hole in numerical
general relativity. A natural strategy would be the expansion in tensor
spherical harmonics in spherical coordinates. Instead, we consider the simpler
and potentially more efficient possibility of a double Fourier expansion on the
sphere for tensors in Cartesian coordinates. As usual for the double Fourier
method, we employ a filter to address time-step limitations and certain
stability issues. We find that a tensor filter based on spin-weighted spherical
harmonics is successful, while two simplified, non-spin-weighted filters do not
lead to stable evolutions. The derivatives and the filter are implemented by
matrix multiplication for efficiency. A key technical point is the construction
of a matrix multiplication method for the spin-weighted spherical harmonic
filter. As example for the efficient parallelization of the double Fourier,
spin-weighted filter method we discuss an implementation on a GPU, which
achieves a speed-up of up to a factor of 20 compared to a single core CPU
implementation.Comment: 33 pages, 9 figure
4-Dimensional General Relativity from the instrinsic spatial geometry of SO(3) Yang--Mills theory
In this paper we derive 4-dimensional General Relativity from three
dimensions, using the intrinsic spatial geometry inherent in Yang--Mills theory
which has been exposed by previous authors as well as as some properties of the
Ashtekar variables. We provide various interesting relations, including the
fact that General Relativity can be written as a Yang--Mills theory where the
antiself-dual Weyl curvature replaces the Yang--Mills coupling constant. We
have generalized the results of some previous authors, covering Einsteins
spaces, to include more general spacetime geometries.Comment: 16 pages. Background material for revised journal articl
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