5,802 research outputs found
Chebyshev, Legendre, Hermite and other orthonormal polynomials in D-dimensions
We propose a general method to construct symmetric tensor polynomials in the
D-dimensional Euclidean space which are orthonormal under a general weight. The
D-dimensional Hermite polynomials are a particular case of the present ones for
the case of a gaussian weight. Hence we obtain generalizations of the Legendre
and of the Chebyshev polynomials in D dimensions that reduce to the respective
well-known orthonormal polynomials in D=1 dimensions. We also obtain new
D-dimensional polynomials orthonormal under other weights, such as the
Fermi-Dirac, Bose-Einstein, Graphene equilibrium distribution functions and the
Yukawa potential. We calculate the series expansion of an arbitrary function in
terms of the new polynomials up to the fourth order and define orthonormal
multipoles. The explicit orthonormalization of the polynomials up to the fifth
order (N from 0 to 4) reveals an increasing number of orthonormalization
equations that matches exactly the number of polynomial coefficients indication
the correctness of the present procedure.Comment: 20 page
Affine symmetry in mechanics of collective and internal modes. Part I. Classical models
Discussed is a model of collective and internal degrees of freedom with
kinematics based on affine group and its subgroups. The main novelty in
comparison with the previous attempts of this kind is that it is not only
kinematics but also dynamics that is affinely-invariant. The relationship with
the dynamics of integrable one-dimensional lattices is discussed. It is shown
that affinely-invariant geodetic models may encode the dynamics of something
like elastic vibrations
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
Kadath: a spectral solver for theoretical physics
Kadath is a library that implements spectral methods in a very modular
manner. It is designed to solve a wide class of problems that arise in the
context of theoretical physics. Several types of coordinates are implemented
and additional geometries can be easily encoded. Partial differential equations
of various types are discretized by means of spectral methods. The resulting
system is solved using a Newton-Raphson iteration. Doing so, Kadath is able to
deal with strongly non-linear situations. The algorithms are validated by
applying the library to four different problems of contemporary physics, in the
fields of gauge field theory and general relativityComment: Accepted to Journal of Computational Physic
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