1,817 research outputs found
A Discretized Fourier Orthogonal Expansion in Orthogonal Polynomials on a Cylinder
We study the convergence of a discretized Fourier orthogonal expansion in
orthogonal polynomials on , where is the closed unit
disk in \RR^2. The discretized expansion uses a finite set of Radon
projections and provides an algorithm for reconstructing three dimensional
images in computed tomography. The Lebesgue constant is shown to be , and convergence is established for functions in
Universal Correlators from Geometry
Matrix model correlators show universal behaviour at short distances. We
provide a derivation for these universal correlators by inserting probe branes
in the underlying effective geometry. We generalize these results to study
correlators of branes and their universal behaviour in the Calabi-Yau crystals,
where we find a role for a generalized brane insertion.Comment: 25 pages, 2 figure
Spectral collocation methods
This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2
Spectral method for matching exterior and interior elliptic problems
A spectral method is described for solving coupled elliptic problems on an
interior and an exterior domain. The method is formulated and tested on the
two-dimensional interior Poisson and exterior Laplace problems, whose solutions
and their normal derivatives are required to be continuous across the
interface. A complete basis of homogeneous solutions for the interior and
exterior regions, corresponding to all possible Dirichlet boundary values at
the interface, are calculated in a preprocessing step. This basis is used to
construct the influence matrix which serves to transform the coupled boundary
conditions into conditions on the interior problem. Chebyshev approximations
are used to represent both the interior solutions and the boundary values. A
standard Chebyshev spectral method is used to calculate the interior solutions.
The exterior harmonic solutions are calculated as the convolution of the
free-space Green's function with a surface density; this surface density is
itself the solution to an integral equation which has an analytic solution when
the boundary values are given as a Chebyshev expansion. Properties of Chebyshev
approximations insure that the basis of exterior harmonic functions represents
the external near-boundary solutions uniformly. The method is tested by
calculating the electrostatic potential resulting from charge distributions in
a rectangle. The resulting influence matrix is well-conditioned and solutions
converge exponentially as the resolution is increased. The generalization of
this approach to three-dimensional problems is discussed, in particular the
magnetohydrodynamic equations in a finite cylindrical domain surrounded by a
vacuum
Gyroscopic polynomials
Gyroscopic alignment of a fluid occurs when flow structures align with the
rotation axis. This often gives rise to highly spatially anisotropic columnar
structures that in combination with complex domain boundaries pose challenges
for efficient numerical discretizations and computations. We define gyroscopic
polynomials to be three-dimensional polynomials expressed in a coordinate
system that conforms to rotational alignment. We remap the original domain with
radius-dependent boundaries onto a right cylindrical or annular domain to
create the computational domain in this coordinate system. We find the volume
element expressed in gyroscopic coordinates leads naturally to a hierarchy of
orthonormal bases. We build the bases out of Jacobi polynomials in the vertical
and generalized Jacobi polynomials in the radial. Because these coordinates
explicitly conform to flow structures found in rapidly rotating systems the
bases represent fields with a relatively small number of modes. We develop the
operator structure for one-dimensional semi-classical orthogonal polynomials as
a building block for differential operators in the full three-dimensional
cylindrical and annular domains. The differential operators of generalized
Jacobi polynomials generate a sparse linear system for discretization of
differential operators acting on the gyroscopic bases. This enables efficient
simulation of systems with strong gyroscopic alignment
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