Numerical evolutions of fields on the 2-sphere using a spectral method based on spin-weighted spherical harmonics

Many applications in science call for the numerical simulation of systems on manifolds with spherical topology. Through the use of integer spin-weighted spherical harmonics, we present a method which allows for the implementation of arbitrary tensorial evolution equations. Our method combines two nu...

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Bibliographic Details
Published in:Classical and quantum gravity 2014-04, Vol.31 (7), p.75019
Main Authors: Beyer, Florian, Daszuta, Boris, Frauendiener, Jörg, Whale, Ben
Format: Article
Language:English
Subjects:
Algorithms
Evolution
Mathematical analysis
Mathematical models
numerical relativity
Quantum gravity
Recursion
Spectra
spectral methods
Spherical harmonics
spin-weighted spherical harmonics
ð-formalism
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Summary:Many applications in science call for the numerical simulation of systems on manifolds with spherical topology. Through the use of integer spin-weighted spherical harmonics, we present a method which allows for the implementation of arbitrary tensorial evolution equations. Our method combines two numerical techniques that were originally developed with different applications in mind. The first is Huffenberger and Wandelt's spectral decomposition algorithm to perform the mapping from physical to spectral space. The second is the application of Luscombe and Luban's method, to convert numerically divergent linear recursions into stable nonlinear recursions, to the calculation of reduced Wigner d-functions. We give a detailed discussion of the theory and numerical implementation of our algorithm. The properties of our method are investigated by solving the scalar and vectorial advection equation on the sphere, as well as the 2 + 1 Maxwell equations on a deformed sphere.
ISSN:0264-9381
1361-6382
DOI:10.1088/0264-9381/31/7/075019