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Polar Spectral Scheme for the Spatially Homogeneous Boltzmann Equation

by E. Fonn and P. Grohs and R. Hiptmair

(Report number 2014-13)

Abstract
We consider the non-linear spatially homogeneous Boltzmann equation, and develop a polar spectral discretization in two dimensions based on Laguerre polynomials, which generalizes previous methods by Ender and Ender [A.Ya.~Ender and I.A.~Ender: Polynomial expansions for the isotropic Boltzmann equation and invariance of the collision integral with respect to the choice of basis functions. Physics of Fluids, 11:2720--2730, 1999] to the case of non-radially symmetric solutions. The method yields sparse approximation for long times and enjoys exponential convergence in the number of degrees of freedom for analytic solutions. A particular implementation exactly conserves mass, momentum and energy. Compared to the Fourier spectral discretization method, we need not truncate the collision operator and, thus, avoid aliasing errors.

Keywords: Boltzmann equartion; spectral Galerkin method; Laguerre polynomials.

@Techreport{FGH14_563,
  author = {E. Fonn and P. Grohs and R. Hiptmair},
  title = {Polar Spectral Scheme for the Spatially Homogeneous Boltzmann Equation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-13},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-13.pdf },
  year = {2014}
}

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