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Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation

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

(Report number 2012-28)

Abstract
The nonlinear spatially homogeneous integro-differential Boltzmann equation is a uniquely challenging task for numerical solvers due to the difficulty of efficiently computing the collision operator. A popular method is to expand the solution in Fourier modes and to truncate the collision operator. We present an approach based on the hyperbolic cross, whereby the performance can be greatly enhanced in some situations, as well as an offset method, which takes advantage of the known equilibrium solutions. Some numerical experiments are presented in two dimensions with Maxwellian kernels. Some error estimates are also given, where it is shown that under reasonable assumptions, the numerical solution converges to the analytical.

Keywords:

@Techreport{FGH12_471,
  author = {E. Fonn and P. Grohs and R. Hiptmair},
  title = {Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-28.pdf },
  year = {2012}
}

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