Hyperbolic Cross Approximation for the Spatially Homogeneous Boltzmann Equation

Eivind Fonn, Philipp Grohs, Ralf Hiptmair

Veröffentlichungen: Beitrag in FachzeitschriftArtikelPeer Reviewed


We consider the nonlinear spatially homogeneous Boltzmann equation and its Fourier spectral discretization in velocity space involving periodic continuation of the density and a truncation of the collision operator. We allow discretization based on arbitrary sets of active Fourier modes with particular emphasis on the family of so-called hyperbolic cross approximations. We also discuss an offset method that takes advantage of the known equilibrium solutions. Extending the analysis in Filbet & Mouhot (2011, Analysis of spectral methods for the homogeneous Boltzmann equation. Trans. Amer. Math. Soc., 363, 1947-1980), we establish consistency estimates for the discrete collision operators and stability of the semidiscrete evolution. Under an assumption of Gaussian-like decay of the discrete solution, we give a detailed bound for H s-Sobolev norms of the error due to Fourier spectral discretization.

Seiten (von - bis)1533-1567
FachzeitschriftIMA Journal of Numerical Analysis
PublikationsstatusVeröffentlicht - Okt. 2015

ÖFOS 2012

  • 101014 Numerische Mathematik
  • 101002 Analysis