A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

G. P. Ghiroldi, L. Gibelli*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed of hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying subsonic slightly rarefied gas flows.

Original languageEnglish
Pages (from-to)568-584
Number of pages17
JournalJournal of Computational Physics
Publication statusPublished - 1 Feb 2014


  • Boltzmann equation
  • Deterministic solution
  • Galerkin method
  • Gaussian quadrature
  • Half- and full-range Hermite polynomials


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