A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization
Provides a new numerical approach for solving the Boltzmann equation, relevant for researchers in rarefied gas dynamics and computational fluid dynamics.
The paper proposes a deterministic Galerkin method for the Boltzmann equation using Gauss-Hermite collocation basis functions, achieving spectral convergence in velocity space. It demonstrates the method on homogeneous relaxation and 2D cavity flow, showing effectiveness for low-speed rarefied gas flows.
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 by 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 low-speed slightly rarefied gas flows.