A discontinuous Galerkin fast spectral method for the multi-species Boltzmann equation

A Discontinuous Galerkin Fast Spectral Method for Multi-Species Full Boltzmann on Streaming Multi-Processors

“…The non-dimensional Boltzmann equation for multi-species, monoatomic gas without external forces can be written as (cf. [7])…”

“…In the present work, we consider the variable soft sphere (VSS) [9] scattering model. It is worth emphasizing that although the VSS collision kernel is adopted in the present work for easy comparison with DSMC solutions, the fast spectral method we use for the collision operator applies straightforwardly to general collision kernels (see [4,6,7]). For Variable Soft-Sphere model [2] in particular, the non-dimensional collision-kernel B (pq) , and the Knudsen number Kn pq are given as…”

“…Given distribution functions f (p) and f (q) of species p and q, dependent only on the velocity coordinate c: discretized uniformly using N 3 points, the method produces Q (pq) (f (p) , f (q) ) at the same grid with O(MN ρ N 3 log N ) complexity, where N ρ ∼ O(N ) is the number of Gauss-Legendre quadrature/discretization points in the radial direction needed for low-rank decomposition , M ≪ N 2 is the number of discretization points on the sphere. The steps (based on [7]) for evaluating Q (pq) can be summarized as:…”

“…The main difficulty of numerically solving the full Boltzmann equation lies in its complicated collision term. Recently, a fast Fourier spectral method for the multi-species Boltzmann collision operator was introduced in [7]. The complexity for a single evaluation of the collision operator is reduced from O(N 6 ) (direct calculation) to O(MN ρ N 3 log N ) (based on a low-rank decomposition strategy), where N is the number of discretization points in each velocity dimension, N ρ ∼ O(N ) is the number of discretization points in the radial direction needed for low-rank decomposition, and M ≪ N 2 is the number of discretization points on the sphere.…”

“…The complexity for a single evaluation of the collision operator is reduced from O(N 6 ) (direct calculation) to O(MN ρ N 3 log N ) (based on a low-rank decomposition strategy), where N is the number of discretization points in each velocity dimension, N ρ ∼ O(N ) is the number of discretization points in the radial direction needed for low-rank decomposition, and M ≪ N 2 is the number of discretization points on the sphere. Based on [6], a discontinuous Galerkin fast spectral (DGFS) method was also proposed in [7] for solving the full multi-species Boltzmann equation. DGFS can produce high order spatially and temporally accurate solutions for low-speed and unsteady flows in micro-systems, and is amenable to excellent nearly-linear scaling characteristics on massively parallel architectures.…”