A spatial discontinuous Galerkin method with rescaled velocities for the Boltzmann equation

Abstract

In this paper we present a numerical method for the Boltzmann equation. It is a spectral discretization in the velocity and a discontinuous Galerkin discretization in physical space. To obtain uniform approximation properties in the mach number, we shift the velocity by the (smoothed) bulk velocity and scale it by the (smoothed) temperature, both extracted from the density distribution. The velocity trial functions are polynomials multiplied by a Maxwellian. Consequently, an expansion with a low number of trial functions already yields satisfying approximation quality for nearly equilibrated solutions. By the polynomial test space, density, velocity and energy are conserved on the discrete level. Different from moment methods, we stabilize the free flow operator in phase space with upwind fluxes. Several numerical results are presented to justify our approach.