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

TL;DR

This paper introduces a novel spectral discontinuous Galerkin method with velocity rescaling for the Boltzmann equation, improving approximation uniformity across Mach numbers and conserving key physical quantities.

Contribution

It develops a new velocity rescaling technique combined with spectral and discontinuous Galerkin discretizations, enhancing accuracy and stability for the Boltzmann equation.

Findings

Achieves accurate solutions with few trial functions near equilibrium

Conserves density, velocity, and energy discretely

Demonstrates numerical stability and effectiveness

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.