A high order space-momentum discontinuous Galerkin method for the Boltzmann equation

Defect correction methods, classic and new (in Ukrainian) A posteriori error analysis of hp-FEM for singularly perturbed problems Defect-based local error estimators for high-order splitting methods involving three linear operators 21/2014 A. Jüngel and N. Zamponi Boundedness of weak solutions to cross-diusion systems from population dynamics 20/2014 A. Jüngel The boundedness-by-entropy principle for cross-diusion systems Abstract. In this paper we present a Discontinuous Galerkin method for the Boltzmann equation. The distribution function f is approximated by a shifted Maxwellian times a polynomial in space and momentum, while the test functions are chosen as polynomials. The first property leads to consistency with the Euler limit, while the second property ensures conservation of mass, momentum and energy. The focus of the paper is on efficient algorithms for the Boltzmann collision operator. We transform between nodal, hierarchical and polar polynomial bases to reduce the inner integral operator to diagonal form. 1. Introduction. In this paper we present a numerical scheme for the Boltz-mann equation. We will concentrate on the development of an efficient realization of the collision operator. The numerical solution of the Boltzmann equation is a huge challenge, due to the high dimensionality (3 spatial + 3 momentum + 1 time variable) and the five fold integral defining the collision operator. Moreover, since the collision operator is closely connected to macroscopic conservation properties of the equation, it's integration has to be carried out with care. Many authors focus on stochastic approaches such as Monte Carlo simulation. These were exploited by Bird [1] and Nanbu [2]. In [3] it was shown – for the space homogeneous situation – that the computational effort for both methods can be bounded by O(N), where N is the number of simulated particles. Typically these methods have to deal with stochastic fluctuation. In deterministic approaches, the complexity of the high dimensional integration for the collision operator is a real challenge. Fourier transformation of the collision integral yields – for certain collision kernels – a significant simplification and moreover offers the possibility to use fast Fourier transform. For Maxwellian gases, the Fourier representation of the collision operator has a relatively simple form [4]. This specific representation was used in [5] to construct an efficient difference scheme. In [6] they extended their ideas to the case of hard sphere interaction. Another attempt using Fourier techniques was made in [7] were a Fourier series expansion of the solution function was combined …