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

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.