P1 approximation of the nonlinear semi-continuous Boltzmann equation
Received: 16 August 1999
Mathematics Subject Classification: 82C40
Abstract: The aim of this paper is to provide and discuss a full velocity discretization of the nonlinear Boltzmann equation governing the evolution of a rarefied single-atomic gas. Based on a semi-continuous version of the Boltzmann equation, a truncated expansion of the angular dependence of the distribution function in terms of spherical harmonics is investigated. For Maxwellian molecules this procedure yields a coupled set of nonlinear partial differential equations, denoted as P1-multigroup approximation. As a consequence of a detailed balance relation, the conservation of particle number, total momentum and energy is established. A Fourier expansion in real space is carried out. In spite of the nonlinearity of the collision term, this further approximation yields a set of coupled ODE consistent with all three conservation equations. The obtained P1-multigroup equations are applied to one-dimensional relaxation problems. This ansatz proves efficient for the study of stationary acoustic waves occurring, for instance, in degenerate four wave mixing (DFWM) experiments.