**WINFRIED KOLLER**

*P1 approximation of the nonlinear semi-continuous Boltzmann equation*

**Pages:** 159-181

**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 P_{1}-*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 P_{1}-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.

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