Particle Simulation of the Boltzmann Equation

Abstract

Validity analysis, existence and uniqueness theorems, and qualitative results on the behavior of the solutions are certainly central to the understanding of rarefied gases. For real physical situations, however, like the flow pattern around an object that moves inside a rarefied gas, we need methods to actually calculate or approximate solutions of the Boltzmann equation. For most situations, it is hopeless to even look for explicit solutions of the Boltzmann equation. On the other hand, the five-dimensional integral in the collision operator makes numerical approximations a difficult topic. Specifically, recall that

$$ Q\left( {f,f} \right)\, = \,\int_{\Re ^3 } {\int_{S^2 } {\left| {n\, \cdot \,\left( {{\rm \xi }\, - \,{\rm \xi }_{\rm *} } \right)} \right|\,\left\{ {f'\,f'_* \, - \,f\,f_* } \right\}dnd{\rm \xi }_{\rm *.} } } $$

Suppose we want to approximate the collision integral by a quadrature formula that requires the evaluation of the integrand at a number of points. Obviously, the integrand must decay fast enough at infinity to give us reasonable accuracy with a finite number of evaluation points.