The phase space PIC method for solving the Boltzmann equation. Part I.

Abstract

The paper describes a scheme for calculating the collision integral in high-accuracy computations of rarefied gas flows. Numerical test computations show that the proposed scheme allows finding numerical solutions of the Boltzmann equation in a wide range of flow velocities, including extremely slow flows. I. Introduction The theoretical fundamentals of the deterministic particle method for solving the Boltzmann equation, which is an extension of the Particle-in-Cell (PIC) method to the velocity space, were described in. 1 Results on homogeneous relaxation of a Maxwellian gas calculated by this method, which can be called the phase space PIC method, were also presented there. Preliminary calculations of one-dimensional rarefied gas flow showed that the deterministic particle method in this case imposes more severe requirements to collision integral calculation than in the case of homogeneous relaxation. Some procedures substantially improving the collision integral calculation accuracy are considered in this paper. This improvement is validated by the results for low-velocity flows calculated with a difference scheme of solving the Boltzmann equation. These procedures also ensure a significantly shorter time needed to calculate the collision integral. The next part of the paper will describe the results calculated by the deterministic particle method with the use of the collision integral calculation scheme presented in this paper.