Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard-sphere molecules

The structure of normal shock waves is investigated on the basis of the standard Boltzmann equation for hard-sphere molecules. This fundamental nonlinear problem in rarefied gas dynamics is analyzed numerically by a newly developed finite-difference method, where the Boltzmann collision integral is computed directly without using the Monte Carlo method. The velocity distribution function, as well as the macroscopic quantities, is accurately obtained. The numerical results are compared with the Mott-Smith and the direct simulation Monte Carlo results in detail. The analytical solution for a weak shock wave based on the standard Boltzmann equation is also presented up to the second order of the shock strength together with its explicit numerical data for hard-sphere molecules.