The velocity distribution function in an infinitely strong shock wave

The structure of an infinitely strong shock wave (i.e., a shock wave with infinitely large upstream Mach number) is investigated on the basis of the Boltzmann equation for hard-sphere molecules. The velocity distribution function is expressed as a sum of a multiple of the Dirac delta function, centered at the upstream bulk velocity, and the remainder. The equation for the latter, which contains the linear collision term linearized around the delta function and the nonlinear collision term, is solved numerically by an accurate finite-difference method after the nonlinear collision term is replaced by the BGK collision model. The result not only confirms the singularity in the remainder observed in the previous Monte Carlo simulation [Cercignani et al., Phys. Fluids 11, 2757 (1999)] but also provides more detailed information about the structure of the singularity.