Solving the Boltzmann equation in the case of passing to the hydrodynamic flow regime

As the hydrodynamic regime is approached, the gas flow is usually accompanied by the formation of narrow highly nonequilibrium zones (Knudsen layers) with the characteristic size on the order of the mean free path λ for molecules. The structure of these zones is determined by fast kinetic processes. In unsteady flows, an initial layer with the time scale on the order of the mean free time τ0 = λ/vT (here, vT is the molecular thermal velocity) occurs as well. In the macroscopic scale l0 @ λ, the flow parameters vary smoothly beyond these zones [1]. From a computational standpoint, solving the Boltzmann equation with steps hx < λ and τ < τ0 is inefficient everywhere over the calculation domain. Moreover, it can result in the early cessation of the iterative process as soon as the error of the numerical method becomes equal to the small difference of two successive approximations. When passing to the macroscopic steps λ ! hx < l0, τ0 ! τ < t0 , where t0 = l0/vT , the problem of a large factor standing ahead the collision integral arises [2]. In terms of the dimensionless variables t = t*/t0, x = x*/l0, ξ = ξ*/vT (here, the asterisk denotes dimensional variables), the Boltzmann equation takes the form