A method for simulating the dynamics of rarefied gas based on lattice Boltzmann equations and the BGK equation

A hybrid method for solving boundary value problems for rarefied gas using the Bhatnagar–Gross–Krook (BGK) model and the lattice Boltzmann equation is studied. One-dimensional boundary value problems subject to membrane-type boundary conditions are considered. In strongly nonequilibrium regions, the BGK model should be used, and in the regions in which the distribution function is close to Maxwell’s one, the lattice Boltzmann equations can be used. On the region boundaries, a matching procedure should be performed; such a procedure is proposed in this paper. Note that the standard lattice Boltzmann models distort the distribution function on the region boundaries, but this distortion has no physical meaning. It is shown that, in order to correctly join the solutions on the region boundaries, the semi-moments of Maxwell’s distribution must be exactly reproduced. For this purpose, novel lattice models of the Boltzmann equation are constructed using the entropy method. Results of numerical computations of the temperature and density profiles for the Knudsen number equal to $0.1$ are presented, and the numerically obtained distribution function at the matching point is compared with the theoretical distribution function. Computation of the matching point is discussed.