Direct methods for solving the Boltzmann equations: Comparisons with direct simulation Monte Carlo and possibilities

The possibilities of direct methods for solving the Boltzmann equation in comparison with direct simulation Monte Carlo are discussed. The general features of these different methods are considered, in particular, from the point of view of application of different variants of discretization in phase space. The advantages and disadvantages of both approaches are clarified. Comparative solutions of some simple problems are given. An important issue concerns anomalous heat transfer and validation of the effect by calculations based on these two methods. The solutions of the stationary one-dimensional heat transfer problem between two infinite plates with nonclassical nonequilibrium reflection from the surface are obtained; the anomalous heat transfer with a temperature gradient and a heat flux having the same sign is observed. One-dimensional and two-dimensional (in the square domain) problems with nonequilibrium “membranelike” boundary conditions are solved numerically; the anomalous heat transfer for all the considered cases is demonstrated.