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Abstract. Macroscopic system of gas dynamic equations, differing from Navier -Stokes and quasi gas dynamic ones, is derived from a stochastic microscopic model of a hard sphere gas in a phase space. The model is diffusive in velocity space and valid for moderate Knudsen numbers. The main peculiarity of our derivation is more accurate velocity averaging due to analytical solving stochastic differential equations with respect to Wiener measure which describe our original meso model. It is shown at an example of a shock wave front structure that our approach leads to larger than Navier -Stokes front widening that corresponds to reality. The numerical solution is performed by a (well suited to high performance computer applications) special "discontinuous" particle method.3121

SUMMARYHeat and mass transfer during crystallization in a square cavity, involving natural laminary convection of liquid metal with temperature-dependent viscosity, are investigated numerically on a grid with 5 1 x 51 nodes. A brief survey of existing techniques for convection/diffusion phase change problems is given. A fixed grid numerical methodology, based on the fictitious regions method, is used for solving the 2-D Stefan problem coupled with the Navier-Stokes equations in the Boussinesq approximation. The viscosity variation is modelled by an exponential form, Y/YO = exp ( -a(T-1)). The influence of the Grashof number and of the parameter o on heat and mass transfer are investigated.