The problems connected with realization of discrete trajectory method of sample particles and some approaches to numerical solution of Cauchy problem for the equations describing the motion of a sample particle and its heat-mass transfer in a fluid or gas flow are considered. Several finite difference schemes taking into account the features of motion of small and large particles as well as finite difference schemes of semianalytic integration for some particular problems are developed. The equations for particle motion in an arbitrary curvilinear frame of reference are given and peculiarities of their integration are discussed.

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