A numerical algorithm based on a shock-capturing scheme on adaptive moving meshes is described. Three-dimensional dynamical adaptive meshes are generated as the result of solving a variational problem. The governing equations are solved in a moving coordinate system therefore the method does not require any interpolation of the solution from one mesh onto another one. The developed algorithm can efficiently be used for the simulation of non-stationary supersonic flows in the atmosphere. As an example, it is applied to the study of the Tunguska meteorite hypersonic impact. For the first time this problem is modelled from the entrance moment up to the interaction with the Earth atmosphere in a three-dimensional formulation. A reasonable correspondence with the data of the observations is obtained. The implementation of the developed numerical algorithm reduces the computational time by the factor of four.
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