A method of extended cells for the formulation of boundary conditions in the numerical integration of the Euler equations in domains with curvilinear boundaries for the cases of one-dimensional and two-dimensional compressible gas flows has been proposed in this study. The proposed method is based on the use of the explicit Godunov type finite volume method on a regular rectangular Cartesian grid. The essence of the extended cell method is that when integrating the basic equations of gas dynamics in a fractional cell, numerical fluxes are calculated through the sides of the new extended cell. This new cell is constructed tangentially to the curved boundary and has a size no less than the cell size in the regular domain. Parameters inside the new cell are calculated as mean integral values over the area of the neighboring regular cells included in it. In this case, when choosing a time step in accordance with the Courant condition, the stability of the method in the main computational domain is preserved both when integrating fractional cells and when integrating regular cells. Thanks to this approach, the proposed method has low requirements for computing resources, the ability to generalize for three-dimensional space and increase the order of accuracy without major modifications of the algorithm. To test the proposed method, solutions were obtained for the generally accepted test problems of gas dynamics: normal and double Mach reflection of a shock wave from a plane wall. The choice of the time step was made in accordance with the Courant condition in regular finite volumes. The results obtained have made it possible to assess the convergence of the proposed method and their comparison with the results of calculations using other methods have shown a good quantitative and qualitative agreement.

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