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The application of a grid-characteristic method is proposed for numerical simulation of the movement of rolling stock along a railway track laid on an earth and a bridge. In the study, the railway track is presented using a dynamic system of equations of the theory of elasticity of hyperbolic type. The grid-characteristic method relies on the characteristic properties of the system under consideration and uses finite-difference schemes of high order of accuracy to obtain a space-time solution. The features of railway structures are considered by changing the boundary conditions and conditions at the contact borders. To simulate the areas of contact of wheel pairs of rolling stock with the rail – the "wheel-rail" system – a previously developed pressure boundary condition is applied, modified so as to consider multi-element rolling stock. As a result of computer simulation of the movement of rolling stock along the railway track of various types, full-wave stress distribution in the track structure is obtained, which makes it possible to predict dangerous sections of rolling stock movement.