An approach for calculating non-stationary thermo-gasdynamic processes in model chambers of solid rocket engines is described. A three-dimensional mathematical model of viscous multicomponent gas mixture is an extended system of Navier-Stokes equations with the component diffusion equations. The computational model and the computer code are based on the developed original mathematical methodology that combines the advantages of splitting method by physical processes, robustness of Godunov's scheme for the convective stage and efficiency of explicit iterative Chebyshev’s scheme for diffusion stage. The code is written in C++ and uses a hybrid three-level parallel structure, including the use of MPI, OpenMP and CUDA technologies.