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A hyperbolic model of a viscous heat-conducting gas is presented, in which the Maxwell – Cattaneo approach is used to hyperbolize the equations, which provides finite wave propagation velocities. In the modified model, instead of the original Stokes and Fourier laws, their relaxation analogues were used and it is shown that when the relaxation times $\tau_\sigma^{}$ и $\tau_w^{}$ tend to The hyperbolized equations are reduced to zero to the classical Navier – Stokes system of non-hyperbolic type with infinite velocities of viscous and heat waves. It is noted that the hyperbolized system of equations of motion of a viscous heat-conducting gas considered in this paper is invariant not only with respect to the Galilean transformations, but also with respect to rotation, since the Yaumann derivative is used when differentiating the components of the viscous stress tensor in time. To integrate the equations of the model, the hybrid Godunov method (HGM) and the multidimensional nodal method of characteristics were used. The HGM is intended for the integration of hyperbolic systems in which there are equations written both in divergent form and not resulting in such (the original Godunov method is used only for systems of equations presented in divergent form). A linearized solver’s Riemann is used to calculate flow variables on the faces of adjacent cells. For divergent equations, a finitevolume approximation is applied, and for non-divergent equations, a finite-difference approximation is applied. To calculate a number of problems, we also used a non-conservative multidimensional nodal method of characteristics, which is based on splitting the original system of equations into a number of one-dimensional subsystems, for solving which a one-dimensional nodal method of characteristics was used. Using the described numerical methods, a number of one-dimensional problems on the decay of an arbitrary rupture are solved, and a two-dimensional flow of a viscous gas is calculated when a shock jump interacts with a rectangular step that is impermeable to gas.

A numerical method is proposed that approximates the equations of the dynamics of a weakly compressible viscous flow in the presence of a polymer component of the flow. The behavior of the flow under the influence of a static external periodic force in a periodic square cell is investigated. The methodology is based on a hybrid approach. The hydrodynamics of the flow is described by a system of Navier – Stokes equations and is numerically approximated by the linearized Godunov method. The polymer field is described by a system of equations for the vector of stretching of polymer molecules $\bf R$, which is numerically approximated by the Kurganov – Tedmor method. The choice of model relationships in the development of a numerical methodology and the selection of modeling parameters made it possible to qualitatively model and study the regime of elastic turbulence at low Reynolds $Re \sim 10^{-1}$. The polymer solution flow dynamics equations differ from the Newtonian fluid dynamics equations by the presence on the right side of the terms describing the forces acting on the polymer component part. The proportionality coefficient $A$ for these terms characterizes the backward influence degree of the polymers number on the flow. The article examines in detail how the flow and its characteristics change depending on the given coefficient. It is shown that with its growth, the flow becomes more chaotic. The flow energy spectra and the spectra of the polymers stretching field are constructed for different values of $A$. In the spectra, an inertial sub-range of the energy cascade is traced for the flow velocity with an indicator $k \sim −4$, for the cascade of polymer molecules stretches with an indicator $−1.6$.