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Currently, different nonlinear numerical schemes of the spatial approximation are used in numerical simulation of boundary value problems for hyperbolic systems of partial differential equations (e. g. gas dynamics equations, MHD, deformable rigid body, etc.). This is due to the need to improve the order of accuracy and perform simulation of discontinuous solutions that are often occurring in such systems. The need for non-linear schemes is followed from the barrier theorem of S. K. Godunov that states the impossibility of constructing a linear scheme for monotone approximation of such equations with approximation order two or greater. One of the most accurate non-linear type schemes are ENO (essentially non oscillating) and their modifications, including WENO (weighted, essentially non oscillating) scemes. The last received the most widespread, since the same stencil width has a higher order of approximation than the ENO scheme. The benefit of ENO and WENO schemes is the ability to maintain a high-order approximation to the areas of non-monotonic solutions. The main difficulty of the analysis of such schemes comes from the fact that they themselves are nonlinear and are used to approximate the nonlinear equations. In particular, the linear stability condition was obtained earlier only for WENO5 scheme (fifth-order approximation on smooth solutions) and it is a numerical one. In this paper we consider the problem of construction and stability for WENO5, WENO7, WENO9, WENO11, and WENO13 finite volume schemes for the Hopf equation. In the first part of this article we discuss WENO methods in general, and give the explicit expressions for the coefficients of the polynomial weights and linear combinations required to build these schemes. We prove a series of assertions that can make conclusions about the order of approximation depending on the type of local solutions. Stability analysis is carried out on the basis of the principle of frozen coefficients. The cases of a smooth and discontinuous behavior of solutions in the field of linearization with frozen coefficients on the faces of the final volume and spectra of the schemes are analyzed for these cases. We prove the linear stability conditions for a variety of Runge-Kutta methods applied to WENO schemes. As a result, our research provides guidance on choosing the best possible stability parameter, which has the smallest effect on the nonlinear properties of the schemes. The convergence of the schemes is followed from the analysis.

For the production of purified final product in chemical engineering used the process of desublimation. For this purpose, the tank is cooled by liquid nitrogen or cold air. The mixture of gases flows inside the tank and is cooled to the condensation or desublimation temperature some components of the gas mixture. The condensed components are deposited on the walls of the tank. The article presents a mathematical model to calculate the cooling air tanks for desublimation of vapours. A mathematical model based on equations of gas dynamics and describes the movement of cooled air in the duct and the heat exchanger with heat exchange and friction. The heat of the phase transition is taken into account in the boundary condition for the heat equation by setting the heat flux. Heat transfer in the walls of the pipe and in the tank wall is described by the nonstationary heat conduction equations. The solution of the system of equations is carried out numerically. The equations of gas dynamics are solved by the method of S. K. Godunov. The heat equation are solved by an implicit finite difference scheme. The article presents the results of calculations of the cooling of two successively installed tanks. The initial temperature of the tanks is equal to 298 K. Cold air flows through the tubing, through the heat exchanger of the first tank, then through conduit to the heat exchanger second tank. During the 20 minutes of tank cool down to operating temperature. The temperature of the walls of the tanks differs from the air temperature not more than 1 degree. The flow of cooling air allows to maintain constant temperature of the walls of the tank in the process of desublimation components from a gas mixture. The results of analytical evaluation of the time of cooling tank and temperature difference between the tank walls and air with the vapor desublimation. Analytical assessment is based on determining the time of heat relaxation temperature of the tank walls. The results of evaluations are satisfactorily coincide with the results of calculations by the present model. The proposed approach allows calculating the cooling tanks with a flow of cold air supplied via the pipeline system.

The conjugate problem of disk movement into gas-filled volume of the spring-type safety valve is solved. The questions of determining the physically correct value of the disk initial lift are considered. The review of existing approaches and methods for solving of such type problems is conducted. The formulation of the problem about the valve actuation when the vessel pressure rises and the mathematical model of the actuation processes are given. A special attention to the binding of physical subtasks is paid. Used methods, numerical schemes and algorithms are described. The mathematical modeling is performed on basе the fundamental system of differential equations for viscous gas movement with the equation for displacement of disk valve. The solution of this problem in the axe symmetric statement is carried out numerically using the finite volume method. The results obtained by the viscous and inviscid models are compared. In an inviscid formulation this problem is solved using the Godunov scheme, and in a viscous formulation is solved using the Kurganov – Tadmor method. The dependence of the disk displacement on time was obtained and compared with the experimental data. The pressure distribution on the disk surface, velocity profiles in the cross sections of the gap for different disk heights are given. It is shown that a value of initial drive lift it does not affect on the gas flow and valve movement part dynamic. It can significantly reduce the calculation time of the full cycle of valve work. Immediate isotahs for various elevations of the disk are presented. The comparison of jet flow over critical section is given. The data carried out by two numerical experiments are well correlated with each other. So, the inviscid model can be applied to the numerical modeling of the safety valve dynamic.

The paper provides a comparative analysis of the results obtained by various approaches to modeling the process of artillery shot. In this connection, the main problem of internal ballistics and its particular case of the Lagrange problem are formulated in averaged parameters, where, within the framework of the assumptions of the thermodynamic approach, the distribution of pressure and gas velocity over the projectile space for a channel of variable cross section is taken into account for the first time. The statement of the Lagrange problem is also presented in the framework of the gas-dynamic approach, taking into account the spatial (one-dimensional and two-dimensional axisymmetric) changes in the characteristics of the ballistic process. The control volume method is used to numerically solve the system of Euler gas-dynamic equations. Gas parameters at the boundaries of control volumes are determined using a selfsimilar solution to the Riemann problem. Based on the Godunov method, a modification of the Osher scheme is proposed, which allows to implement a numerical calculation algorithm with a second order of accuracy in coordinate and time. The solutions obtained in the framework of the thermodynamic and gas-dynamic approaches are compared for various loading parameters. The effect of projectile mass and chamber broadening on the distribution of the ballistic parameters of the shot and the dynamics of the projectile motion was studied. It is shown that the thermodynamic approach, in comparison with the gas-dynamic approach, leads to a systematic overestimation of the estimated muzzle velocity of the projectile in the entire range of parameters studied, while the difference in muzzle velocity can reach 35%. At the same time, the discrepancy between the results obtained in the framework of one-dimensional and two-dimensional gas-dynamic models of the shot in the same range of change in parameters is not more than 1.3%.

A spatial gas-dynamic formulation of the backpressure problem is given, which describes the change in pressure in front of an accelerating projectile as it moves along the barrel channel. It is shown that accounting the projectile’s front, considered in the two-dimensional axisymmetric formulation of the problem, leads to a significant difference in the pressure fields behind the front of the shock wave, compared with the solution in the framework of the onedimensional formulation of the problem, where the projectile’s front is not possible to account. It is concluded that this can significantly affect the results of modeling ballistics of a shot at high shooting velocities.