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We study the computational properties of a parametric class of finite-volume schemes with customizable dissipative properties with splitting by physical processes into Lagrangian, Eulerian, and the final stages (the hybrid large-particle method). The method has a second-order approximation in space and time on smooth solutions. The regularization of a numerical solution at the Lagrangian stage is performed by nonlinear correction of artificial viscosity. Regardless of the grid resolution, the artificial viscosity value tends to zero outside the zone of discontinuities and extremes in the solution. At Eulerian and final stages, primitive variables (density, velocity, and total energy) are first reconstructed by an additive combination of upwind and central approximations weighted by a flux limiter. Then numerical divergent fluxes are formed from them. In this case, discrete analogs of conservation laws are performed.

The analysis of dissipative properties of the method using known viscosity and flow limiters, as well as their linear combination, is performed. The resolution of the scheme and the quality of numerical solutions are demonstrated by examples of two-dimensional benchmarks: a gas flow around the step with Mach numbers 3, 10 and 20, the double Mach reflection of a strong shock wave, and the implosion problem. The influence of the scheme viscosity of the method on the numerical reproduction of a gases interface instability is studied. It is found that a decrease of the dissipation level in the implosion problem leads to the symmetric solution destruction and formation of a chaotic instability on the contact surface.

Numerical solutions are compared with the results of other authors obtained using higher-order approximation schemes: CABARET, HLLC (Harten Lax van Leer Contact), CFLFh (CFLF hybrid scheme), JT (centered scheme with limiter by Jiang and Tadmor), PPM (Piecewise Parabolic Method), WENO5 (weighted essentially non-oscillatory scheme), RKGD (Runge –Kutta Discontinuous Galerkin), hybrid weighted nonlinear schemes CCSSR-HW4 and CCSSR-HW6. The advantages of the hybrid large-particle method include extended possibilities for solving hyperbolic and mixed types of problems, a good ratio of dissipative and dispersive properties, a combination of algorithmic simplicity and high resolution in problems with complex shock-wave structure, both instability and vortex formation at interfaces.

For a non-homogeneous model transport equation with source terms, the stability analysis of a linear hybrid scheme (a combination of upwind and central approximations) is performed. Stability conditions are obtained that depend on the hybridity parameter, the source intensity factor (the product of intensity per time step), and the weight coefficient of the linear combination of source power on the lower- and upper-time layer. In a nonlinear case for the non-equilibrium by velocities and temperatures equations of gas suspension motion, the linear stability analysis was confirmed by calculation. It is established that the maximum permissible Courant number of the hybrid large-particle method of the second order of accuracy in space and time with an implicit account of friction and heat exchange between gas and particles does not depend on the intensity factor of interface interactions, the grid spacing and the relaxation times of phases (K-stability). In the traditional case of an explicit method for calculating the source terms, when a dimensionless intensity factor greater than 10, there is a catastrophic (by several orders of magnitude) decrease in the maximum permissible Courant number, in which the calculated time step becomes unacceptably small.

On the basic ratios of Riemann’s problem in the equilibrium heterogeneous medium, we obtained an asymptotically exact self-similar solution of the problem of interaction of a shock wave with a layer of gas-suspension to which converge the numerical solution of two-velocity two-temperature dynamics of gassuspension when reducing the size of dispersed particles.

The dynamics of the shock wave in gas and its interaction with a limited gas suspension layer for different sizes of dispersed particles: 0.1, 2, and 20 ìm were studied. The problem is characterized by two discontinuities decay: reflected and refracted shock waves at the left boundary of the layer, reflected rarefaction wave, and a past shock wave at the right contact edge. The influence of relaxation processes (dimensionless phase relaxation times) to the flow of a gas suspension is discussed. For small particles, the times of equalization of the velocities and temperatures of the phases are small, and the relaxation zones are sub-grid. The numerical solution at characteristic points converges with relative accuracy $O \, (10^{-4})$ to self-similar solutions.

We present a physico-mathematical statement of coupled geometrical and gas dynamics problem of intrachamber processes simulation and calculation of main internal ballistics characteristics of solid rocket motors in axisymmetric approximation. Method and numerical algorithm of solving the problem are described in this paper. We track the propellant burning surface using the level set method. This method allows us to implicitly represent the surface on a fixed Cartesian grid as zero-level of some function. Two-dimensional gas-dynamics equations describe a flow of combustion products in a solid rocket motor. Due to inconsistency of domain boundaries and nodes of computational grid, presence of ghost points lying outside the computational domain is taken into account. For setting the values of flow parameters in ghost points, we use the inverse Lax – Wendroff procedure. We discretize spatial derivatives of level set and gas-dynamics equations with standard WENO schemes of fifth and third-order respectively and time derivatives using total variation diminishing Runge –Kutta methods. We parallelize the presented numerical algorithm using CUDA technology and further optimize it with regard to peculiarities of graphics processors architecture.

Created software package is used for calculating internal ballistics characteristics of nozzleless solid rocket motor during main firing phase. On the base of obtained numerical results, we discuss efficiency of parallelization using CUDA technology and applying considered optimizations. It has been shown that implemented parallelization technique leads to a significant acceleration in comparison with central processes. Distributions of key parameters of combustion products flow in different periods of time have been presented in this paper. We make a comparison of obtained results between quasione-dimensional approach and developed numerical technique.

First-order optimization methods are workhorses in a wide range of modern applications in economics, physics, biology, machine learning, control, and other fields. Among other first-order methods accelerated and momentum ones obtain special attention because of their practical efficiency. The heavy-ball method (HB) is one of the first momentum methods. The method was proposed in 1964 and the first analysis was conducted for quadratic strongly convex functions. Since then a number of variations of HB have been proposed and analyzed. In particular, HB is known for its simplicity in implementation and its performance on nonconvex problems. However, as other momentum methods, it has nonmonotone behavior, and for optimal parameters, the method suffers from the so-called peak effect. To address this issue, in this paper, we consider an averaged version of the heavy-ball method (AHB). We show that for quadratic problems AHB has a smaller maximal deviation from the solution than HB. Moreover, for general convex and strongly convex functions, we prove non-accelerated rates of global convergence of AHB, its weighted version WAHB, and for AHB with restarts R-AHB. To the best of our knowledge, such guarantees for HB with averaging were not explicitly proven for strongly convex problems in the existing works. Finally, we conduct several numerical experiments on minimizing quadratic and nonquadratic functions to demonstrate the advantages of using averaging for HB. Moreover, we also tested one more modification of AHB called the tail-averaged heavy-ball method (TAHB). In the experiments, we observed that HB with a properly adjusted averaging scheme converges faster than HB without averaging and has smaller oscillations.

In a number of fundamental and practical problems, it is necessary to describe the dynamics of the motion of complexshaped particles in a high-speed gas flow. An example is the movement of coal particles behind the front of a strong shock wave during an explosion in a coal mine. The paper is devoted to numerical simulation of the dynamics of translational and rotational motion of a square-shaped body, as an example of a particle of a more complex shape than a round one, in a supersonic flow behind a passing shock wave. The formulation of the problem approximately corresponds to the experiments of Professor V. M. Boiko and Professor S. V. Poplavski (ITAM SB RAS).

Mathematical model is based on the two-dimensional Euler equations, which are solved in a region with varying boundaries. The defining system of equations is integrated using an explicit scheme and the Cartesian grid method which was developed and verified earlier. The computational algorithm at the time integration step includes: determining the step value, calculating the dynamics of the body movement (determining the force and moment acting on the body; determining the linear and angular velocities of the body; calculating the new coordinates of the body), calculating the gas parameters. To calculate numerical fluxes through the edges of the cell intersected by the boundaries of the body, we use a two-wave approximation for solving the Riemann problem and the Steger – Warming scheme.

The movement of a square with a side of 6 mm was initiated by the passage of a shock wave with a Mach number of 3,0 propagating in a flat channel 800 mm long and 60 mm wide. The channel was filled with air at low pressure. Different initial orientation of the square relative to the channel axis was considered. It is found that the initial position of the square with its side across the flow is less stable during its movement than the initial position with a diagonal across the flow. In this case, the calculated results qualitatively correspond to experimental observations. For the intermediate initial positions of a square, a typical mode of its motion is described, consisting of oscillations close to harmonic, turning into rotation with a constant average angular velocity. During the movement of the square, there is an average monotonous decrease in the distance between the center of mass and the center of pressure to zero.

The article is devoted to modeling the shallow water hydrodynamic processes using the example of the Azov Sea. The article presents a mathematical model of the hydrodynamics of a shallow water body, which allows one to calculate three-dimensional fields of the velocity vector of movement of the aquatic environment. Application of regularizers according to B.N.Chetverushkin in the continuity equation led to a change in the method of calculating the pressure field, based on solving the wave equation. A discrete finite-difference scheme has been constructed for calculating pressure in an area whose linear vertical dimensions are significantly smaller than those in horizontal coordinate directions, which is typical for the geometry of shallow water bodies. The method and algorithm for solving grid equations with a tridiagonal preconditioner are described. The proposed method is used to solve grid equations that arise when calculating pressure for the three-dimensional problem of hydrodynamics of the Azov Sea. It is shown that the proposed method converges faster than the modified alternating triangular method. A parallel implementation of the proposed method for solving grid equations is presented and theoretical and practical estimates of the acceleration of the algorithm are carried out taking into account the latency time of the computing system. The results of computational experiments for solving problems of hydrodynamics of the Sea of Azov using the hybrid MPI + OpenMP technology are presented. The developed models and algorithms were used to reconstruct the environmental disaster that occurred in the Sea of Azov in 2001 and to solve the problem of the movement of the aquatic environment in estuary areas. Numerical experiments were carried out on the K-60 hybrid computing cluster of the Keldysh Institute of Applied Mathematics of Russian Academy of Sciences.

This article presents the results of an analytical and computer study of the chaotic evolution of a regular velocity field generated by a large-scale harmonic forcing. The authors obtained an analytical solution for the flow stream function and its derivative quantities (velocity, vorticity, kinetic energy, enstrophy and palinstrophy). Numerical modeling of the flow evolution was carried out using the OpenFOAM software package based on incompressible model, as well as two inhouse implementations of CABARET and McCormack methods employing nearly incompressible formulation. Calculations were carried out on a sequence of nested meshes with 64², 128², 256², 512², 1024² cells for two characteristic (asymptotic) Reynolds numbers characterizing laminar and turbulent evolution of the flow, respectively. Simulations show that blow-up of the analytical solution takes place in both cases. The energy characteristics of the flow are discussed relying upon the energy curves as well as the dissipation rates. For the fine mesh, this quantity turns out to be several orders of magnitude less than its hydrodynamic (viscous) counterpart. Destruction of the regular flow structure is observed for any of the numerical methods, including at the late stages of laminar evolution, when numerically obtained distributions are close to analytics. It can be assumed that the prerequisite for the development of instability is the error accumulated during the calculation process. This error leads to unevenness in the distribution of vorticity and, as a consequence, to the variance vortex intensity and finally leads to chaotization of the flow. To study the processes of vorticity production, we used two integral vorticity-based quantities — integral enstrophy ($\zeta$) and palinstrophy $(P)$. The formulation of the problem with periodic boundary conditions allows us to establish a simple connection between these quantities. In addition, $\zeta$ can act as a measure of the eddy resolution of the numerical method, and palinstrophy determines the degree of production of small-scale vorticity.

This work is devoted to the numerical study of high-speed mixing layers of compressible flows. The problem under consideration has a wide range of applications in practical tasks and, despite its apparent simplicity, is quite complex in terms of modeling. Because in the mixing layer, as a result of the instability of the tangential discontinuity of velocities, the flow passes from laminar flow to turbulent mode. Therefore, the obtained numerical results of the considered problem strongly depend on the adequacy of the used turbulence models. In the presented work, this problem is studied based on the two-fluid approach to the problem of turbulence. This approach has arisen relatively recently and is developing quite rapidly. The main advantage of the two-fluid approach is that it leads to a closed system of equations, when, as is known, the long-standing Reynolds approach leads to an open system of equations. The paper presents the essence of the two-fluid approach for modeling a turbulent compressible medium and the methodology for numerical implementation of the proposed model. To obtain a stationary solution, the relaxation method and Prandtl boundary layer theory were applied, resulting in a simplified system of equations. In the considered problem, high-speed flows are mixed. Therefore, it is also necessary to model heat transfer, and the pressure cannot be considered constant, as is done for incompressible flows. In the numerical implementation, the convective terms in the hydrodynamic equations were approximated by the upwind scheme with the second order of accuracy in explicit form, and the diffusion terms in the right-hand sides of the equations were approximated by the central difference in implicit form. The sweep method was used to implement the obtained equations. The SIMPLE method was used to correct the velocity through the pressure. The paper investigates a two-liquid turbulence model with different initial flow turbulence intensities. The obtained numerical results showed that good agreement with the known experimental data is observed at the inlet turbulence intensity of $0.1 < I < 1 \%$. Data from known experiments, as well as the results of the $k − kL + J$ and LES models, are presented to demonstrate the effectiveness of the proposed turbulence model. It is demonstrated that the two-liquid model is as accurate as known modern models and more efficient in terms of computing resources.

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 Exner equation in conjunction phenomenological sediment transport models is widely used for mathematical modeling non-cohesive river bed. This approach allows to obtain an accurate solution without any difficulty if one models evolution of simple shape bed. However if one models evolution of complex shape bed with unstable soil the numerical instability occurs in some cases. It is difficult to detach this numerical instability from the natural physical instability of bed.

This paper analyses the causes of numerical instability occurring while modeling evolution of complex shape bed by using the Exner equation and phenomenological sediment rate models. The paper shows that two kinds of indeterminateness may occur while solving numerically the Exner equation closed by phenomenological model of sediment transport. The first indeterminateness occurs in the bed area where sediment transport is transit and bed is not changed. The second indeterminateness occurs at the extreme point of bed profile when the sediment rate varies and the bed remains the same. Authors performed the closure of the Exner equation by the analytical sediment transport model, which allowed to transform the Exner equation to parabolic type equation. Analysis of the obtained equation showed that it’s numerical solving does not lead to occurring of the indeterminateness mentioned above. Parabolic form of the transformed Exner equation allows to apply the effective and stable implicit central difference scheme for this equation solving.

The model problem of bed evolution in presence of periodic distribution of the bed shear stress is carried out. The authors used the explicit central difference scheme with and without filtration method application and implicit central difference scheme for numerical solution of the problem. It is shown that the explicit central difference scheme is unstable in the area of the bed profile extremum. Using the filtration method resulted to increased dissipation of the solution. The solution obtained by using the implicit central difference scheme corresponds to the distribution law of bed shear stress and is stable throughout the calculation area.