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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.

One of the numerical methods for solving problems of scattering of electromagnetic waves by systems formed by parallel oriented cylindrical elements — two-dimensional photonic crystals — is considered. The method is based on the classical method of separation of variables for solving the wave equation. Тhe essence of the method is to represent the field as the sum of the primary field and the unknown secondary scattered on the elements of the medium field. The mathematical expression for the latter is written in the form of infinite series in elementary wave functions with unknown coefficients. In particular, the field scattered by N elements is sought as the sum of N diffraction series, in which one of the series is composed of the wave functions of one body, and the wave functions in the remaining series are expressed in terms of the eigenfunctions of the first body using addition theorems. From satisfying the boundary conditions on the surface of each element we obtain systems of linear algebraic equations with an infinite number of unknowns — the required expansion coefficients, which are solved by standard methods. A feature of the method is the use of analytical expressions describing diffraction by a single element of the system. In contrast to most numerical methods, this approach allows one to obtain information on the amplitude-phase or spectral characteristics of the field only at local points of the structure. The absence of the need to determine the field parameters in the entire area of space occupied by the considered multi-element system determines the high efficiency of this method. The paper compares the results of calculating the transmission spectra of two-dimensional photonic crystals by the considered method with experimental data and numerical results obtained using other approaches. Their good agreement is demonstrated.

A method for studying the dispersion characteristics of photonic crystals — media with a dielectric constant that varies periodically in space — is considered. The method is based on the representation of the wave functions and permittivity of a periodic medium in the form of Fourier series and their subsequent substitution into the wave equation, which leads to the formulation of the dispersion equation. Using the latter, for each value of the wave vector it is possible determined a set of eigen frequencies. Each of eigen frequency forms a separate dispersion curve as a continuous function of the wave number. The Fourier expansion coefficients of the permittivity, which depend on the vectors of the reciprocal lattice of the photonic crystal, are determined on the basis of data on the geometric characteristics of the elements that form the crystal, their electrophysical properties and the density of the crystal. The solution of the dispersion equation found makes it possible to obtain complete information about the number of modes propagating in a periodic structure at different frequencies, and about the possibility of forming band gaps, i.e. frequency ranges within which wave propagation through a photonic crystal is impossible. The focus of this work is on the application of this method to the analysis of the dispersion properties of metallic photonic crystals. The difficulties that arise in this case due to the presence of intrinsic dispersion properties of the metals that form the elements of the crystal are overcome by an analytical description of their permittivity based on the model of free electrons. As a result, a dispersion equation is formulated, the numerical solution of which is easily algorithmized. That makes possible to determine the dispersion characteristics of metallic photonic crystals with arbitrary parameters. Obtained by this method the results of calculation of dispersion diagrams, which characterize two-dimensional metal photonic crystals, are compared with experimental data and numerical results obtained using the method of self-consistent equations. Their good agreement is demonstrated.

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.

The work investigates the process of self-consistent relaxation of the region of disturbances created in a rarefied binary low-temperature plasma by a stationary charged ball or cylinder with an absorbing surface. A feature of such problems is their self-consistent kinetic nature, in which it is impossible to separate the processes of transfer in phase space and the formation of an electromagnetic field. A mathematical model is presented that makes it possible to describe and analyze the state of the gas, electric and thermal fields in the vicinity of the body. The multidimensionality of the kinetic formulation creates certain problems in the numerical solution, therefore a curvilinear system of nonholonomic coordinates was selected for the problem, which minimizes its phase space, which contributes to increasing the efficiency of numerical methods. For such coordinates, the form of the Vlasov kinetic equation has been justified and analyzed. To solve it, a variant of the large particle method with a constant form factor was used. The calculations used a moving grid that tracks the displacement of the distribution function carrier in the phase space, which further reduced the volume of the controlled region of the phase space. Key details of the model and numerical method are revealed. The model and the method are implemented as code in the Matlab language. Using the example of solving a problem for a ball, the presence of significant disequilibrium and anisotropy in the particle velocity distribution in the disturbed zone is shown. Based on the calculation results, pictures of the evolution of the structure of the particle distribution function, profiles of the main macroscopic characteristics of the gas — concentration, current, temperature and heat flow, and characteristics of the electric field in the disturbed region are presented. The mechanism of heating of attracted particles in the disturbed zone is established and some important features of the process of formation of heat flow are shown. The results obtained are well explainable from a physical point of view, which confirms the adequacy of the model and the correct operation of the software tool. The creation and testing of a basis for the development in the future of tools for solving more complex problems of modeling the behavior of ionized gases near charged bodies is noted.

The work will be useful to specialists in the field of mathematical modeling, heat and mass transfer processes, lowtemperature plasma physics, postgraduate students and senior students specializing in the indicated areas.

The objective of this study is to determine the best formations for the joint movement of a group of small robots in an aquatic environment. Estimation of drag of the flow is a traditional and well-known area of research, but it is not always valid to extend the conclusions made for a single robot to a group of similar devices due to the physical effects that appear during joint movement, such as a wave shadow. For these reasons, it is necessary to study the hydrodynamic characteristics of certain robot formations as a stable structure. The hydrodynamic parameters of systems with two main types of propulsion were studied: locomotive (fishtails) and propellers. Formations similar in structure to schools of fish were mainly considered, and then their applicability for robots of different types was assessed. The relationship between the speed of movement of the group and the drag of each of its participants was also studied. Mathematical modeling of the flow around a group of robots was performed using the finite volume method using two software packages (FlowVision and OpenFoam). Robots with a screw propeller interfere with each other when packed into tight formations, and for the locomotive case, being in the disturbance zone, on the contrary, is preferable. Also, with poorly streamlined bodies, flows separating from the surface can turn into narrow turbulent jets that greatly interfere with the rear robots. It has been established that wake effect reduces energy costs only at low speeds of movement — about 5 cm/s; at high speeds, movement in columns becomes difficult for the rear robots. No large difference in frontal resistance was found between a single robot and a group for a fish-like tail. The studies made it possible to develop and substantiate recommendations for optimizing robot designs for group movement.

Efficient method for numerical solving of the steady transport equation in x-y-z-geometry has been suggested. The equation is being solved on hexagonal mesh, reflecting real structure of the reactor active zone cross-section. Method of characteristics is used, that inherits all the outcomes from the two-dimensional r-z-geometry calculation. Two variants of the method of characteristics have been applied for solving the transport equation in a cell: method of short characteristics and its conservative modification. It has been confirmed that in three-dimensional geometry conservative method has advantage over pure characteristic and it produces highly accurate solution, especially for quasi-diffusion tensor components.

Based on contents of phytoplankton pigments, fluorescence samples and some physico-chemical characteristics of the Rybinsk reservoir waters, searching for connections between biological and physicalchemical characteristics is working out. The standard methods of statistical analysis (correlation, regression), methods of description of connection between qualitative classes of characteristics, based on deviation of the studied characteristics distribution from independent distribution, are studied. A method of searching for boundaries of quality classes by criterion of maximum connection coefficient is offered.

Molecular dynamics simulation using the embedded-atom method is applied to study the structural evolution of the particle diameter of 40 Å during the quenching process. Was carried comparative analysis of the structural reconstruction for the particle and the bulk models. Was a reduction in temperature of the beginning and end of the transformation of the particle. In formation of a percolation cluster from interpenetrating and contacting icosahedrons, for model of the particle, it is involved for 10 percent of atoms more, than for model of a bulk.