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This paper presents definition and solution problem of correct conditions on the boundary, separating subdomains for hyperbolic linear equation systems. The solution algorithm is demonstrated by means of an example system of elastodynamic equations for two spatial variables. Stated approach can be easily expanded on systems of first-order linear hyperbolic equations with random number of spatial variables.

Calculations of radiation transport in the shockwave layer of a descent space vehicle cause essential difficulties due to complex multi-resonance dependence of the absorption macroscopic cross sections from the photon energy. The convergence of two approximate spectrum averaging methods to the results of exact pointwise spectrum calculations is investigated. The first one is the well known multigroup method, the second one is the Lebesgue averaging method belonging to methods of the reduction of calculation points by means of aggregation of spectral points which are characterized by equal absorption strength. It is shown that convergence of the Lebesgue averaging method is significantly faster than the multigroup approach as the number of groups is increased. The only 100–150 Lebesgue groups are required to achieve the accuracy of pointwise calculations even in the shock layer at upper atmosphere with sharp absorption lines. At the same time the number of calculations is reduced by more than four order. Series of calculations of the radiation distribution function in 2D shock layer around a sphere and a blunt cone were performed using the local flat layer approximation and the Lebesgue averaging method. It is shown that the shock wave radiation becomes more significant both in value of the energy flux incident on the body surface and in the rate of energy exchange with the gas-dynamic flow in the case of increasing of the vehicle’s size.

The paper deals with a heat wave motion problem for a degenerate second-order nonlinear parabolic equation with power nonlinearity. The considered boundary condition specifies in a plane the motion equation of the circular zero front of the heat wave. A new numerical-analytical algorithm for solving the problem is proposed. A solution is constructed stepby- step in time using difference time discretization. At each time step, a boundary value problem for the Poisson equation corresponding to the original equation at a fixed time is considered. This problem is, in fact, an inverse Cauchy problem in the domain whose initial boundary is free of boundary conditions and two boundary conditions (Neumann and Dirichlet) are specified on a current boundary (heat wave). A solution of this problem is constructed as the sum of a particular solution to the nonhomogeneous Poisson equation and a solution to the corresponding Laplace equation satisfying the boundary conditions. Since the inhomogeneity depends on the desired function and its derivatives, an iterative solution procedure is used. The particular solution is sought by the collocation method using inhomogeneity expansion in radial basis functions. The inverse Cauchy problem for the Laplace equation is solved by the null field method as applied to a circular domain with a circular hole. This method is used for the first time to solve such problem. The calculation algorithm is optimized by parallelizing the computations. The parallelization of the computations allows us to realize effectively the algorithm on high performance computing servers. The algorithm is implemented as a program, which is parallelized by using the OpenMP standard for the C++ language, suitable for calculations with parallel cycles. The effectiveness of the algorithm and the robustness of the program are tested by the comparison of the calculation results with the known exact solution as well as with the numerical solution obtained earlier by the authors with the use of the boundary element method. The implemented computational experiment shows good convergence of the iteration processes and higher calculation accuracy of the proposed new algorithm than of the previously developed one. The solution analysis allows us to select the radial basis functions which are most suitable for the proposed algorithm.

One of the destroying natural processes is the initiation of the regional seismic activity. It leads to a large number of human deaths. Much effort has been made to develop precise and robust methods for the estimation of the seismic stability of buildings. One of the most common approaches is the natural frequency method. The obvious drawback of this approach is a low precision due to the model oversimplification. The other method is a detailed simulation of dynamic processes using the finite-element method. Unfortunately, the quality of simulations is not enough due to the difficulty of setting the correct free boundary condition. That is why the development of new numerical methods for seismic stability problems is a high priority nowadays.

The present work is devoted to the study of spatial dynamic processes occurring in geological medium during an earthquake. We describe a method for simulating seismic wave propagation from the hypocenter to the day surface. To describe physical processes, we use a system of partial differential equations for a linearly elastic body of the second order, which is solved numerically by a grid-characteristic method on parallelepiped meshes. The widely used geological hypocenter model, called the “double-couple” model, was incorporated into this numerical algorithm. In this case, any heterogeneities, such as geological layers with curvilinear boundaries, gas and fluid-filled cracks, fault planes, etc., may be explicitly taken into account.

In this paper, seismic waves emitted during the earthquake initiation process are numerically simulated. Two different models are used: the homogeneous half-space and the multilayered geological massif with the day surface. All of their parameters are set based on previously published scientific articles. The adequate coincidence of the simulation results is obtained. And discrepancies may be explained by differences in numerical methods used. The numerical approach described can be extended to more complex physical models of geological media.

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.

Seismic survey process is the common method of prospecting and exploration of deposits: oil and natural gas. Invented at the beginning of the XX century, it has received significant development and is currently used by almost all service oil companies. Its main advantages are the acceptable cost of fieldwork (in comparison with drilling wells) and the accuracy of estimating the characteristics of the subsurface area. However, with the discovery of non-traditional deposits (for example, the Arctic shelf, the Bazhenov Formation), the task of improving existing and creating new seismic data processing technologies became important. Significant development in this direction is possible with the use of numerical simulation of the propagation of seismic waves in realistic models of the geological medium, since it is possible to specify an arbitrary internal structure of the medium with subsequent evaluation of the synthetic signal-response.

The present work is devoted to the study of spatial dynamic processes occurring in geological medium containing fractured inclusions in the process of seismic exploration. The authors constructed a three-dimensional model of a layered massif containing a layer of fluid-saturated cracks, which makes it possible to estimate the signal-response when the structure of the inhomogeneous inclusion is varied. To describe physical processes, we use a system of equations for a linearly elastic body in partial derivatives of the second order, which is solved numerically by a grid-characteristic method on hexahedral grid. In this case, the crack planes are identified at the stage of constructing the grid, and further an additional correction is used to ensure a correct seismic response for the model parameters typical for geological media.

In the paper, three-component area seismograms with a common explosion point were obtained. On their basis, the effect of the structure of a fractured medium on the anisotropy of the seismic response recorded on the day surface at a different distance from the source was estimated. It is established that the kinematic characteristics of the signal remain constant, while the dynamic characteristics for ordered and disordered models can differ by tens of percents.

The review is based on some of the authors ’early works of particular scientific, methodological and practical interest and the greatest attention is paid to recent works, where quite detailed numerical studies of not only single, but also double and multiple explosions in a wide range of heights and environmental conditions have been performed . Since the shock wave of a powerful explosion is one of the main damaging factors in the lower atmosphere, the review focuses on both the physical analysis of their propagation and their interaction. Using the three-dimensional algorithms developed by the authors, the effects of interference and diffraction of several shock waves, which are interesting from a physical point of view, in the absence and presence of an underlying surface of various structures are considered. Quantitative characteristics are determined in the region of their maximum values, which is of known practical interest. For explosions in a dense atmosphere, some new analytical solutions based on the small perturbation method have been found that are convenient for approximate calculations. For a number of conditions, the possibility of using the self-similar properties of equations of the first and second kind to solve problems on the development of an explosion has been shown.

Based on numerical analysis, a fundamental change in the structure of the development of the perturbed region with a change in the height of the explosion in the range of 100–120 km is shown. At altitudes of more than 120 km, the geomagnetic field begins to influence the development of the explosion; therefore, even for a single explosion, the picture of the plasma flow after a few seconds becomes substantially three-dimensional. For the calculation of explosions at altitudes of 120–1000 km under the guidance of academician A. Kholodov. A special three-dimensional numerical algorithm based on the MHD approximation was developed. Numerous calculations were performed and for the first time a quite detailed picture of the three-dimensional flow of the explosion plasma was obtained with the formation of an upward jet in 5–10 s directed in the meridional plane approximately along the geomagnetic field. After some modification, this algorithm was used to calculate double explosions in the ionosphere, spaced a certain distance. The interaction between them was carried out both by plasma flows and through a geomagnetic field. Some results are given in this review and are described in detail in the original articles.

In this work, we consider a special approximation of the general perturbation formula for the electromagnetic field by a set of electrically small inhomogeneities located in the domain of interest. The problem considered in this paper arises in many applications of technical electrodynamics, radar technologies and subsurface remote sensing. In the general case, it is formulated as follows: at some point in the perturbed domain, it is necessary to determine the amplitude of the electromagnetic field. The perturbation of electromagnetic waves is caused by a set of electrically small scatterers distributed in space. The source of electromagnetic waves is also located in perturbed domain. The problem is solved by introducing the far field approximation and through the formulation for the scatterer radar cross section value. This, in turn, allows one to significantly speed up the calculation process of the perturbed electromagnetic field by a set of a spherical inhomogeneities identical to each other with arbitrary electrophysical parameters. In this paper, we consider only the direct scattering problem; therefore, all parameters of the scatterers are known. In this context, it may be argued that the formulation corresponds to the well-posed problem and does not imply the solution of the integral equation in the generalized formula. One of the features of the proposed algorithm is the allocation of a characteristic plane at the domain boundary. All points of observation of the state of the system belong to this plane. Set of the scatterers is located inside the observation region, which is formed by this surface. The approximation is tested by comparing the results obtained with the solution of the general formula method for the perturbation of the electromagnetic field. This approach, among other things, allows one to remove a number of restrictions on the general perturbation formula for E-filed analysis.

The paper presents an analysis of the nonlinear equation of spin wave transport by the method of characteristics. The conclusion of a new mathematical model of spin wave propagation is presented for the solution of which the characteristic is applied. The behavior analysis of the behavior of the real and imaginary parts of the wave and its amplitude is performed. The phase portraits demonstrate the dependence of the desired function on the nonlinearity coefficient. It is established that the real and imaginary parts of the wave oscillate by studying the nature of the evolution of the initial wave profile by the phase plane method. The transition of trajectories from an unstable focus to a limiting cycle, which corresponds to the oscillation of the real and imaginary parts, is shown. For the amplitude of the wave, such a transition is characterized by its amplification or attenuation (depending on the nonlinearity coefficient and the chosen initial conditions) up to a certain threshold value. It is shown that the time of the transition process from amplification (attenuation) to stabilization of the amplitude also depends on the nonlinearity parameter. It was found out that at the interval of amplification of the amplitude of the spin wave, the time of the transition process decreases, and lower amplitude values correspond to higher parameters of nonlinearity.

The influence of the process of initiating a rapid local heat release near surface streamlined by supersonic gas (air) flow on the separation region that occurs during a fast turn of the flow was investigated. This surface consists of two planes that form obtuse angle when crossing, so that when flowing around the formed surface, the supersonic gas flow turns by a positive angle, which forms an oblique shock wave that interacts with the boundary layer and causes flow separation. Rapid local heating of the gas above the streamlined surface simulates long spark discharge of submicrosecond duration that crosses the flow. The gas heated in the discharge zone interacts with the separation region. The flow can be considered two-dimensional, so the numerical simulation is carried out in a two-dimensional formulation. Numerical simulation was carried out for laminar regime of flow using the sonicFoam solver of the OpenFOAM software package.

The paper describes a method for constructing a two-dimensional computational grid using hexagonal cells. A study of grid convergence has been carried out. A technique is given for setting the initial profiles of the flow parameters at the entrance to the computational domain, which makes it possible to reduce the computation time by reducing the number of computational cells. A method for non-stationary simulation of the process of rapid local heating of a gas is described, which consists in superimposing additional fields of increased pressure and temperature values calculated from the amount of energy deposited in oncoming supersonic gas flow on the corresponding fields of values obtained in the stationary case. The parameters of the energy input into the flow corresponding to the parameters of the electric discharge process, as well as the parameters of the oncoming flow, are close to the experimental values.

During analyzing numerical simulation data it was found that the initiation of rapid local heating leads to the appearance of a gas-dynamic perturbation (a quasi-cylindrical shock wave and an unsteady swirling flow), which, when interacting with the separation region, leads to a displacement of the separation point downstream. The paper considers the question of the influence of the energy spent on local heating of the gas, and of the position on the streamlined surface of the place of heating relative to the separation point, on the value of its maximum displacement.