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This work is dedicated to application of bicompact schemes to numerical solution of evolutionary hyperbolic equations. The main advantage of this class of schemes lies in combination of two beneficial properties: the first one is spatial approximation of high even order on a stencil that always occupies only one mesh cell; the second one is spectral resolution which is better in comparison to classic compact finite-difference schemes of the same order of spatial approximation. One feature of bicompact schemes is considered: their spatial approximation is rigidly tied to Cartesian meshes (with parallelepiped-shaped cells in three-dimensional case). This feature makes rather challenging any application of bicompact schemes to problems with complex computational domains as treated in the framework of unstructured meshes. This problem is proposed to be solved using well-known methods for treating complex-shaped boundaries and their corresponding boundary conditions on Cartesian meshes. The generalization of bicompact schemes on problems in geometrically complex domains is made in case of gas dynamics problems and Euler equations. The free boundary method is chosen as a particular tool to introduce the influence of arbitrary-shaped solid boundaries on gas flows on Cartesian meshes. A brief description of this method is given, its governing equations are written down. Bicompact schemes of fourth order of approximation in space with locally one-dimensional splitting are constructed for equations of the free boundary method. Its compensation flux is discretized with second order of accuracy. Time stepping in the obtained schemes is done with the implicit Euler method and the third order accurate $L$-stable stiffly accurate three-stage singly diagonally implicit Runge–Kutta method. The designed bicompact schemes are tested on three two-dimensional problems: stationary supersonic flows with Mach number three past one circular cylinder and past three circular cylinders; the non-stationary interaction of planar shock wave with a circular cylinder in a channel with planar parallel walls. The obtained results are in a good agreement with other works: influence of solid bodies on gas flows is physically correct, pressure in control points on solid surfaces is calculated with the accuracy appropriate to the chosen mesh resolution and level of numerical dissipation.

An essential class of problems describes physical processes occurring in non-convex domains containing a corner greater than 180 degrees on the boundary. The solution in a neighborhood of a corner is singular and its finding using classical approaches entails a loss of accuracy. In the paper, we consider stationary, linearized by Picard’s iterations, Navier – Stokes equations governing the flow of a incompressible viscous fluid in the convection form in $L$-shaped domain. An $R_\nu$-generalized solution of the problem in special sets of weighted spaces is defined. A special finite element method to find an approximate $R_\nu$-generalized solution is constructed. Firstly, functions of the finite element spaces satisfy the law of conservation of mass in the strong sense, i.e. at the grid nodes. For this purpose, Scott – Vogelius element pair is used. The fulfillment of the condition of mass conservation leads to the finding more accurate, from a physical point of view, solution. Secondly, basis functions of the finite element spaces are supplemented by weight functions. The degree of the weight function, as well as the parameter $\nu$ in the definition of an $R_\nu$-generalized solution, and a radius of a neighborhood of the singularity point are free parameters of the method. A specially selected combination of them leads to an increase almost twice in the order of convergence rate of an approximate solution to the exact one in relation to the classical approaches. The convergence rate reaches the first order by the grid step in the norms of Sobolev weight spaces. Thus, numerically shown that the convergence rate does not depend on the corner value.

The development of the Splitting Method for Incompressible Fluid flows (SMIF) during last 50 years is described. The hybrid explicit finite difference scheme of method SMIF is based on Modified Central Difference Scheme (MCDS) and Modified Upwind Difference Scheme (MUDS) with special switch condition depending on the velocity sign and the signs of the first and second differences of transferred functions. Application of this method for solving of some tasks (the spatial flow around a sphere and a circular cylinder for homogeneous and stratified fluids in a wide range of dimensionless parameters of the problem, including the transitional regimes (2D–3D transition, laminar-turbulent transition in the boundary layer); a plane problem of fluid flows with a free surface; a dynamics of vortex pair in a water; a collapse of spots in stratified fluid; the air-, heat-, and mass transfer in «clean rooms») is demonstrated.

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.

A mathematical model is formulated for the coastal slope erosion of sandy channel, which occurs under the action of a passing flood wave. The moving boundaries of the computational domain — the bottom surface and the free surface of the hydrodynamic flow — are determined from the solution of auxiliary differential equations. A change in the hydrodynamic flow section area for a given law of change in the flow rate requires a change in time of the turbulent viscosity averaged over the section. The bottom surface movement is determined from the Exner equation solution together with the equation of the bottom material avalanche movement. The Exner equation is closed by the original analytical model of traction loads movement. The model takes into account transit, gravitational and pressure mechanisms of bottom material movement and does not contain phenomenological parameters.

Based on the finite element method, a discrete analogue of the formulated problem is obtained and an algorithm for its solution is proposed. An algorithm feature is control of the free surface movement influence of the flow and the flow rate on the process of determining the flow turbulent viscosity. Numerical calculations have been carried out, demonstrating qualitative and quantitative influence of these features on the determining process of the flow turbulent viscosity and the channel bank slope erosion.

Data comparison on bank deformations obtained as a result of numerical calculations with known flume experimental data showed their agreement.

Based on the modeling method using a mesh system, an algorithm is implemented for solving a unsteady problem with moving bodies The algorithm takes into account the movement and rotation of bodies according to a given law of motion. The algorithm is applied to analysis the flow around an infinite composed of cylinders with an elliptical cross-section, which either move across the flow or rotate with a change in the angle of attack. To simulate the flow of bodies with a sharp edge, characteristic of the profiles of gas turbine machines, an algorithm for constructing a mesh of type C with the inclusion of a certain area behind the profile is implemented. The program for modeling the flow near the profile is implemented within the framework of models of Euler equations, Navier – Stokes equations in the approximation of a thin layer with laminar viscosity and turbulent viscosity in the framework of an algebraic viscosity model. The program has also been adapted to solve the problems of internal gas dynamics of turbomachines. For this purpose, the method of setting the boundary conditions at the entrance and exit from the calculated area from the velocity to the pressure drop, as well as at the lateral boundaries from the free flow to the periodicity, was changed. This made it possible to simulate the flow of gas in the inter-blade channels of compressors and turbines of gas turbine engines. To refine the algorithm, a series of calculations of the aerodynamic parameters of several turbine cascades in various subsonic and supersonic modes and their comparison with the experiment were carried out. Calculations of turbine grating parameters were carried out within the framework of the inviscid and viscous gas model. The calculation and experiment were compared by the distribution of gas parameters near the profile, as well as by the energy losses of the flow in the cascade. Calculations have shown the applicability and correctness of the program to solve this class of problems. To test the program on the problems of external subsonic aerodynamics, calculations of the aerodynamic characteristics of an isolated airfoil in an undisturbed flow were performed. The results obtained allow us to assert the applicability of the hybrid grid method to various classes of problems of applied gas dynamics.

The article contains the description of developed mathematical models of fractures which can be used for numerical solution of exploration seismology problems with use of grid-characteristic method on unstructured triangular and tetrahedral meshes. The base of developed models is the concept of infinitely thin fracture. This fracture is represented by contact boundary. Such approach significantly reduces the consumption of computer resources by the absence of the mesh definition inside of fracture necessity. By the other side it lets state the fracture discretely in integration domain, therefore one can observe qualitative new effects which are not available to observe by use of effective models of fractures, actively used in computational seismic.

The main target in the development of models have been getting the most accurate result. Developed models thet can receive the response close to the actual response of the existing fracture in geological environment. We considered fluid-filled fractures, glued and partially glued fractures, and also fractures with dynamical friction force. Fracture behavior determinated by the nature of condition on the border.

Empty fracture was represented as free boundary condition. This condition give us opportunity for total reflection of wave fronts from fracture. Fluid-filling provided the condition for sliding on the border. Under this condition, there was a passage of longitudinal and total reflection of converted waves. For the real fractures, which has unequal distance between the borders has been proposed the model of partially glued fracture. At different points of the fracture's boundary were sat different conditions. Almost the same effect is achieved by using a fracture model of dynamic friction condition. But its disadvantage is the inabillity to specify the proportion of fracture's glued area due to the friction factor can take values from zero to infinity. The model of partially glued fracture is devoid of this disadvantage.

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.

The paper is devoted to the study of wave and relaxation effects during the pulsed outflow of a gas mixture with a high content of solid particles from a cylindrical channel during its initial partial filling. The problem is formulated in a two-speed two-temperature formulation and was solved numerically by the hybrid large-particle method of the second order of approximation. The numerical algorithm is implemented in the form of parallel computing using basic Free Pascal language tools. The applicability and accuracy of the method for wave flows of concentrated gas-particles mixtures is confirmed by comparison with test asymptotically accurate solutions. The calculation error on a grid of low detail in the characteristic flow zones of a two-phase medium was 10^-6 . . . 10^-5.

Based on the wave diagram, the analysis of the physical pattern of the outflow of a gas suspension partially filling a cylindrical channel is performed. It is established that, depending on the degree of initial filling of the channel, various outflow modes are formed. The first mode is implemented with a small degree of loading of the high-pressure chamber, at which the left boundary of the gas-particles mixture crosses the outlet section before the arrival of the rarefaction wave reflected from the bottom of the channel. At the same time, the maximum value of the mass flow rate of the mixture is achieved. Other modes are formed in cases of a larger initial filling of the channel, when the rarefaction waves reflected from the bottom of the channel interact with the gas suspension layer and reduce the intensity of its outflow.

The influence of relaxation properties with changing particle size on the dynamics of a limited layer of a gas-dispersed medium is studied. Comparison of the outflow of a limited gas suspension layer with different particle sizes shows that for small particles (the Stokes number is less than 0.001), an anomalous phenomenon of the simultaneous existence of shock wave structures in the supersonic and subsonic flow of gas and suspension is observed. With an increase in the size of dispersed inclusions, the compaction jumps in the region of the two-phase mixture are smoothed out, and for particles (the Stokes number is greater than 0.1), they practically disappear. At the same time, the shock-wave configuration of the supersonic gas flow at the outlet of the channel is preserved, and the positions and boundaries of the energy-carrying volumes of the gas suspension are close when the particle sizes change.

Modeling of channel processes in the study of coastal channel deformations requires the calculation of hydrodynamic flow parameters that take into account the existence of secondary transverse currents formed at channel curvature. Three-dimensional modeling of such processes is currently possible only for small model channels; for real river flows, reduced-dimensional models are needed. At the same time, the reduction of the problem from a three-dimensional model of the river flow movement to a two-dimensional flow model in the cross-section assumes that the hydrodynamic flow under consideration is quasi-stationary and the hypotheses about the asymptotic behavior of the flow along the flow coordinate of the cross-section are fulfilled for it. Taking into account these restrictions, a mathematical model of the problem of the a stationary turbulent calm river flow movement in a channel cross-section is formulated. The problem is formulated in a mixed formulation of velocity — “vortex – stream function”. As additional conditions for problem reducing, it is necessary to specify boundary conditions on the flow free surface for the velocity field, determined in the normal and tangential direction to the cross-section axis. It is assumed that the values of these velocities should be determined from the solution of auxiliary problems or obtained from field or experimental measurement data.

To solve the formulated problem, the finite element method in the Petrov – Galerkin formulation is used. Discrete analogue of the problem is obtained and an algorithm for solving it is proposed. Numerical studies have shown that, in general, the results obtained are in good agreement with known experimental data. The authors associate the obtained errors with the need to more accurately determine the circulation velocities field at crosssection of the flow by selecting and calibrating a more appropriate model for calculating turbulent viscosity and boundary conditions at the free boundary of the cross-section.