It is proposed to introduce into Probability Theory courses such a new moment characteristic of random variable as Senatov moment. Naturalness of this proposal is confirmed by three views of appearance of Senatov moments. Introducing of them will answer the question about what is analogue of Taylor series of function for density.

Introduction. This article is devoted to the consideration of options for symmetrization of two-layer implicit iterative methods for solving grid equations that arise when approximating boundary value problems for two-dimensional elliptic equations. These equations are included in the formulation of many problems of hydrodynamics, hydrobiology of aquatic systems, etc. Grid equations for these problems are characterized by a large number of unknowns — from 10⁶ to 10¹⁰, which leads to poor conditionality of the corresponding system of algebraic equations and, as a consequence, to a significant increase in the number of iterations, necessary to achieve the specified accuracy. The article discusses a method for reducing the number of iterations for relatively simple methods for solving grid equations, based on the procedure of symmetrized traversal of the grid region.

Materials and Methods. The methods for solving grid equations discussed in the article are based on the procedure of symmetrized traversal along the rows (or columns) of the grid area.

Results. Numerical experiments have been performed for a model problem — the Dirichlet difference problem for the Poisson equation, which demonstrate a reduction in the number of iterations compared to the basic algorithms of these methods.

Discussion and Conclusions. This work has practical significance. The developed software allows it to be used to solve specific physical problems, including as an element of a software package.

Introduction. The problem of modelling the propagation of elastic waves is of great practical importance when conducting seismic exploration. Based on it, a model of the environment under study is being built. At the same time, the quality of the constructed model is determined by the accuracy of solving the modelling problem, which ensures constantly increasing requirements for modelling accuracy. For accurate modelling, it is important to correctly describe and take into account the boundaries of the media. At the same time, the quality of the constructed model is determined by the accuracy of solving the modelling problem, which ensures constantly increasing requirements for modelling accuracy.

Materials and Methods. We have studied a modification of the grid-characteristic method on rectangular grids using overset grids to describe the interface of media of complex shape. This approach has previously been used to describe the earth’s surface when conducting simulations on land. This paper describes its application in modelling the relief of the ocean shelf.

Results. The use of the overset grid reduces the modelling error, the number of parasitic waves and artifacts and makes it possible to get a more visual picture.

Discussion and Conclusions. Overset grids can be used to describe the interface of media in modelling seismic exploration of the ocean shelf. Their use makes it possible to increase the accuracy of modelling and reduce the number of artifacts compared to using only one grid.

Introduction.

Seismic exploration in conditions of heterogeneity of the environment is an urgent topic for the oil and gas industry. Consequently, the development of numerical methods for solving the direct problem of seismic exploration remains relevant as a necessary link in the development and improvement of methods for solving the inverse problem. The Schonberg thin crack model has performed well in the numerical solution of problems requiring explicit consideration of geological inhomogeneities.

Materials and Methods. In this paper, we consider a modification of the grid-characteristic method using superimposed grids. The presented approach makes it possible to conduct computational experiments, explicitly taking into account fractured inhomogeneities with arbitrary spatial orientation. For this, in addition to the basic regular computational grid, there is the concept of superimposed grids. Inhomogeneities, such as cracks, are described within the framework of the superimposed grid and, in turn, have no restrictions associated with the main grid. Thus, by performing an interpolation operation between the superimposed main grids, we can bypass the requirement of alignment of cracks and edges of the main grid.

Results. The proposed approach made it possible to study the dependence of the anisotropy of the seismic response of a fractured cluster on the dispersion of the angles of inclination of the cracks.

Discussion and Conclusions. A modification of the grid-characteristic method using superimposed grids is proposed to explicitly account for fractured inhomogeneities in a heterogeneous geological environment.

Introduction. Prediction of suspension deposition zones is required to assess and minimize the negative impact on the ecosystem of the reservoir during dredging within the framework of large-scale engineering projects, prediction of suspension deposition zones is required to assess and minimize the negative impact on the ecosystem of the reservoir. It is necessary to build a mathematical model that takes into account many factors that have a significant impact on the accuracy of forecasts. The aim of the work is to construct a mathematical model of transport of multicomponent suspension, taking into account the composition of the soil (different diameter of the suspension particles), the flow velocity of the water flow, the complex geometry of the coastline and bottom, wind stresses and friction on the bottom, turbulent exchange, etc.

Materials and Methods. A mathematical model for the transport of a multicomponent suspension and an approximation of the proposed continuous model with the second order of accuracy with respect to the steps of the spatial grid are described, considering the boundary conditions of the Neumann and Robin type. The approximation of the hydrodynamics model is obtained based on splitting schemes by physical processes, which guarantee fulfillment mass conservation for discrete model.

Results. The proposed mathematical model formed the basis of the developed software package that allows to simulate the process of sedimentation of a multicomponent suspension. The results of the work of the software package on the model problem of sedimentation of a three-component suspension in the process of soil dumping during dredging are presented.

Discussions and Conclusions. The mathematical model of transport of three-component suspension is described and software implemented. The developed software allows to simulate the process of deposition of suspended particles of various diameters on the bottom, and to evaluate its effect on the bottom topography and changes in the bottom composition. The developed software package also allows to analyze the process of sediment movement in the case of resuspension of multicomponent bottom sediments of the reservoir, which causes secondary pollution of the reservoi

Introduction. Two-dimensional hydrodynamic models have proven their ability to adequately describe the processes of runoff and transportation in rivers, lakes, estuaries, deltas and seas. Practice shows that even where significant three-dimensional effects are expected, for example, with wind flows, a two-dimensional approach can work effectively. However, in some cases, the two-dimensional model does not accurately reflect the actual flow structures. For example, in shallow waters with complex bathymetry, heterogeneous terrain and dynamics can lead to a non-uniform velocity profile. The aim of the study is to develop a basis for determining in which cases a two-dimensional model averaged in depth is sufficient for modelling hydrodynamic processes in shallow waters like the Azov Sea, and in which cases it is advisable to use a three-dimensional model to obtain accurate results.

Materials and Methods. Local analytical solutions have been obtained for the propagation of the predominant singular progressive wave in a shallow, well-mixed reservoir. Advective terms and Coriolis terms are neglected, the vortex viscosity is assumed to be constant, and the lower friction term is linearized. Special attention is paid to the latter, since the characteristics of the models significantly depend on the method of determining the coefficients of lower friction. The analytical method developed in the study shows that certain combinations of higher flow velocities (u ≈˃ 1 m/s) and water depths (d ˃ 50 m) can cause significant differences between the results of the depth-averaged model and the model containing vertical information.

Results. The results obtained are verified by numerical simulation of stationary and non-stationary periodic flows in a schematized rectangular basin. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modelling averaged in depth. Both simulations show good compliance with analytical solutions.

Discussion and Conclusions. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear effects — nonlinearities caused by higher-order terms in the equations of motion, i.e. terms of advective acceleration and friction, and nonlinear effects caused by geometric nonlinearities, this is due, for example, to different water depths and reservoir widths, which will be important when modelling a real sea.