The development of a computational method is not a simple matter and boils down to replacing the differential operator with a difference one. To construct it, it is necessary to correctly set a mathematical problem that is adequate to the physical one under consideration. In addition, the algorithm must meet some other requirements. Therefore, to create a numerical algorithm requires not only ingenuity and imagination, but also a deep understanding of the reasons why these requirements are caused.

Systems of partial differential equations of hyperbolic type are used to describe the unsteady behavior of continuous media. To solve these problems, characteristic methods were developed in such a way as to take into account the corresponding properties of hyperbolic equations and to be able to build a so-called characteristic irregular grid adapting to the solution of the problem. Methods of end-to-end counting have been developed that take into account the properties of systems of hyperbolic equations — inverse methods of characteristics or grid-characteristic methods.

In grid-characteristic methods, a regular computational grid is used, not a solvable initial system is approximated on it, but compatibility conditions along characteristic lines with interpolation of the desired functions at the points of intersection of characteristics with a coordinate line on which the data is already known. The obtained characteristic form of the gas dynamics equations makes it possible to understand how to set the boundary conditions correctly.

The construction of a numerical method is not a simple matter and is not reduced to the formal replacement of derivatives by approximating their difference relations (for example, using finite differences). When developing the method, it is necessary to take into account the physical side of the problem being solved. At the same time, the method must meet certain requirements, the understanding of which is necessary during its development.

Introduction. The criterion for the success of an advertising campaign is the maximum profit from sales, taking into account the costs of its implementation, while the sale of the same type of goods is sales occur in a competitive environment. The article examines a model for predicting mass sales of two similar products depending on the tactics of an advertising campaign. First of all, the distribution of funds between its separate types is considered: expenses for advertising paper products, banners and advertising in electronic media (EMM).

Materials and methods. The model is formulated in the form of a Cauchy problem for a system of two ordinary differential equations with nonlinear right-hand sides, taking into account: the total number of potential solvent buyers of the first and second goods; the intensity of the advertising campaign, mainly through EMM, the positive impact on sales of the interaction of those who have already bought the first or second type of goods with potential buyers, as well as informal (at the level of buyers) anti-advertising.

The results of the study. A solution is given for the case of constant coefficients determined by the above factors for the corresponding Cauchy problem in closed form.

Discussion and conclusions. The results obtained can be used to replay model situations of advertising organization in order to determine the conditions for extracting the greatest profit from sales minus advertising costs.

Introduction. The article is devoted to one of the problems in the development of oil and gas fields - the construction of correct geological models of the subsurface space. Researchers from various scientific groups around the world have proposed various ways to improve the accuracy of the computer simulations used in this process. The purpose of this study is to assess the degree of influence of the day surface relief and the anisotropy of the upper part of the geological section on the recorded seismic signal using a realistic model of the Orenburg field as an example.

Materials and methods. A seismogeological model describing the Lower Permian interval of the Orenburg geological section is considered. According to well data, the elastic properties of geological formations were estimated: density and propagation velocities of longitudinal and transverse waves. There is a high contrast of P-wave velocities estimated from sonic logs. The reservoir in this model is confined to the lower layers. It is composed of sulfate-carbonate media, uniform in density and acoustic properties. Using the grid-characteristic method, zero-offset synthetic seismograms were calculated. The choice of structural curvilinear computational grids made it possible to correctly consider the relief of the day surface.

Research results. In this work, two different models were compared. The first model included the anisotropy of the upper part of the section and the topography of the day surface. In the second model, the upper boundary of the computational domain was flat, and the entire medium was considered within the framework of an isotropic linear elastic model. The analysis of synthetic seismograms showed that the anisotropy inherent in this model does not significantly affect the recorded seismic wave field. However, considering the relief of the day surface significantly shifts the times of arrival of reflected waves.

Discussion and Conclusion. The algorithm presented in the paper can be used to verify the field data processing graph, since the assessment of the anisotropy of the medium is a standard step in building a velocity model. The presented approach can be extended to 3D models of realistic dimensions.

Introduction. Reliable prediction of indicators of turbulent flows is a very difficult task, which is explained by the exceptional physical complexity of turbulence, in particular its probabilistic nature, a wide space-time spectrum and a fundamentally three-dimensional non-stationary nature. Despite conducting a wide range of studies focused on the problem under consideration, they did not fully reflect the totality of various factors and processes affecting the structure and parameters of vertical turbulent mixing. This indicates the need for a systematic analysis of the problem and modeling of such complex formalized systems. The aim of the work is to construct a scenario of changes in hydrodynamic wave processes of the coastal zone, based on an improved mathematical model of wave processes.

Materials and methods. The article is devoted to the study of spatial-three-dimensional wave processes in shallow water bodies, taking into account the features of turbulent exchange depending on the source and localization in the column of liquid, as well as the study of the influence of regular wave processes on turbulent exchange and vertically using a mathematical model of wave processes based on the system of Navier-Stokes equations, including three equations of motion in the with dynamically changing geometry of the computational domain.

The results of the study. Based on the developed software package, a scenario of changes in hydrodynamic wave processes of the coastal zone is constructed, the formation of vortex structures is predicted. Discussion and conclusions. The separation of the wave flow into a near-surface macroturbulent layer caused by wave motion and a lower layer with background hydrodynamic turbulence is proved, the strength and intensity of turbulence changed synchronously with wave oscillations, demonstrating a pronounced asymmetry of turbulence generation throughout the water column.

Introduction. The problem of creating a set of criteria for practically substantiated computational modeling of a number of complex staged biophysical processes with pronounced stages and critical transformations, for example, aggressive invasions, is discussed. Known models have a variety of behavior with the occurrence of bifurcations according to the same scenarios, the appearance of cycles, the coexistence of which is determined by Sharkovskii’s theorem. In the limit of complication of cyclic behavior in such models, they often encounter chaotization of the trajectory, but with the existence of an infinite number of periodicity windows. The conditions for an infinite cascade of bifurcations for iterations are determined by the fulfillment of the conditions of Singer’s theorem. The purpose of this work is to show that most of the nonlinear effects associated with chaotization scenarios do not have an ecological interpretation, but we will propose ways to exclude non-interpretable parametric ranges.

Materials and methods. Using methods for estimating the stability of stationary states and cyclic trajectories using Singer’s theorem on the criterion for the occurrence of bifurcations for iterative models, we analyze interconnected nonlinear effects. The phenomena are considered on the example of cascades of the appearance of cycles of the period p = 2ⁱ + 1, i→∞ and a cascade of cycles p = 2ⁱ – 1, i→0 of “doubling” or “halfing” the period, which occur in ecological models often used to optimize fishing.

Results. It is confirmed that the coexistence of nonlinear effects turns out to be contradictory if the simulation results are interpreted in the field of biocybernetics, on the basis of model and real examples. Iterative models generate unnecessary non-linear modes of behavior, when predicting the dynamics of invasions or harvesting bioresources, taking into account the regulatory impact, for example, in the case of the well-known Feigenbaum scenario. It has been established that bifurcations connected in one scenario have no explanation in ecological reality and are not reflected in the observed biophysical systems. These mathematical artifacts are common to several biophysical models that are very different in their theoretical foundations. Chaotization in real population dynamics has somewhat different properties than can be obtained in a cascade of period doubling bifurcations. The formation of a non-attractive chaotic set in the form of a strange repeller is more consistent with the dynamics of the development of fast invasions.

Discussion and conclusions. It is shown that to describe the transformations of biosystemic processes with external influence, as the collapse of a commercial population, it is adequate to use models with the emergence of alternative attractors. These models correspond better to the transitions between the states of populations under the influence of fishing than models with the implementation of cascades of bifurcations of cycles, strange Cantor attractors and chaos regimes in the form of a continuum of unstable trajectories of all periods. The most promising are hybrid models of the life cycle with developmental stages for essential interpretation in ecology and forecasting of biosystems, as they allow to determine the parametric ranges of functioning and exclude unacceptable ranges of parameters where excessive nonlinear effects occur, which have no justification for population processes. The analysis of the adequacy criteria is based on degradation scenarios for a complexly structured sturgeon population in the Volga basin, cod off the coast of Canada, outbreaks of invasive insects, and the spread of the invasive ctenophore Mnemiopsis leidy in the Caspian Sea.

Introduction. The main idea of the grid-characteristic method is to take into account the characteristic properties, systems of hyperbolic equations, and the finite velocity of propagation of perturbations in the simulated media.

Materials and methods. The simplest hyperbolic equation is a one-dimensional linear transfer equation. To increase the order of approximation of the grid-characteristic scheme to the second, you can use the Bim-Warming scheme. If we use a four-point pattern, we get a central Lax-Vendroff scheme. Difference schemes for the linear transfer equation can be obtained using the method of indefinite coefficients.

Results. The grid-characteristic scheme admits a conservative variant, which is relevant if there are discontinuities (shock waves, shock waves) inside the integration domain, while the original system of equations for a matrix with constant coefficients, in partial derivatives, should be presented in a divergent form.

Discussion and conclusions. When numerically solving a three-dimensional problem, the construction is performed similarly, in the case of upper and lower bounds, after scalar multiplication of the scheme by eigenvectors, relations approximating the compatibility conditions with the first order of accuracy are obtained.

Introduction. The formation of salt deposits and oilfield equipment’s corrosion in most oil fields has become particularly relevant due to the increase in the volume of oil produced and the increase in its water content in recent years. The deposition of salts in the formation and wells leads to a decrease in the permeability of the oil reservoir, the flow rate of wells. The aim of the work is to use machine learning algorithms to simulate the effects of an electromagnetic field on the processes of salt deposition and corrosion. Prediction of experimental results will allow faster and more accurate experiments to establish the influence of electromagnetic fields on the processes of corrosion and salt deposition.

Materials and methods. Three groups of data were used, to train the models, differing in the composition of the studied initial model salt solution: the waters of the Vyngapurovsk’s and Priobsk’s deposits, as well as tap water. The following machine learning models were used: linear regression with Elastic-Net regularization, the k-nearest neighbors algorithm, the decision tree, the random forest and a fully connected neural network.

Results. The processes of electromagnetic field influence on the formation of salt deposits and corrosion of oilfield equipment were simulated with the help of machine learning algorithms. Python program has been developed to predict the output results of experiments. Modeling with various models and their parameters is carried out.

Discussion and conclusions. It was found that the decision tree and the random forest have the best accuracy of predictions, from the experiments conducted. This is due to the fact that there is too little data in the training samples. With the increase in the number of observations, it is worth using neural networks of various architectures.

Introduction. Рolyarylene phthalides (PAF) are widely used in optoelectronics today. The reactions occurring during the synthesis of polyarylene phthalides have a complex character, which has not yet been described using mathematical models. In this regard, it is impossible to use PAF in many processes. Рolyarylene phthalides have luminescence, good optical and electrophysical properties. The elucidation of the mechanisms of the occurrence of luminescent states of PAF is of both fundamental and practical interest. The elucidation of the mechanisms of the occurrence of luminescent states of PAF is of both fundamental and practical interest. Due to the complexity of calculating the kinetics of the luminescence intensity of polyarylene phthalides using known mathematical models, the aim of the study was to build a system using machine learning methods that predicts luminescence values depending on temperature and heating time.

Materials and methods. Experimental data have been prepared for calculations, the use of “random forest” and “gradient boosting” methods has been justified, a method for selecting hyperparameters of these models has been selected and the expediency of its use has been justified, optimal models have been constructed and predictions have been obtained.

The results of the study. An algorithm for predicting the luminescence intensity of polyarylene phthalides has been developed. Using machine learning methods based on experimental data, the key hyperparameters of the system were determined and the average accuracy of predicting values was achieved — 80 %.

Discussion and conclusions. High-accuracy forecasts will allow predicting how products containing polyarylene phthalides will react to external influences. The paper presents two methods for solving the problem, as they showed the best results.

Introduction. The lack of stationarity in the time series’ development, small samples, the outliers presence, jumps do not allow to find estimates of model parameters that have good properties of statistical estimates and, as a result, to find reliable forecasts, both in the development trend of the process and in its numerical expression. The means to solve these problems is the use of ordinal or robust statistics. Scientific monographs and special chapters in books on mathematical statistics contain a deep and extensive theory on the study of the properties of ordinal statistics, which are the justification for their application in forecasting methods. The aim of the work is to develop and verify method for obtaining one-step forecasts of trends in the development of time series based on stable statistical estimates.

Materials and methods. The article presents the results of the development of a method for obtaining one-step forecasts of trends in the development of time series based on the construction of confidence intervals of a selective stable HodgesLehman estimate based on Walsh averages. In particular, the Hodges-Lehman median is used to solve the problem of small samples obtained during the procedure of shifting the time series window. The proposed method is considered in detail in the article: the basic definitions, the theoretical justification of the method, calculation formulas, a detailed description of the algorithm, formulas for calculating the quality metric of forecasts are given.

The results of the study. The method was implemented in computational experiment using the example of forecasting the Urals crude oil’s spot price. The article presents the computational experiment’s results. The parameters of the proposed method can be configured to obtain reliable one-step forecasts.

Discussion and conclusions. The method proposed in the article has shown its effectiveness on experimental data and can be used as an independent method for constructing one-step forecasts of trends in the development of time series. Further development of the method involves the improvement of computational procedures, verification of the method in case of jumps in the dynamics of the time series.