The work is devoted to modeling the movement of liquid and gas in a porous medium containing gas hydrate, taking into account the force of gravity. Taking gravity into account is necessary to study a number of problems associated with the rise of gas during the decomposition of gas hydrates. Such problems include studying the causes of large-scale gas flows from the ocean floor, studying the degree of influence of gas hydrates on geology and ecology, and determining the optimal methods for developing gas hydrate deposits. For numerical modeling of filtration problems in formations of complex geological and lithological structure, the support operator method is used, which makes it possible to carry out calculations on irregular grids, allowing for a detailed approximation of curved heterogeneous formations. The developed algorithms are tested on the problem of gravitational differentiation of gas and water in a porous medium taking into account gas hydrates. Analysis of the calculation results showed that the method can be used to solve problems of gas hydrate fluid dynamics in complex reservoir systems considering gravity.

This work presents a mathematical model for solving three-dimensional radiation problems of magnetohydrodynamics. An implicit fully conservative difference scheme is used to solve the system of differential equations. Two methods are used to solve the system of difference equations: the method of separate and the method of combined solution of equations, which are split by physical processes. A software implementation of the developed numerical algorithms is carried out, and calculations are performed modeling the compression of plasma by a magnetic field. The time dynamics of the parameters of matter and the magnetic field are studied. During the calculation process, at its various stages, both numerical methods used in the program are involved. The results obtained correspond to the physics of the process. В данной работе представлена математическая модель для решения трехмерных радиационных задач магнитной гидродинамики. Для решения системы дифференциальных уравнений применена неявная полностью консервативная разностная схема. Используется два метода решения системы разностных уравнений: метод раздельного и метод комбинированного решения уравнений, которые расщеплены по физическим процессам. Произведена программная реализация разработанных численных алгоритмов, выполнены расчеты, моделирующие сжатие плазмы магнитным полем. Изучалась динамика по времени параметров вещества и магнитного поля. В процессе расчета на его различных стадиях были задействованы оба используемых в программе численных метода. Полученные результаты соответствуют физике процесса.

The work is devoted to modeling the processes of spraying the nanoparticles transported by supersonic cold gas flow on substrates. This problem is relevant for the implementation of many nanotechnologies, for example, for the production of ultra-high resolution video systems, nanolithography technologies, etc. Experimental work in this area is based on theoretical analysis and mathematical modeling methods. Both the spraying process itself and the possibilities of controlling the quality of the resulting surface are studied. For this, a multiscale approach is often used combining models of continuum mechanics and particle models. Such combination makes it possible to describe the macroscopic properties of the gas flow carrying nanoparticles and its interaction with the substrate at the level of individual atoms and molecules. In this work, two scale levels are used. The macroscopic model is based on the equations of quasigasdynamics, the microscopic model is based on the equations of molecular dynamics. These two descriptions are used both separately and together. To analyze the full cycle of the spraying process, an original set of algorithms for pairing these models is proposed. The approach was tested on the example of spraying the nickel nanoclusters on a substrate of the same material and showed its efficiency.

The work discusses a web laboratory designed for a supercomputer modeling of spraying processes. The main problems it solves are the unification of interaction with various remote supercomputers and the automation of computational experiments through an interactive graphical user web interface. The work describes the architecture and the main technology stack used to build the laboratory, and also provides the results of embedding a specific application.

This study presents a thermodynamically equilibrium filtration model for the gas-hydrate phase-equilibrium (invariant) and hydrate-thaw zones with two components (Н2О, gas), taking into account the ice-water phase transition. The Н2О components (liquid water and ice) and the gas outside the hydrate form a water-ice and gas mixture in a porous medium. The study of hydrates in the permafrost zone is of significant practical importance for understanding the processes related to climate change. To implement the model proposed in this study, the method of splitting by physical processes is applied; and the system is transformed to a block form with the separation of the dissipative and hyperbolic parts. The developed mathematical model is common for the entire area of the process and makes it possible to study gas-hydrate and water-ice phase transitions due to the use of the original enthalpy form of the piezoconductivity equation. The change in the enthalpy value in the internal process of the phase transformation of the water-ice mixture makes it possible to model the internal evolution of phase transitions, in particular, the volume fractions of the water-ice structure. This model is discretized. Discrete algorithms are integrally consistent, which allows maintaining the exact balance of the mass components (Н2О, gas) and the total internal energy of the entire system at the difference level. A software implementation is developed, with the help of which a series of calculations are carried out. The calculations show that in the water-ice zone, over time, a phase ice-hydrate transformation occurs with the melting of ice, which is energetically compensated by the formation of a hydrate.

The process of two-phase filtration in a carbonate formation of fractured-pore type is considered. A mathematical model in a spatially two-dimensional formulation is proposed, a numerical method for solution and a parallel algorithm for its implementation are developed. The mathematical model is based on the Buckley-Leverett approach. The reservoir takes into account the exchange of fluids between low-permeability pores and natural fracturing, specified within the framework of the dual porosity model. The numerical algorithm is based on the use of the finite difference method and the splitting scheme by physical processes. To speed up calculations, a parallel algorithm based on two-dimensional domain decomposition is used. Numerical experiments were carried out, which showed that the developed algorithm is highly efficient and allows one to calculate the necessary characteristics of the modeled physical process.

This paper studies the development of a multiscale approach to calculate the gas flows near solid surfaces taking into account microscopic effects. In this line of research, the problem of setting boundary conditions on the surface of a solid body is considered, taking into account the effects preliminarily calculated at the atomic-molecular level. The main aim of this paper is to formulate macroscopic boundary equations that take into account the processes on the surface of a solid body around which there is a flow of gas. The macroscopic model is based on a system of quasi-gasdynamic (QGD) equations in the volume and the thermal conductivity equation in the near-surface layer of the streamlined body. The system is supplemented with real gas state equations and dependencies of the kinetic coefficients of the QGD equations on temperature and pressure, obtained on the basis of molecular dynamics calculations. To test the proposed boundary equations, the problem of the gas flow around a blunt body is considered. Dry air is selected as the gas. Nickel is chosen as the body coating. Calculations are carried out for two values of the inlet velocity. They confirm the qualitative correctness of the developed boundary model and the entire modeling technology.

The work is devoted to the application of irregular computational grids to problems of underground fluid dynamics in the presence of gas hydrates taking into account gravity. Discretization of filtration equations is carried out using the support operator method. The obtained algorithms are tested on the problem of gravitational differentiation of gas and water in a porous medium taking into account gas hydrates. One of the criteria used in testing is the horizontality of the gas-water contact in the stationary case. The calculations are divided into two stages. At the first stage, a regular rectangular grid is used, which makes it possible to check compliance with this criterion. At the second stage, calculations are carried out on an irregular grid for a system of heterogeneous layers of complex geometric structure corresponding to real geological structures. The results obtained correspond to the physics of the process. The approach can be used to study the migration of deep gas released during the dissociation of gas hydrates to the earth's surface.

The article considers the methodology of mathematical modeling of the process of hydrocarbon fluid isomerization in a catalytic reactor used for the synthesis of organic fuels. A model for calculating the parameters of fluid flow in a mixed medium "open space – porous body" has been developed, taking into account the properties of the porous material and the main chemical transformations characterizing hydroisomerization. The model includes the Navier-Stokes equations regularized on the basis of the quasi-hydrodynamic approach, averaged over a representative elementary volume, and a system of convection-diffusion equations for calculating the evolution of the concentrations of raw materials and the reaction product. In the spatially two-dimensional case, a numerical algorithm for solving the problem has been developed; its software implementation has been performed. Trial calculations of the model problem have been carried out, which have shown the correctness of the developed numerical approach and the operability of the created software code.

An approach for describing the metric properties of a difference mesh for discretizing repeated rotational operations of vector analysis as applied to modeling electromagnetic fields is proposed. Based on the support operator method, integral-consistent operations (gradient, divergence and curl) are constructed, which are necessary to obtain estimates of the convergence of difference schemes for repeated rotational operations designed to solve specific problems of magnetohydrodynamics. Using smooth solutions of a model magnetostatic problem with first-order accuracy, the convergence of the difference schemes constructed in this work with a zero eigenvalue of the spectral problem is proved. In this case, no restrictions are imposed on the difference tetrahedral mesh, except for its nondegeneracy. Calculation of electromagnetic fields for a three-dimensional problem of magnetic hydrodynamics in a two-temperature approximation with the full set of spatial components of velocity and electromagnetic fields is presented. The dynamics of electromagnetic fields is developed against the background of rotational diffusion of the magnetic field vector.

This study proposes a spatial two-component (H2O, CH4), three-phase (hydrate, free water, and gas) filtration model, taking into account the dissociation of gas hydrates, based on splitting by physical processes, using a nonclassical law of motion (taking its nonlinearity into account). The presented mathematical model makes it possible to calculate two-dimensional flows in areas with an irregular strata structure. With its help, it is possible to carry out both profile and areal calculations, taking into account the complex geometry of sedimentary basins. When testing it to solve problems of the theory of filtration in sedimentary basins, the method of support operators is applied and implemented. This method makes it possible to calculate filtration processes in media with discontinuous physical properties, which is achieved by using irregular meshes. As a result, it becomes possible to model the shear zones and obtain a numerical solution under conditions of different scales of the problem. At the same time, on meshes with large cells, where there are discontinuities in the material properties, a qualitative approximation of the transfer of the saturation and gradients of the thermodynamic quantities is preserved. The constructed mesh model also approximates the identities of the support operator method on different time layers. Based on the developed computing technology, a software package is created, whose tools are capable of solving two-dimensional problems of multiphase and multicomponent modeling of gas hydrate dissociation processes in the porous environment of sedimentary basins of a lithologically complex structure on meshes of an irregular structure. To test the software package, model calculations of piezoconductive processes in a three-phase medium with hydrated solid-phase inclusions in a two-dimensional case on irregular meshes are carried out. The calculations show a decrease in the depression value within the spatial regions when using nonlinear filtration laws of motion in the medium compared to the classical Darcy law, which makes it possible to correctly describe the physics of low-permeability reservoirs.

Spraying the nanoparticles on a substrate is an actual and promising technology in many industries. The theoretical study of this process in various conditions is often implemented using computer modeling. In this work, a comprehensive methodology for modeling spraying processes is presented. The methodology is based on direct atomic-molecular calculations. Parallel technologies are used for the computer implementation of the methodology, which allow obtaining results with a given level of resolution and accuracy. Various aspects of the developed technology and computation results are discussed on the example of the interaction of nanoparticles with a substrate consisting of nickel atoms.

Problem of modeling the interaction of metal nanoclusters in very large-scale integrated circuits (VLSIC) interconnects is considered in the context of improving manufacturing technologies for promising microelectronics instruments and devices. This problem is relevant when analyzing the performance and durability of VLSIC, the main components of which are obtained using epitaxy, sputtering and lithography methods. At the current stage of development, a feature of such technologies is the transition of the VLSIC element base to the nanometer range, which significantly increases the quality requirements for all their components.The work presents a supercomputer technology for atomistic modeling, which is proposed to be applied to the numerical analysis of the degradation problem for ultra-thin interconnects of VLSIC elements. It includes the construction of a mathematical model, its parallel numerical implementation and test calculations. The result of the work is a numerical analysis of the interaction processes between copper and tantalum nanoclusters, which are typical components of interconnects. The data obtained as a result of the analysis are consistent with theoretical ideas about the processes occurring in such microsystems.

The work provides a description of the spatial filtration problem in a three-phase hydrate equilibrium zone. A mathematical model is presented for studying two-dimensional fluid flows taking into account the solid hydrate phase and the irregular structure of formations. A non-classical form of the motion law is used, applicable at low permeability and low pressure drops. Efficient computational algorithms based on the support operator method are proposed that make it possible to separate the hyperbolic and dissipative subsystems of the problem. The algorithms are implemented on meshes of irregular structure to model the two-dimensional multiphase processes of gas hydrate dissociation. Testing is carried out on model piezoconductive processes with saturation transfer, where it is shown that depression processes are less expressed when using a nonlinear law of motion.

Предложен подход к численному моделированию неизотермической задачи фильтрации в трещиновато-пористой среде, основанный на методе расщепления по физическим процессам. В задаче учитывается наличие двухфазной жидкости и двойной пористости у коллектора. Применение метода расщепления по физическим процессам позволяет упростить алгоритм решения, при этом сохранив эквивалентность к консервативной разностной аппроксимации исходных уравнений и обеспечив устойчивость решения задачи. При численном решении используются аппроксимации дифференциальных операторов, полученные в рамках метода конечных разностей. Реализация численного алгоритма основывается на методе матричной прогонки. Апробация метода и его верификация выполнена в серии вычислительных экспериментов, исходные данные для которых взяты из исследований промысловых скважин на российских нефтяных месторождениях. The work proposes an approach to numerical modeling based on splitting by physical processes for a non-isothermal problem of filtration in a fractured-porous medium. The task is complicated by the presence of a two-phase fluid and dual porosity of the reservoir. The system of equations defining the model is complex and is described by a system of strongly non-linear partial differential equations. The use of the splitting method according to physical processes makes it possible to simplify the solution algorithm while maintaining the equivalence to the conservative difference approximation of the original equations and ensuring the stability of the problem solution. In the numerical solution, the approximations of differential operators obtained in the framework of the finite difference method are used. The implementation of the numerical algorithm is based on the matrix sweep method. To test the method, a series of computational experiments were carried out. Calculations have shown that the developed methodology is correct and allows one to simulate the dynamic operating conditions of wells.

This paper studies the convergence of methods of a combined and separate solution of difference equations groups, splitted by physical processes, applied to a family of completely conservative difference schemes (CCDSs) of two-dimensional magnetohydrodynamics (MHD). Estimates are obtained for the convergence of iterative processes for the entire family of CCDSs, both for the method of a separate and combined solution of groups of difference equations. These results are obtained for the first time; previously, such estimates were obtained only for a purely implicit difference scheme. The validity of the estimates obtained in this study is confirmed by the numerical calculations. Based on the estimates obtained in this study, recommendations are developed for any CCDS, whose numerical method is more appropriate to use to solve the system of difference equations. Depending on the ratio of the parameters of the substance and the electromagnetic field at each moment of time, the estimates obtained in this study, even for calculating one physical problem of two-dimensional MHD, make it possible to choose the optimal numerical method for each time integration step, which leads to a significant reduction in the computational time of the problem. This can be quite important, especially when conducting a large-scale computational experiment. Thus, the results obtained in this study have not only an interesting theoretical but also an important practical value.

The paper is devoted to the development and computer implementation of a numerical method for modeling the interaction of metal nanoclusters with a substrate at the mesoscopic level. This research as a whole is relevant in connection with the development of nanotechnologies for obtaining extremely thin metal coatings by various spraying methods. From a practical point of view, its relevance is determined by the lack of adequate mathematical models at the mesoscopic level to describe processes in the submicron size range. This work presents the mathematical model of a metallic medium consisting of spherical nanoclusters and a parallel numerical algorithm for its implementation. The model includes Maxwell's equations of electrodynamics to describe the evolution of the electromagnetic field, as well as the averaged equations of Newtonian dynamics to describe the motion of individual nanoclusters and the electron gas surrounding them. The numerical algorithm is based on the method of grids and the integration of the equations of motion of the particles. The algorithm is parallelized with respect to both space and particles. We devised a set of parallel programs and carried out preliminary model calculations. Nickel is used as the material for the nanoclusters and the substrate. The conducted numerical experiments show the efficiency of the proposed computer model.

In the present work, the support operator method for spatial problems of elasticity theory is used to construct a finite-difference approximation of elastic forces on staggered Lagrangian meshes. For displacement vectors on irregular meshes, as applied to difference schemes, for problems of elasticity theory, the corresponding discrete operations have been developed. Taking into account the energy balance of the medium, the presented families of integrally consistent approximations of vector analysis operations are sufficient for numerical modeling of these processes. The resulting forces acting on the nodal domains of the medium are obtained explicitly in two-dimensional and three-dimensional geometry. Calculations are given for the propagation of sound waves in an aluminum three-dimensional orthogonal plate due to an end impact. On the example of numerical calculations, the invariance of the elastic force and energy characteristics of the medium during solid-state rotations is confirmed.

Работа посвящена анализу принципов построения веб-лабораторий, предназначенных для суперкомпьютерного математического моделирования сложных физических процессов и явлений. Основными целями подобных цифровых платформ являются: автоматизация вычислительных экспериментов, формирование базы знаний и обеспечение совместной работы исследователей в выбранной предметной области. В работе рассмотрены существующие сегодня решения и принципы построения предметно-ориентированных платформ. На основе выполненного анализа была разработана и практически реализована веб-лаборатория, связанная с решением задач напыления. В работе приведены архитектура и детали программной реализации платформы. Основными ее преимуществами являются возможность динамического встраивания проблемно-ориентированных приложений и удаленных вычислительных ресурсов. Реализованная веб-лаборатория была протестирована посредством проведения серии вычислительных экспериментов по модельным задачам сверхзвукового холодного газодинамического напыления. Также в работе предлагаются направления для последующего развития оригинальной веб-лаборатории. The work is devoted to the analysis of the principles of constructing the web laboratories intended for supercomputer mathematical modeling of complex physical processes and phenomena. The main goals of such digital platforms are the automation of computational experiments, formation of a knowledge base and ensuring collaboration of researchers in a subject area. The work discusses the solutions that exist today and the principles for constructing subject-oriented platforms. Based on the analysis performed, a web laboratory related to solving spraying problems was developed and implemented. The work provides the architecture and details of the software implementation of the platform. Its main advantages are the ability to dynamically integrate problem-oriented applications and remote computing resources. The implemented web laboratory was tested through a series of computational experiments on model problems of supersonic cold gas-dynamic spraying. The work suggests directions for the subsequent development of the original web laboratory.

The work is devoted to the development of multiscale approaches for modeling the processes of supersonic cold gas dynamic spraying of nanoparticles on the substrates. Modeling of gas dynamic spraying processes involves solving two practical problems: a) controlled transportation of nanoclusters to the spraying place in the general gas flow; b) analysis of the interaction of nanoclusters with the substrate surface in its boundary layer. A combination of these problems is considered, which is implemented on the basis of a multiscale calculation of a two-phase flow of a gaseous medium with the inclusion of finely dispersed solid metal particles. The work proposes a new multiscale computing technology that combines macroscopic, microscopic and mesoscopic descriptions of the physical processes under study at the model level. Within its framework, two two-scale approaches are used, taking into account, respectively, macro- and micro- and macro- and mesolevels of spatial detail. The macroscopic components of the final mathematical model are based on modified quasigasdynamic equations for analyzing the flow of a two-phase multicomponent gaseous medium, as well as Maxwell’s equations near the substrate surface for calculating the effect of electromagnetic fields on the solid phase. Newton’s dynamics equations are used to describe processes on micro- and mesoscopic scales. In the first case, these are the equations of molecular dynamics, written taking into account the pressure forces in the gas phase and mechanical stresses in the solid phase. In the second case, these are the equations of particle dynamics, taking into account mainly the Lorentz force. For the numerical implementation of the macroscopic components of the model, the grid method of finite volumes is used; for the micromodel, the Verlet scheme is used; for the mesomodel, a symmetric scheme is used that approximates the equations of Newton’s electrodynamics. The aims of the work were a physically substantiated formulation of all model components and preliminary calculations of the motion of a nickel nanocluster accelerated by a supersonic nitrogen flow near a nickel substrate.