Home – Home – Jornals – Numerical Analysis and Applications 2021 number 3

This paper discusses the contribution of two outstanding petroleum geologists, Academicians I.M. Gubkin and A.A. Trofimuk, to the development of the oil and gas complex of the Soviet Union and Russia. It presents a brief description of their biographies and the main results of their scientific, pedagogical, production, and organizational activities. The paper is a detailed preface to the special issue of the Russian Geology and Geophysics Journal dedicated to the 150th birth anniversary of I.M. Gubkin and the 110th birth anniversary of A.A. Trofimuk. A brief description of the papers published in the special issue is given.

This study discusses the evolution of the composition of dispersed organic matter from the Bazhenov Formation (West Siberian petroleum basin) and the products of its catagenetic transformation on the basis of the balance and kinetic approaches to modeling of the catagenetic transformation of organic matter and its individual components, primarily kerogen. The results show that the variations in the elemental composition of kerogen and the extent of generation of both hydrocarbons and nonhydrocarbons can be quantitatively described using a simplified kinetic model. Preliminary estimates of the model parameters are given for the averaged Bazhenov-type kerogen. It is shown that numerical modeling of the catagenetic transformation of dispersed organic matter confirms the validity of the recognition of the main phase (zone) of oil generation and the main phase (zone) of gas generation.

A comprehensive study of Domanik deposits of the Timan-Pechora Basin has been carried out. The examined composition of hydrocarbon biomarkers, chemical structure of kerogen, carbon isotope composition, and rock lithology, the Rock-Eval pyrolysis data, and the contents of bitumen and C_org in the rocks give an insight into the geochemical processes in the oil window in the Domanik deposits, which took place at T_max = 435-450 ºC. The bitumen coefficient β^CB is maximum in this temperature interval, reaching 30 %. The obtained data on the distribution of polycyclic biomarkers in the Domanik rocks and the bituming and Rock-Eval pyrolysis data allowed determining the boundary values of biomarker maturity coefficients in the study of the maturation of organic matter of the rocks. The carbon isotope composition of bitumen fractions in the Domanik rocks is considered, and the bimodal distribution of the δ¹³C values of the bitumen is shown.

The homologous series of high-molecular weight dimethylalkanes (HMWDMAs) with either odd- or even-numbered carbon chains in the range from C_19-20 to C_30-31 have been identified in organic matter from recent and partially lithified deposits of Siberia and the Russian Platform by chromatography-mass spectrometry. The first homologous series is represented by even-numbered 3,4-HMWDMAs followed by the alternation of odd-numbered 3,5-HMWDMAs, even-numbered 3,6-HMWDMAs, and odd-numbered 3,7-HMWDMAs. The most abundant are 3,7-dimethylalkanes. The microbial origin of high-molecular weight dimethylalkanes is the most likely explanation for their presence in the fossil organic matter. The precursors of HMWDMAs might have been tetra- and diether lipids of archaea and bacteria. It is assumed that HMWDMAs and other immature hydrocarbons from great depths (SV-27 and SG-6 superdeep boreholes) result from the decomposition of asphaltenes, which occluded the related compounds inside their structure during the early stages of generation and carried them unchanged throughout the «oil window».

We substantiate certain ideas concerning the key role of fluid-geodynamic processes in the evolvement of hydrocarbon accumulations at great depths in the Earth’s crust. The presented geodynamic model of oil and gas accumulation is based on updated ideas of the structure of the Earth’s tectosphere, which includes plate, preplate, and folded complexes. The model makes clearer the spatial scale of the organic-matter transformation into hydrocarbons of the oil series. In the bottom layers of the Earth’s crust, we predict the existence of a special stagnation type of water drive systems with the following distinguishing features: (a) different scales of manifestation, from local to regional; (b) a limited water exchange with the external environment; (c) the absence of persistent drainage horizons (beds and interbeds); (d) alignment of hydrodynamic potentials in terms of depths and laterals; (e) certain growth in the role of lithohydrochemical and organic chemical factors in the development of the void space of the fluid-host medium. In their inner space, systems with difficult water exchange can control the evolvement and preservation of autoclave hydrocarbon systems for a long time, the key feature of the autoclave systems being spatial coincidence (localization) of the processes of oil and gas generation and accumulation. We assume that in the settings of all-round compression, hydrodynamic instability, and no drainage, the position of productive zones must be controlled by foci of low pore (reservoir) pressures rather than local hypsometric highs. We present results of the prediction for the development of water drive stagnation systems in the subsalt deposits of the Caspian depression. A prediction for reservoir pressures was made for the sedimentary cover at great (and supergreat) depths. It can be regarded as a necessary component of prediction of oil and gas potential because it makes it possible to delineate new (previously unknown) commercial zones of hydrocarbon accumulation.

The Vendian deposits of the Siberian Platform are represented by four regional horizons (from bottom to top): Nepa, Vilyuchan, Tira, and Danilovka. The TDS of brines in the Nepa and Vilyuchan horizons varies from 170.3 to 470.1 g/L³, and the TDS in the overlying Tira and Danilovka horizons, from 84.6 to 583.1 g/L³. Brines of Cl Na, ClNa-Ca, Cl Ca-Na, Cl Ca, and Cl Ca-Mg types have been identified. The pH values of the brines vary from 3.0 to 7.0; acidic and weakly acidic brines (pH = 3.0-5.5) prevail. The redox potential Eh varies from -440 mV (highly reducing conditions) to +130 mV (oxidizing conditions), averaging -176 mV. Areal regional hydrogeochemical zoning is controlled by the outer (on the periphery of the Siberian Platform, where infiltration of meteoric waters took place) and inner feeding areas. The outer feeding areas bear infiltrogenic brines of different ages (group I), mostly of Cl-Na composition. They are characterized by TDS ≤ 280 g/L³, r Na/ r Cl ≤ 1.55, Cl/Br ≤ 2512, and average Ca/Cl = 0.3. The average values of the integrated brine metamorphization index S (according to S.L. Shvartsev) range from 50 to 200. In the intraplatform feeding areas, there are several phases of brines that underwent gravitational sinking into the Vendian horizons in the periods when Cambrian salt-bearing basins existed on the platform. These brines were metamorphosed during filtration along the fracture and fault zones. The intrusion of traps in the late Permian-early Triassic was accompanied by hydrothermal activity. The waters of the enclosing Cambrian dolomite horizons were heated to 800-1000 ºC; their subsequent interaction with carbonate rocks and evaporites (in particular, chlorides) resulted in extrastrong saturated brines. These brines were also subjected to gravitational sinking into the Vendian reservoirs, which was the main cause of the inverse vertical hydrogeochemical zoning in the hydrogeologic subsalt formation. The brine metamorphism processes and temperature changes in the rock horizons during the intrusion of traps and their subsequent cooling led to a significant alteration of the enclosing rocks under interaction with brines and to salinization of the reservoirs, an increase in the total salinity of the brines, and catagenetic changes in their composition. The brines closest to the maximum saturation with halite were found in reservoir zones with the highest salinization and near faults and boundaries of intrusive bodies. During the pre-Vilyuchan, pre-Dnieper, pre-Tira, and pre-Danilovka sedimentation breaks, infiltrogenic waters penetrated into the Vendian sediments. Geochemically, the sedimentogenic, infiltrogenic, and metamorphic strong brines (group II) have a predominantly Cl-Ca or Cl-Ca-Mg composition with TDS > 350 g/L³. They are characterized by low r Na/ r Cl ratios (on average, 0.3), Cl/Br ≤ 100, high Ca/Cl values (on average, 0.4), and S > 250.

The general Vendian stratigraphic scale of Siberia, with the uncertain age of the Vendian base ranging from 600 to 630-640 Ma in most of recent publications, remains worse constrained than the Cambrian scale, in which the boundaries of epochs and stages have been well defined. However, the imperfect classical stratigraphic division has been compensated by data on the cyclicity of the Vendian-Cambrian sedimentary section. The Vendian stratigraphy of the Siberian Platform and the related deposition history with cycles of sedimentation and gaps, as well as the hierarchy of sedimentation processes, can be inferred from the succession of alternating clastic, carbonate, and salt units. The cyclicity of geologic processes and their recurrence are attributed to periodic oscillatory motions of the crust. The ranks of these motions correlate with the cyclicity of sedimentary strata, including regocyclites, nexocyclites, and halcyclites separated by gaps. Each Vendian long-period oscillatory motion begins with a regocyclite and ends with a regional-scale gap. The Cambrian section includes one pre-Mayan regional gap at the end of the early Cambrian long-period cycle. Cambrian regocyclites are composed of carbonate subformations and formations in the lower part and alternating salt and carbonate beds in the upper part.

We present a model of the stratigraphic and lateral distribution of Cambrian deposits in the Vilyui hemisyneclise, based on an analysis of drilling data and interpretation of seismic data. The study shows a series of formations and sequences penetrated by wells (Syugdzher saddle, Khorgochum monocline, Ygyatta depression, Tyukyan-Chybyda monocline, Arbai-Sinyaya megaswell, etc.). In the areas where the Cambrian was not penetrated by wells, the distribution of Cambrian deposit was inferred based on the available seismic data. The distribution of the Kuonamka Horizon formed by Cambrian organic-rich rocks is characterized in detail. These are the Kumakh and Sinyaya-Kutorgina sequences and the Inikan and Kuonamka formations. It has been found that the Kuonamka Horizon was deposited during two stages, Botomian and Toyonian-early Mayan. The horizon is overlain by younger deposits of the Mayan Stage, with characteristic cross-bedding structures. Schemes of facies zoning of the Cambrian for Botomian, Toyonian-early Mayan, and middle Mayan times were constructed based on the most recent geological and geophysical understanding of the Vilyui hemisyneclise.

The paper considers the regularities in the structure and conditions of formation of Domanik carbonaceous carbonate-siliceous productive deposits. They are shown to occur in the stratigraphic interval from middle Frasnian to upper Famennian. The highly persistent structure of their sections for many kilometers within specific sedimentation zones and the drastic changes in the structure at the boundaries of the zones are justified. Lithological classification of rocks is considered. Methods and results of combined paleontological, lithological, geochemical, and petrophysical core studies are presented, as well as interpretation of well logging data and seismic-survey materials used in the assessment of oil resources and identification of oil- and gas-promising zones and areas. The conducted studies have proved significant oil resources in the Domanik productive sequence; their extraction might compensate for the decline in oil production from conventional pools.

The paper considers the seismogeologic, structural, and tectonic features of Neoproterozoic-Paleozoic and Mesozoic sedimentary complexes in the Arctic regions of the Siberian Platform. Based on the results of deep drilling, the geologic structure of the study area was analyzed, and the key sections of Neoproterozoic-Paleozoic deposits of the Anabar-Khatanga and Lena-Anabar oil and gas areas (OGA) were compiled. Analysis of geological and geophysical materials showed the existence of a sedimentary basin up to 14-16 km in thickness on the continental margin of the Siberian Platform, with five regional seismogeologic megacomplexes in its section: Riphean, Vendian, lower-middle Paleozoic, Permian, and Mesozoic. Based on the results of a complex interpretation of CDP seismic-survey and deep-drilling data, a structural and tectonic analysis was performed, structural maps were compiled for all reference stratigraphic levels, and a conclusion has been drawn about the similarity of the structural plans of the Riphean top and overlying sedimentary complexes. Using a structural map along the Permian top, a tectonic map of the study area was compiled, which corresponds to the current state of study. The results of numerical modeling of the salt diapir formation processes are presented, and the types of anticlinal structures, potential oil- and gas-promising objects, are considered.

The paper presents results of regional paleogeographic reconstructions of the West Siberian sedimentary basin in the Late Bajocian-Bathonian. Regional paleogeographic maps of the Yu₄, Yu₃ and upper part of the Yu₂ reservoir units were constructed and described for the first time ever. The implemented approach provided insights into the evolution of paleolandscapes and highlighted the deposition features of the Upper Tyumen Subformation and Malyshev Formation in the different parts of the West Siberian basin. The compilation of paleogeographic maps was based on the electrofacies analysis performed for individual parts of the Malyshev stratigraphic horizon, with regard to the core description materials, paleontological, sedimentological, geochemical data and other. The paleogeographic control of the reservoir’s formation in the Bathonian regional reservoir is discussed.

The paper presents estimates of recent numerical visibility forecast methods in the atmosphere. METAR reports for 786 airports in Europe and central Russia for the period 01.02.2020-01.08.2020 were used as visibility standards. The estimates were summarized in three ranges: poor visibility (0-1500 m), satisfactory (1500-3000), and good (3000 or more) for certain regions of Western Europe and the European territory of Russia. It is shown that the forecast of poor visibility by all the considered methods for up to 36-48 hours can be satisfactory. For a longer period, the accuracy drops noticeably, and the forecast using the method presented by the authors shows high accuracy for low visibility in most regions. The rest of the methods make it possible to predict good visibility.

A method for estimating cross sections of excitation transfer in collisions with atoms of rare-earth metals (REM) is presented. The cases of excitation transfer in collisions of REM atoms in unscreened resonant states 6 s 6 p with REM atoms in the ground state, as well as REM atoms in screened resonant states 5 d 6 s ² with inert gas atoms are considered. The determination of the cross sections is based on the fact of collisional population of the upper laser levels from close-lying resonant levels excited by electron impact in REM vapor lasers. The quantities required for calculating the cross sections are determined from the active laser medium parameters and average lasing power. The advantages and disadvantages of the method suggested in comparison with methods based on optical excitation of metal atoms and recording non-resonant fluorescence are discussed.

The values of 13 partial and 15 effective cross sections of excitation transfer in Dy-Dy, Tm-Er, Tm-Tm, and Tm-He collisions are estimated. The active media of Dy, Tm, and Er-Tm vapor lasers with buffer gas He were used as collisional ones. The determination of the cross sections is based on the fact of collisional population of the upper laser levels from close-lying resonant levels excited by electron impact in discharge. The quantities required for calculating the cross sections were determined from the laser medium parameters. The partial cross sections of collisional excitation transfer, depending on the excitation cross sections of resonant levels, type of collisions and energy defect between resonant and upper laser levels are ~10^-16-10^-14 cm², whereas the effective cross sections are ~ 10^-18 - 10^-15 cm².

The Nekrasov integral equation is obtained which describes the steady flow of an ideal incompressible fluid with a free surface above an uneven bottom with a wavy profile. A numerical method has been developed for solving this equation at the coordinates of the bottom profile defined in the plane of the complex variable z = x + iy .

The joint flow of blood and embolization composition inside an arteriovenous malformation is modeled with the use of a one-dimensional model of two-phase filtration based on real clinical data. Numerical simulations are performed by a monotonic modification of the CABARET scheme. Optimal regimes of embolization for real patients are found with the use of a modified particle swarm method, which is a numerical method of global optimization.

The model relating the polymorphic transformation of a crystalline substance under shock wave loading to the change in its elastic energy is further considered, now by an example of boron nitride. The results obtained show that the model provides a reliable description of the martensite phase transition in boron nitride subjected to a shock wave. It is found that the shock adiabat of boron nitride regardless of its structure has an inflection at the shock wave velocity D » 6.2 km/s. The nature of this inflection is not clear yet.

A non-homogeneous system of one-dimensional conservation laws is proposed, which describes propagation of large-amplitude bottom (subsurface) internal waves in multilayer stratified shallow water in the Boussinesq approximation. The model can be applied to layered flows of a stably stratified fluid and is hyperbolic for moderate velocity shear in the layers. Steady solutions of equations of motion are studied, and conditions for the formation of solitary waves of the first mode are formulated. The model is verified by means of comparing the results predicted by this model with the results of actual observations and calculations of two-dimensional equations of motion. Propagation of unsteady nonlinear wave packets in a multilayer fluid is numerically simulated.

The paper considers the evolution of a pulse signal in the annular gap between a diagnostic probe and an open well surrounded by a low-permeability fractured formation. Fractures are located along the well, and the well and the fractured-porous medium are filled with the same acoustically compressible fluid. The problem is solved numerically by the fast Fourier transform. Dispersion equations are obtained that describe the propagation of damped traveling waves in the gap, taking into account fluid filtration through longitudinal fractures. Analysis was performed of the influence of the filtration characteristics of the reservoir, hydraulic fractures, and the width of the gap between the probe body and the borehole wall on the phase velocity, the attenuation coefficient, and the evolution of pulse signals.

This paper describes a mathematical model and features of gas hydrate formation during the injection of natural gas with a given composition into a porous reservoir containing the same gas and water in the initial state. Numerical solutions of the axisymmetric problem are constructed, which describe the distributions of the main parameters in the reservoir both with the frontal surface and with the volumetric region of phase transitions. The influence of the parameters of the injected gas and porous medium on the mode and rate of gas hydrate formation is investigated.

A numerical study of the cooling of a water-saturated porous medium in a cylindrical heat-insulated vessel with a vertical cooling element is performed. The effect of convective heat transfer on the cooling a porous medium saturated with water is carried out, taking into account the density inversion at various values of the porous medium permeability. It is shown that the effect of a maximum water density during the convective motion of water in porous media in the vicinity of zero temperature should be taken into account. The presence of a maximum water density in highly permeable soils restructures the flow and reduces the convective transfer rate, which slows down the cooling of the system.

This paper describes the generation of an induction discharge of atmospheric pressure with ferromagnetic enhancement of magnetic coupling between an inductor and a discharge. It is shown that intensifying a magnetic flux that connects the inductor and the plasma makes it possible to efficiently generate an electrodeless discharge in a low-frequency (10-100 kHz) range and greatly simplifies the use of the inductively coupled plasma. This study also presents new experimental data on the dependence of the electric field strength and the thermal efficiency of a low-frequency induction discharge of atmospheric pressure on the discharge current strength and the flow rate of a plasma-forming gas (argon or air). It is shown how the criterion for generating a low-frequency induction discharge is related to its thermal efficiency is shown, and methods for reducing the heat losses of a low-frequency induction discharge of atmospheric pressure are analyzed.

The ideal gas submodel invariant with respect to time translation and space translation along one direction has four integrals in the case of vortex motion. For the stream function and specific volume, a system of nonlinear differential equations of the third order with one arbitrary element containing the equation of state and arbitrary functions of integrals. Equivalence transformations for an arbitrary element were found. The problem of group classification was solved. The optimal system of dissimilar subalgebras for the Lei algebras from group classification was obtained. Examples of invariant solutions describing vortex gas flow with variable entropy, including a point source or stock are considered. Analogs of simple waves were obtained for some two-dimensional subalgebras.

An inclined impact of a spherical segment with a curvature radius of 106.5 mm onto a water layer 20 mm thick is studied up to the instant of the body-liquid contact. The range of the impact angle with respect to the horizontal plane is 90-15oC, and the vertical component of velocity for all angles is 10 or 20 mm/s. The measurements are performed by a synthetic schlieren method. The marker displacements are measured by a PTV algorithm. Distributions of the liquid layer thickness in space and time along two horizontal axes are obtained. It is shown that a change in the impact angle does not alter the dynamics of crater deepening and expansion at the initial stage; the crater remains axisymmetric. It is found that the rear slope of the crater becomes steeper in the case with the maximum deviation of the angle from the vertical line, presumably, because of the local decrease in pressure. The characteristics of capillary waves generated by the impact are independent of the impact angle.

This article focuses on the effect of frequency dispersion on the wave runup height and the characteristics of the surface waves reflected from the coastal slope. The calculations are performed within the framework of nonlinear dispersive and dispersionless shallow water models using the original boundary conditions on the movable water-land boundary. We consider the problem of solitary wave runup on flat coastal slopes with parameters close to the characteristics of one of the Kamchatka bays, for which the authors obtained probabilistic assessments of the tsunami hazard. The obtained results show that the maximum runup and amplitudes of the reflected waves are overestimated by the dispersionless model by 100%.

A model of the initial stage of surface treatment of a material by a particle beam is presented. The model takes into account the mutual influence of elastic, thermal, and diffusion waves, as well as the formation of a new phase in the surface layer of the substrate. Examples of problem solution for different combinations of model parameters are shown; the dynamics of process development under the action of two consecutive pulses is illustrated. Possible partial variants of the model are given.

Inverse problems of design of special shells that serve for concentrating heat fluxes are considered. With the use of an optimization method and geometric discretization, these problems are reduced to finite-dimensional extreme problems, where the thermal conductivity coefficients of the shell sectors play the role of control parameters. The chosen set of control parameters ensures obtaining thermal concentrators, which can be simply implemented and provide high efficiency in the class of devices under consideration.

A group of point transformations permitted by the three-dimensional Kadomtsev-Petviashvili equation is calculated. An example of an invariant solution is given. Exact solutions for the equation under study in the form of double waves are found. The resulting solutions are expressed in terms of elementary functions and describe the interaction of a pair of solitons. Smooth bounded rational solutions are also constructed.

This paper describes the use of a multiphase model of poroelasticity meant for the brain substance and based on medical data to study the displacement of a ventricle wall in the brain and the value of pressure on it. The dependence of these values on the parameters of the model in normal-pressure hydrocephalus is studied. It is shown for the brain geometry with different levels of detail that the simplified geometry of the brain allows one to accurately estimate critical pressures and displacements in the case of more complex geometry.

The problem of the propagation of plane sound waves of small amplitudes in a mixture of an isothermal carrier gas and solid particles of various sizes is formulated on the basis of a multi-liquid macroscopic model of the medium. In the model, the dispersed phase is considered as N fractions of monodisperse particles, and the dynamics of each fraction is described using the equations of a continuous medium in which does internal pressure is absent. The fractions exchange momenta with the carrier gas, but not with each other. The whole mixture is acted upon by the total pressure determined by the motion of gas molecules, and the dust particles are considered buoyant. An analytical solution of the problem is obtained using the Fourier method and analysis of variance. In the general case for an arbitrary value of the relaxation time, the solution is found numerically using the developed and published code. In special cases (infinitely small time of velocity relaxation or relaxation equilibrium and infinitely long time of velocity relaxation or frozen equilibrium), the effective velocity of sound in the gas-dust medium is determined and used to obtain simple analytical representations of the solution of the problem.

The problem of constructing solutions to a system of two coupled nonlinear parabolic equations of the reaction-diffusion type is considered. Solutions in the form of diffusion waves propagating over zero background with a finite speed are investigated. The theorem on the construction of exact solutions by reduction to the Cauchy problem for a system of ordinary differential equations is proved. A time-step numerical technique for solving the reaction-diffusion system using radial basis function expansion is proposed. The same approach is used to solve systems of ordinary differential equations that determine the exact solutions of the reaction-diffusion system. Numerical analysis and estimation of the accuracy of solutions to a system of ordinary differential equations are carried out. These solutions are applied to verify the obtained time-step solutions of the original system.

Data are presented on time-local solvability of problems with free boundaries for a system of equations of magnetohydrodynamics of an ideal compressible fluid. A problem with a free plasma-vacuum boundary and a problem with boundary conditions on a contact discontinuity are considered. A scheme is given for proving the local existence and uniqueness of smooth solutions of these problems with and without account for surface tension.

Equations of filtration of a two-component miscible fluid in a porous medium are derived with the use of the method of two-scale homogenization of the system of the Navier-Stokes and Cahn-Hilliard equations. The case of strong miscibility is considered.

A steady problem of trapped solitary waves in supercritical flows of a stratified fluid above an uneven bottom is considered. For gently sloping low-amplitude obstacles, a family of approximate two-parameter solutions is constructed, which correspond to internal solitary waves in the limit of a zero height of the obstacle. It is numerically demonstrated that the number of approximate solutions significantly depends on the bottom shape.

In this paper, we investigate the problem of control of a complex object, described by a large ODE system of a block structure with unseparated boundary conditions between blocks. The controls in the right-hand sides of the equations and the values of the source parameters in the boundary conditions are to be optimized. We propose to apply the first order optimization methods for the numerical solution to the optimal control problem, using functional gradient formulas participating in the obtained necessary optimality conditions. Special schemes of the sweep method for the solution to the direct and conjugate boundary value problems, having a block structure, and unseparated non-local boundary conditions are offered. This method takes into account special features of ODE systems and boundary conditions, allows the transfer of boundary conditions for each block and each boundary condition in the block independent of each other. The obtained results of numerical experiments in solving the test problem and their analysis are given.

Currently the finite element method is most widespread technique for solving problems of mechanics of a deformable solid body. Its shortcomings are well-known: approximating a displacement by a piecewise-linear function, we obtain the tension to be discontinuous. At the same time, it is necessary to notice that most problems of mechanics of a deformable solid body are described by the elliptic type equations which have smooth decisions. It seems to be relevant to develop algorithms which would take this smoothness into account. The idea of such algorithms belongs to K.I. Babenko. This idea was stated in the early seventies of the last century. A long-lasting application of this technique in elliptic tasks to eigenvalues has proved their high performance to the author of this study. However, in this technique the matrix of the finite-dimensional task turns out to be not symmetric but only close to that to be symmetrized. Below, the application when sampling the Bubnov-Galyorkina method, this defect is eliminated. Let us note that the symmetry of the matrix of the finite-dimensional task is important when studying the stability. Unlike classical difference methods and the finite element method where the dependence of the convergence ratio on the number of nodes of the grid is power, we have an exponential decrease of the error.

This paper sets a theoretical foundation for applications of fractal interpolation functions (FIFs). We construct rational cubic spline FIFs (RCSFIFs) with a quadratic denominator involving two shape parameters. The elements of the iterated function system (IFS) in each subinterval are identified befittingly so that the graph of the resulting C¹-RCSFIF lies within a prescribed rectangle. These parameters include, in particular, conditions on the positivity of the C¹-RCSFIF. The problem of visualization of constrained data is also addressed when the data is lying above a straight line, the proposed fractal curve is required to lie on the same side of the line. We illustrate our interpolation scheme with some numerical examples.

An experimental study of the solvers efficiency of 2D boundary value problems on subgrids of quasistructured rectangular grids was carried out. A solver is understood as a solution method and its software implementation. Three solvers are considered: one direct solver -- the Buneman cyclic reduction method and two iterative ones: the Peaceman-Rachford method and the method of successive over relaxation. Characteristic features of the studies are: 1) the subgrids contain a small number of nodes, namely 8 х 8, 16 х16, 32 х 32, 64 х 64; 2) the efficiency is estimated not only for single calculations, but also mainly for series of calculations, in each of which several repetitions of solving the problem with different boundary conditions on the same subgrid are carried out. Based on serial calculations, a combined method is proposed, and recommendations on the use of solvers are given.

For the numerical solution of a system of linear algebraic equations with a three-diagonal matrix, a recursive version of the Cramer method is proposed. This method does not require additional restrictions on the system matrix, similar to those formulated for the sweep method. The results of numerical experiments are presented on a large set of test problems, a comparative analysis of the effectiveness of the proposed methodology and the corresponding algorithms is given.

An explicit representation of filter banks for constructing the wavelet transform of spaces of linear minimal splines on non-uniform grids on a segment is obtained. The decomposition and reconstruction operators are constructed, their mutual inverse is proved. The relations connecting the corresponding filters are established. The approach to constructing the spline wavelet decompositions used in this paper is based on approximation relations as the initial structure for constructing spaces of minimal splines and calibration relations to prove the embedding of the corresponding spaces. The advantages of the approach proposed, due to rejecting the formalism of the Hilbert spaces, are in the possibility of using non-uniform grids and fairly arbitrary non-polynomial spline wavelets.