## About

210

Publications

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Additional affiliations

September 1994 - May 2004

**Novosibirsk State Technical University, Novosibirsk, Russia**

Position

- Professor on part-time basis

Description

- I have presented for the magistrates a lecture course entitled: "Difference Methods for Solving Continuum Mechanics Problems". The materials of this lecture course were published (in Russian) in two my textbooks.

June 1972 - present

## Publications

Publications (210)

The explicit symplectic difference schemes are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplec-tic schemes of high order of accuracy using parametric Gröbner bases, resultants, and permutations of variables is proposed. The implementation of...

The explicit symplectic difference schemes with a number of stages from 1 to 5 are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using Gröbner bases is proposed. It is shown that it is possible to signif...

This is the poster of the international workshop on Computer Algebra in Scientific Computing to be held at Gebze Technical University on August 22-26, 2022.

The Runge--Kutta--Nystr\"{o}m (RKN) explicit symplectic difference
schemes with the number of stages from 1 to 5 for the numerical
solution of molecular dynamics problems described by the systems
with separable Hamiltonians have been considered. All schemes have
been compared in terms of the accuracy and stability with the use
of \Gr bases. For eac...

The Runge--Kutta--Nystr\"{o}m (RKN) explicit symplectic four-stage
schemes for the numerical solution of molecular dynamics problems
described by the systems with separable Hamiltonians have been
considered. In the case of the zero Vandermonde determinant, 20
schemes are obtained using the Gr\"{o}bner basis technique. Four
invertible (symmetric)...

The explicit symplectic difference schemes with a number of stages from 1 to 5 are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using Gröbner bases is proposed. It is shown that it is possible to signif...

This book constitutes the proceedings of the 23rd International Workshop on Computer Algebra in Scientific Computing, CASC 2021, held in Sochi, Russia, in September 2021. The 24 full papers presented together with 1 invited talk were carefully reviewed and selected from 40 submissions. The papers cover theoretical computer algebra and its applicati...

The Runge–Kutta–Nyström (RKN) explicit symplectic four-stage schemes for the numerical solution of molecular dynamics problems described by the systems with separable Hamiltonians have been considered. In the case of the zero Vandermonde determinant, 20 schemes are obtained using the Gröbnerbasis technique. Four invertible (symmetric) schemes are a...

The problem of the acceleration of the iterative process of numerical solution by the collocation and least squares (CLS) method of boundary value problems for partial differential equations is considered. For its solution, it is proposed to apply simultaneously three ways to accelerate the iterative process: preconditioner, multigrid algorithm, an...

This text was reviewed in July 2020 by three reveiwers. It has been published in SEptember 2020 in the following Proceedings:
Vorozhtsov, E.V., Kiselev, S.P.: Comparative study of the accuracy of higher-order difference schemes for molecular dynamics problems using the computer algebra means. In: Computer Algebra in
Scientific Computing. CASC 2020....

The Runge–Kutta–Nyström (RKN) explicit symplectic difference schemes with the number of stages from 1 to 5 for the numerical solution of molecular dynamics problems described by the systems with separable Hamiltonians have been considered. All schemes have been compared in terms of the accuracy and stability with the use of Gröbner bases. For each...

Рассматривается проблема ускорения итерационного процесса численного решения методом коллокаций и наименьших квадратов (КНК) краевых задач для уравнений с частными производными. Для ее решения предложено применять одновременно три способа ускорения итерационного процесса: предобуславливатель, многосеточный алгоритм и метод Крылова. Предложен метод...

This book constitutes the refereed proceedings of the 22nd International Workshop on Computer Algebra in Scientific Computing, CASC 2020, held in Linz, Austria, in September 2020. The conference was held virtually due to the COVID-19 pandemic.
The 34 full papers presented together with 2 invited talks were carefully reviewed and selected from 41 su...

The international conference on Computer Algebra in Scientific Computing will be held at Johannes Kepler University in Linz, Austria on September 14-18, 2020. For details, please visit http://www.casc-conference.org/

The preconditioner, multigrid algorithm, and the Krylov method are applied for accelerating the iteration process of solving the Navier--Stokes equations by the method of collocations and least residuals (CLR). These methods have been used simultaneously in their combination and separately. Their capabilities and efficiency have been verified by a...

To increase the accuracy of computations by the method of collocations and least squares (CLS) a generalization of this method is proposed for the case of a non-uniform logically rectangular grid. The main work formulas of the CLS method on non-uniform grid, including the formulas implementing the prolongation operator on a non-uniform grid at the...

This book constitutes the refereed proceedings of the 21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019, held in Moscow, Russia, in August 2019.
The 28 full papers presented together with 2 invited talks were carefully reviewed and selected from 44 submissions. They deal with cutting-edge research in all major disci...

A p-version of the collocation method for the numerical solution of the Fredholm integral equations of the second kind has been proposed and implemented. In the given implementation, the possibilities have been realized for the variation of the polynomial degree in the polynomial representation of the approximate solution of equations and the varia...

Partial differential equations (PDEs) play an important role in the natural sciences and technology, because they describe the way systems (natural and other) behave. The inherent suitability of PDEs to characterizing the nature, motion, and evolution of systems, has led to their wide-ranging use in numerical models that are developed in order to a...

This book is intended for mathematicians, physicists, and engineers, as well as for postgraduate students, and for anyone involved with numeric solutions for real-world physics problems. In particular, it deals with those schemes that are used to solve complex physical problems in areas such as gas dynamics, heat and mass transfer, catastrophe theo...

To increase the accuracy of computations by the method of collocations and least residuals (CLR) it is proposed to increase the number of degrees of freedom with the aid of the following two techniques: an increase in the number of basis vectors and the integration of the linearized partial differential equations (PDEs) over the subcells of each ce...

In the work, we consider the problem of accelerating the iteration process of the numerical solution of boundary-value problems for partial differential equations (PDE) by the method of collocations and least residuals (CLR). To solve this problem, it is proposed to combine simultaneously three techniques of the iteration process acceleration: the...

This book constitutes the proceedings of the 19th International Workshop on Computer Algebra in Scientific Computing, CASC 2017, held in Beijing, China, in September 2017.
The 28 full papers presented in this volume were carefully reviewed and selected from 33 submissions. They deal with cutting-edge research in all major disciplines of Computer A...

We derive in this chapter the governing differential equations of continuum mechanics with the aid of the above definitions and the conservation laws for the mass, momentum, energy, and momentum moment written for finite volumes of a continuum. The differential equations of continuum mechanics (equations of continuity, momentum, and energy) represe...

Fluid mechanics is a branch of science dealing with the study of flows of continua under the action of external forces. It has a rich history and unchanging core of materials, but is constantly expanding and evolving as new methods, applications, and computational tools are developed.
This text presents the basic concepts and methods of fluid mech...

This chapter deals with one-dimensional stationary and nonstationary as well as planar and three-dimensional stationary gas flows. The theories for the Laval nozzle and normal and oblique shock waves are presented. The Becker’s solution for the shock wave structure is given. The solution of a simple wave type is obtained with the aid of the method...

This chapter deals with incompressible ideal fluid flows. While choosing the material, the authors aimed at presenting those reults that have now become the classical results and are widely used in the current research work of the aerohydrodynamicists¹⁻⁹. In particular, the Bernoulli and Lagrange integrals are derived in Section 4.1. They enable on...

This chapter is devoted to the viscous fluid flows, which are described by the Navier-Stokes equations. We derive the Navier-Stokes equations in the Cartesian, cylindrical, and spherical coordinate systems and consider their exact solutions at small Reynolds numbers. We present the Prandtl’s theory of boundary layer, which is va of boundary layer,...

The purpose of the present chapter is to provide a systematic introduction of the basic concepts and definitions of the tensors of strains and stress. The presentation begins with a section, in which we briefly present the elements of tensor analysis. Tensor analysis enables one to present in a simple and elegant form the fundamantals of continuum...

In this chapter, we consider the similarity and dimensional methods, including the construction of self-similar solutions. We also present the theory of weak discontinuities (the characteristics) and strong discontinuities (shock waves and tangential discontinuities).

The fundamentals of the mechanics of multiphase media are presented here for the first time within the framework of a course in fluid mechanics. This new branch of mechanics has appeared comparatively recently, about 40 years ago, in connection with the development of aerospace technology, nuclear power, and new technologies. At present, the genera...

The computer algebra system (CAS) Mathematica has been applied for constructing the optimal iteration processes of the Gauss–Seidel type at the solution of PDE’s by the method of collocations and least residuals. The possibilities of the proposed approaches are shown by the examples of the solution of boundary-value problems for the 2D Navier–Stoke...

This book constitutes the proceedings of the 18th International Workshop on Computer Algebra in Scientific Computing, CASC 2016, held in Bucharest, Romania, in September 2016.
The 32 papers presented in this volume were carefully reviewed and selected from 39 submissions. They deal with cutting-edge research in all major disciplines of Computer Alg...

A version of the method of collocations and least residuals is proposed for the numerical solution of the Poisson equation in polar coordinates on non-uniform grids. By introducing general curvilinearcoordinates the original Poisson equation is reduced to the Beltrami equation. A uniform grid is used in curvilinear coordinates. The grid non-uniform...

Реализованы несколько новых вариантов известных способов ускорения процессов итерационного решения дискретных задач, которые возникают при решении численными методами краевых задач для уравнений с частными производными (PDE). В частности, предложено и реализовано комбинированное применение операции продолжения метода Федоренко и метода Крылова. В к...

In the present work, the computer algebra system (CAS) is applied for constructing a new version of the analytic-numerical method of collocations and least residuals (CLR) for solving the Burgers equation and the Korteweg-de-Vries-Burgers equation. The CAS is employed at all stages from writing, deriving, and verifying the formulas of the method to...

This paper represents a survey of the investigation carried out at the Institute of Theoretical & Applied Mechanics of the Siberian Division of the USSR Academy of Sciences. The questions of construction of efficient difference schemes are considered for the numerical integration of a compressible viscous heat-conducting gas based on splitting in t...

The method of collocations and least residuals (CLR), which was proposed previously for the numerical solution of two-dimensional Navier–Stokes equations governing the stationary flows of a viscous incompressible fluid, is extended here for the three-dimensional case. The solution is sought in the implemented version of the method in the form of an...

In the present work, the computer algebra system (CAS) is applied for constructing a new version of the method of collocations and least residuals (CLR) for solving the 3D incompressible Navier-Stokes equations. The CAS is employed at all stages from writing, deriving, and verifying the formulas of the method to their translation into arithmetic op...

Proceedings of CASC 2013, Computer Algebra in Scientific Computing, 15th International Workshop, Berlin, Germany, September 9-13, 2013

The method of collocations and least squares, which was previously proposed for the numerical solution of the two-dimensional Navier---Stokes equations governing steady incompressible viscous flows, is extended here for the three-dimensional case. The derivation of the collocation and matching conditions is carried out in symbolic form using the CA...

In this paper, some unsolved problems of the Mathematica software package are documented. At first, it is shown, using a number of examples, that the processing (simplification) of rational-fractional expressions involving powers in the general form, has been implemented in Maple more carefully than in Mathematica. Then, an error in Mathematica is...

Übersicht über Veröffentlichungen und Software / Publications and Software-Downloads
Dissertation, Habilitationsschrift, Monographien und Lehrbücher / PhD Thesis, Habilitation Thesis, Textbooks and Monographs
Herausgebertätigkeit / Editorial Work
Wissenschaftliche Veröffentlichungen in referierten Zeitschriften / Publications in Refereed Journals...

Ideas Underlying Catastrophe TheoryReduction of the von Neumann Analysis to a Canonical Problem of Catastrophe TheoryNumerical Determination of a Segment of the Stability Region BoundaryDirect Use of the Resultant for the Determination of Boundary PointsAutomatic Generation of FORTRAN SubroutinesSome Practical Applications

Formulation of a Search for Stability Region Boundaries of Difference Schemes in Terms of Optimization TheorySymbolic Computation of Algebraic ExpressionsNumerical Realization of the Optimization Method
Some Practical Applications

Basic DefinitionsSymbolic Algorithm for the Scalar Equation CaseSymbolic Computation of Differential Approximations of Schemes with Fractional StepsLocal Approximation Study of Difference Operators on Nonorthogonal Curvilinear Spatial GridsDifferential Approximation and Stability of Difference Schemes

Theoretical Background
The Case of One Spatial VariableThe Case of Two Spatial VariablesFormulation of the Optimization ProblemSymbolic StagesNumerical Realization of the Optimization Method
Two Methods for Determining the Stability Region's BoundaryComputational ExamplesBibliographic NotesConcluding Remarks

Half TitleTitleCopyrightContentsPreface

Basic DefinitionsStability and Accuracy FunctionalsA Search Algorithm for Maximally Stable Difference Schemes and Its Computer ImplementationApplication to One-Dimensional ProblemsJameson's Scheme for the Two-Dimensional Advection EquationPeyret-Taylor's Family of Schemes for the Two-Dimensional Advection-Diffusion Equation

General Organizational Scheme of a Process for Detection of Stability Regions of Difference SchemesDetection of Boundaries by Digital Image SegmentationTracing Contour SegmentsRefinement of Boundary Point PositionsExtraction of Singular PointsSome Practical Applications

Preliminary Discussion of Stability and ApproximationComputer Algebra SystemsA Brief Review of the Contents of ChaptersStability, Approximation, and ConvergenceA Survey of Methods for the Stability Analysis of Difference SchemesAlgebraic Criteria for Localization of Polynomial ZerosDetermination of the Maximal Time Step from Stability Analysis Resu...

General Structure of the Symbolic-Numerical MethodThe Case of Diagonalizable Amplification MatricesScheme CheckerSymbolic Stages of the Method
Generation of a FORTRAN Program by Computer AlgebraComputation of the Coordinates of Points of a Stability Region BoundaryImproved Accuracy of Numerical ResultsExamples of Stability Analyses of Difference Sc...

Various aspects of cloud computing applications for scientific research, applied design, and remote education are described in this paper. An analysis of the different aspects is performed based on the experience from the “SciShop.ru” Computer Simulation Center. This analysis shows that cloud computing technology has wide prospects in scientific re...

The general description of infrastructure and content of SciShop.ru computer
simulation center is given. This resource is a new form of knowledge generation
and remote education using modern Cloud Computing technologies.

Proceedings of CASC 2011, Computer Algebra in Scientific Computing, 13th International Workshop, Kassel, Germany, September 5-9, 2011

We propose to derive the explicit multistage methods of the Runge-Kutta
type for ordinary differential equations (ODEs) with the aid of the
expansion of grid functions into the Lagrange-Burmann series. New
explicit first- and second-order methods are derived, which are applied
to the numerical integration of the Cauchy problem for a moderately
stif...

Proceedings of CASC 2010, Computer Algebra in Scientific Computing, 12th International Workshop, Tsakhkadzor, Armenia, September 6-12, 2010

We analyse the known approximate analytic solution of the problem of gas flow induced by the disc rotation inside a closed
casing. It is shown that this solution is inapplicable because of the negative thickness of the boundary layer in the shaft
neighborhood. Several new analytic solutions are obtained for the flow parameters inside the boundary l...

We investigate the stability of the modified difference scheme of Kim and Moin for numerical integration of two-dimensional
incompressible Navier–Stokes equations by the Fourier method and by the method of discrete perturbations. The obtained analytic-form
stability condition gives the maximum time steps allowed by stability, which are by factors f...

We present the results of the numerical modelling of the interaction of a shock wave with a cloud of finite size particles. The computations were carried out within the framework of continuum/discrete model with the use of the techniques of digital diagnostics and pattern recognition. The shock wave and vortex formation behind the cloud of particle...

Automatic generation of smooth, non-overlapping meshes on arbitrary regions is the well-known problem. Considered as optimization task the problem may be reduced to nding a minimizer of the weighted combination of so-called length, area, and orthogonality functionals. Un- fortunately, it has been shown that on the one hand, certain weights of the i...

The problem of the numerical modelling of the damping effect of the spiral compensators of percussion–rotary drilling devices is considered. The Roe first-order difference method has been adapted for the computation of a three-dimensional flow of a barotropic compressible fluid on a spatial curvilinear grid. As a result, the distributions of the so...

We propose a new algorithm for the generation of orthogonal grids on regions bounded by arbitrary number of polynomial inequalities.
Instead of calculation of the grid nodes positions for a particular region, we perform all calculations for general polynomials
given with indeterminate coefficients. The first advantage of this approach is that the c...