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Rings is an open-source library, written in Java and Scala programming languages, which implements basic concepts and algorithms from computational commutative algebra. The goal of the Rings library is to provide a high-performance... more
Rings is an open-source library, written in Java and Scala programming languages, which implements basic concepts and algorithms from computational commutative algebra. The goal of the Rings library is to provide a high-performance implementation packed into a lightweight library (not a full-featured CAS) with a clean application programming interface (API), which meets modern standards of software development. Polynomial arithmetic, GCDs, factorization, and Gröbner bases are implemented with the use of modern fast algorithms. Rings provides a simple API with a fully typed hierarchy of algebraic structures and algorithms for commutative algebra. The use of the Scala language brings a quite novel powerful, strongly typed functional programming model allowing to write short, expressive, and fast code for applications.
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Comparison Maps for Relatively Free Resolutions.- A Symbolic-Numeric Approach for Solving the Eigenvalue Problem for the One-Dimensional Schrodinger Equation.- Reducing Computational Costs in the Basic Perturbation Lemma.- Solving... more
Comparison Maps for Relatively Free Resolutions.- A Symbolic-Numeric Approach for Solving the Eigenvalue Problem for the One-Dimensional Schrodinger Equation.- Reducing Computational Costs in the Basic Perturbation Lemma.- Solving Algorithmic Problems on Orders and Lattices by Relation Algebra and RelView.- Intervals, Syzygies, Numerical Grobner Bases: A Mixed Study.- Application of Computer Algebra for Construction of Quasi-periodic Solutions for Restricted Circular Planar Three Body Problem.- Efficient Preprocessing Methods for Quantifier Elimination.- Symbolic and Numerical Calculation of Transport Integrals for Some Organic Crystals.- On the Provably Tight Approximation of Optimal Meshing for Non-convex Regions.- Providing Modern Software Environments to Computer Algebra Systems.- The Instability of the Rhombus-Like Central Configurations in Newton 9-Body Problem.- Algorithmic Invariants for Alexander Modules.- Sudokus and Grobner Bases: Not Only a Divertimento.- Simplicial Perturbation Techniques and Effective Homology.- Numerical Study of Stability Domains of Hamiltonian Equation Solutions.- Numeric-Symbolic Computations in the Study of Central Configurations in the Planar Newtonian Four-Body Problem.- A Symbolic-Numerical Algorithm for Solving the Eigenvalue Problem for a Hydrogen Atom in Magnetic Field.- On Decomposition of Tame Polynomials and Rational Functions.- Newton Polyhedra and an Oscillation Index of Oscillatory Integrals with Convex Phases.- Cellular Automata with Symmetric Local Rules.- Parallel Laplace Method with Assured Accuracy for Solutions of Differential Equations by Symbolic Computations.- On Connection Between Constructive Involutive Divisions and Monomial Orderings.- A Symbolic-Numeric Approach to Tube Modeling in CAD Systems.- Inequalities on Upper Bounds for Real Polynomial Roots.- New Domains for Applied Quantifier Elimination.- Algorithms for Symbolic Polynomials.- Testing Mersenne Primes with Elliptic Curves.
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It is our deepest regret to inform you that Vladimir Petrovich Gerdt, Professor, Head of the Algebraic and Quantum Computing Group of the Scientific Department of Computational Physics of the Laboratory of Information Technologies (LIT)... more
It is our deepest regret to inform you that Vladimir Petrovich Gerdt, Professor, Head of the Algebraic and Quantum Computing Group of the Scientific Department of Computational Physics of the Laboratory of Information Technologies (LIT) at the Joint Institute of Nuclear Research (JINR) in Dubna, Oblast Moscow, Russia, died on January 5th, 2021 at the age of 73, following complications caused by COVID-19. Vladimir Gerdt was born on January 21, 1947 in the town of Engels, Saratov region of the USSR. He began his scientific activity at JINR in November 1971, after graduating from the Physics Department of Saratov State University, first in the Department of Radiation Safety, and from February 1977 on in the Laboratory of Computer Technology and Automation, which was renamed in the year 2000 to Laboratory of Information Technologies, where he was engaged in the deployment of analytical computing software systems on the computers of the JINR Central Research Center, their development and application for solving physical problems. Starting in 1983, he was the head of the Computer Algebra Research Group (renamed in 2007 to Algebraic and Quantum Computing Group) at LIT. In 1976, Vladimir Gerdt successfully defended his Ph.D. thesis (for Kandidat nauk/Kandidat nauk) in the field Theoretical and Mathematical Physics, and in 1992, his doctoral dissertation (for Doktor nauk/Doktor nauk, D.Sc.) in the field Application of Computer Technology, Mathematical Modeling, and Mathematical Methods for Scientific Research. In 1997, he was awarded the academic title of Professor. Vladimir Gerdt started his career with work on the integrability analysis of nonlinear evolution equations using symmetries, and he never ceased to be interested in symmetry methods for di↵erential equations. Later, the involution analysis of polynomial systems and systems of di↵erential or di↵erence equations became a core theme in his research. Together with some of his students, he developed the theory of involutive bases out of the Janet–Riquier theory. As one application of these results to physics, he was always interested in the study of systems with constraints, in particular in Dirac theory. Another application in recent years consisted of designing structure preserving numerical methods for partial di↵erential equations. In the last years, he also revived the Thomas decomposition for polynomial di↵erential equations and applied it to numerous problems. One of his last significant results was an algorithmic solution of the linearization problem for ordinary di↵erential equations. For many years, Vladimir Gerdt also worked in the field of quantum computing. His group at JINR developed methods and computer algebra programs for studying quantum information processes and for modeling quantum systems. In particular, they applied a combination of computational invariant theory and involutive bases to a qualitative and quantitative study of entangled quantum states. Vladimir Gerdt was the author or co-author of more than 240 scientific papers (a listing is available at his CV at JINR), and he was a leading expert in the field of symbolic and algebraic computation. He devoted a lot of e↵ort and energy to train young researchers in these modern scientific areas. He was a professor at the Department of Distributed Information Computing Systems of Dubna State University, where, under his supervision, seven PhD theses were defended.
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By using the Fourier method we study the stability of a three-stage finite volume Runge–Kutta time stepping scheme approximating the 2D Euler equations on curvilinear grids. By combining the analytic and numeric stability investigation... more
By using the Fourier method we study the stability of a three-stage finite volume Runge–Kutta time stepping scheme approximating the 2D Euler equations on curvilinear grids. By combining the analytic and numeric stability investigation results we obtain an analytic formula for stability condition. The results of numerical solution of a number of internal and external fluid dynamics problems are presented,
Research Interests: Mathematics, Computer Science, Fluid Dynamics, Mathematical Sciences, Physical sciences, and 11 moreEuler Equations, Finite Volume, Correctness, Euler Lagrange Equation, Numerical Stability, Numerical Solution, Curvilinear Coordinates, Euler Method, Runge Kutta, Runge Kutta Methods, and Euler Equation
We describe a symbolic-numerical method for the Fourier stability analyses of difference initial-value problems approximating the initial-value problems for hyperbolic or parabolic PDEs. The Fourier method is reduced to the algebra of the... more
We describe a symbolic-numerical method for the Fourier stability analyses of difference initial-value problems approximating the initial-value problems for hyperbolic or parabolic PDEs. The Fourier method is reduced to the algebra of the resultants. We further use the REDUCE computer algebra system for the symbolic computation of the resultant and for the generation of a FORTRAN function to compute the value of the resultant. Basing on this FORTRAN function we further construct a binary function to characterize the stability and instability points. Using this function we generate a bilevel digital picture, and the stability region boundaries are then detected in this picture with the aid of the efficient algorithm proposed previously by Pavlidis (1982). The above symbolic-numerical method has enabled us to obtain for the first time the stability regions of the considered Jameson's schemes as applied to the two-dimensional advection-diffusion equation. Analytical formulas are proposed for the approximation of the boundaries of the obtained stability regions. It is shown that these formulas underestimate insignificantly the actual sizes of the stability regions. Therefore, these formulas can efficiently be used in practical computations by Jameson's schemes.
Research Interests: Mathematics, Applied Mathematics, Partial Differential Equations, Symbolic Computation, Stability, and 11 moreNumerical Method, Computation, Fortran, Mathematical Sciences, Maple Computer Algebra System, Physical sciences, Efficient Algorithm for ECG Coding, PARTIAL DIFFERENTIAL EQUATION, Numerical Stability, Advection diffusion equation, and Difference Scheme
ABSTRACT It is proposed to use the deterministic discriminant methods of the pattern recognition theory for a solution of the problem of classifying the discontinuities into several types (shock waves, contact discontinuities, etc.) in... more
ABSTRACT It is proposed to use the deterministic discriminant methods of the pattern recognition theory for a solution of the problem of classifying the discontinuities into several types (shock waves, contact discontinuities, etc.) in the numerical solutions of two-dimensional gas dynamic problems obtained by finite-difference shock-capturing schemes. The formulae for the computation of the features are derived. Three different schemes for the classification of singularities in gas flows are presented. The efficiency of these schemes is illustrated by the example of a number of applied two-dimensional problems, including transonic potential flow around an airfoil, the diffraction of a strong shock wave on a wedge and supersonic flow in a wind tunnel with a lower-wall step.
Research Interests: Mechanical Engineering, Mathematics, Applied Mathematics, Image Processing, Pattern Recognition, and 15 moreAerodynamics, Computation, Computation Fluid Dynamics, Gas Dynamics, Wind Tunnel, Finite Difference, Supersonic flow, Interdisciplinary Engineering, Airfoil, Shock Wave, Potential Flow, Numerical Solution, Gas Flow, Supersonic speed, and Difference Scheme
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Introduction to Mathematica General Information about Mathematica Symbolic Computations with Mathematica Numerical Computations with Mathematica Finite Difference Methods for Hyperbolic PDEs Construction of Difference Schemes for the... more
Introduction to Mathematica General Information about Mathematica Symbolic Computations with Mathematica Numerical Computations with Mathematica Finite Difference Methods for Hyperbolic PDEs Construction of Difference Schemes for the Advection Equation The Notion of Approximation Fourier Stability Analysis Elementary Second-Order Schemes Algorithm for Automatic Determination of Approximation Order of Scalar Difference Schemes Monotonicity Property of Difference Schemes TVD Schemes The Construction of Difference Schemes for Systems of PDEs Implicit Difference Schemes Von Neumann Stability Analysis in the Case of Systems of Difference Equations Difference Initial- and Boundary-Value Problems Construction of Difference Schemes for Multidimensional Hyperbolic Problems Determination of Planar Stability Regions Curvilinear Spatial Grids Answers to the Exercises Finite Difference Methods for Parabolic PDEs Basic Types of Boundary Conditions for Parabolic PDEs Simple Schemes for the One-Dimensional Heat Equation Difference Schemes for Advection-Diffusion Equation Runge-Kutta Methods Finite Volume Method The Adi Method Approximate Factorization Scheme Dispersion Answers to the Exercises Numerical Methods for Elliptic PDEs Boundary-Value Problems for Elliptic PDEs A Simple Elliptic Solver Pseudo-Unsteady Methods The Finite Element Method Numerical Grid Generation Local Approximation Study of Finite Volume Operators on Arbitrary Grids Local Approximation Study of Difference Schemes on Logically Rectangular Grids Answers to the Exercises Appendix Glossary of Programs Index Each Chapter also includes a list of references.
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Рассмотрены явные симплектические разностные схемы Рунге–Кутты–Нистрема (RKN) с числом стадий от 1 до 5 для численного решения задач молекулярной динамики, описываемых системами с распадающимися гамильтонианами. Для числа стадий 2 и 3... more
Рассмотрены явные симплектические разностные схемы Рунге–Кутты–Нистрема (RKN) с числом стадий от 1 до 5 для численного решения задач молекулярной динамики, описываемых системами с распадающимися гамильтонианами. Для числа стадий 2 и 3 параметры RKN-схем получены с помощью техники базисов Гребнера. Для числа стадий 4 и 5 новые схемы най дены с применением метода численной оптимизации Нелдера–Мида. В частности, для числа стадий 4 получены четыре новые схемы. Для числа стадий 5 получены три новые схемы в дополнение к четырем схемам, известным в литературе. Для каждого конкретного числа стадий найдена схема, являющаяся наилучшей с точки зрения минимума ведущего члена погрешности аппроксимации. Верификация схем осуществлена на задаче, имеющей точное решение. Показано, что симплектическая пятистадийная RKN-схема обеспечивает более точное сохранение баланса полной энергии системы частиц, чем схемы более низких порядков точности. Исследования устойчивости схем выполнены с помощью программно...
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Предлагается строить явные разностные схемы третьего порядка точности для гиперболических законов сохранения с применением разложений сеточных функций в ряды Лагранжа--Бюрмана. Результаты тестовых расчетов для случаев одномерного... more
Предлагается строить явные разностные схемы третьего порядка точности для гиперболических законов сохранения с применением разложений сеточных функций в ряды Лагранжа--Бюрмана. Результаты тестовых расчетов для случаев одномерного уравнения переноса и многомерных уравнений Эйлера невязкого сжимаемого газа подтверждают третий порядок точности построенных схем. Получены квазимонотонные профили численных решений. It is proposed to construct several explicit third-order difference schemes for the hyperbolic conservation laws using the expansions of grid functions in Lagrange-Burmann series. The results of test computations for the one-dimensional advection equation and multidimensional Euler equations governing the inviscid compressible gas flows confirm the third order of accuracy of the constructed schemes. The quasi-monotonous profiles of numerical solutions are obtained.
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ABSTRACT
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We propose a method for the construction of arbitrary multiply connected stability domains of difference schemes approximating the systems of partial differential equations with constant coefficients. The method is based on using the... more
We propose a method for the construction of arbitrary multiply connected stability domains of difference schemes approximating the systems of partial differential equations with constant coefficients. The method is based on using the Fourier method and the means of the pattern recognition theory and does not require any a priori information on the shape of the domain sought for. The complete automation of the method is achieved by combining the means of computer algebra (symbolic manipulations on computer) and digital image processing. The efficiency of the method presented is demonstrated at the practical examples of difference schemes for problems of fluid dynamics.
Research Interests: Mathematics and Appendix
The Runge–Kutta–Nystrom (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... more
The Runge–Kutta–Nystrom (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 Grobner bases. For each specific number of stages, the schemes are found, which are the best in terms of accuracy and stability. The efficiency parameter of RKN schemes has been introduced by analogy with the efficiency parameter for Runge–Kutta schemes and the values of this parameter have been computed for all considered schemes. The verification of schemes has been done by solving a problem having the exact solution. It has been shown that the symplectic five-stage RKN scheme ensures a more accurate conservation of the total energy of a system of particles than the schemes of lower accuracy orders. All investigations of the accuracy and stability of schemes have been carried out in the analytic form with the aid of the computer algebra system (CAS) Mathematica.
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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... more
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–Stokes equations.
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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... more
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 cell of a spatial computational grid. The implementation of these modifications, however, leads to the necessity of increasing the amount of symbolic computations needed for obtaining the work formulas of the new versions of the CLR method. The computer algebra system (CAS) Mathematica has proved to be successful at the execution of all these computations. It is shown that the proposed new symbolic-numeric versions of the CLR method possess a higher accuracy than the previous versions of this method. Furthermore, the version of the CLR method, which employs the integral form of collocation equations, needs a much lesser number of iterations ...
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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... more
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 cell of a spatial computational grid. The implementation of these modifications, however, leads to the necessity of increasing the amount of symbolic computations needed for obtaining the work formulas of the new versions of the CLR method. The computer algebra system (CAS) Mathematica has proved to be successful at the execution of all these computations. It is shown that the proposed new symbolic-numeric versions of the CLR method possess a higher accuracy than the previous versions of this method. Furthermore, the version of the CLR method, which employs the integral form of collocation equations, needs a much lesser number of iterations for its convergence than the “differential” CLR method.
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Spinor analysis is applied to study the stability of finite-difference schemes in square norm. Necessary and sufficient stability conditions are obtained in terms of the coefficients in the decomposition of a amplification matrix over... more
Spinor analysis is applied to study the stability of finite-difference schemes in square norm. Necessary and sufficient stability conditions are obtained in terms of the coefficients in the decomposition of a amplification matrix over Pauli spin matrices. Formulas are derived for the coefficients in the spin expansion of an arbitrary power of the amplification matrix. Conditions are found for the asymptotic stability of a difference scheme. Decomposition of 4 x 4 transition matrices over the Dirac matrices is considered. Stability conditions are obtained for a class of difference schemes that includes the four-layer splitting schemes.
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ABSTRACT 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... more
ABSTRACT 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 stiff ODE system. It turns out that the L 2 norm of the error of the solution obtained by the new numerical second-order method is 50 times smaller than in the case of the classical second-order Runge-Kutta method.
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ABSTRACT 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... more
ABSTRACT 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 layer of the casing motionless base. To reduce further the discrepancy between the analytic solution and the direct difference solution of three-dimensional Navier–Stokes equations it is proposed to account for the viscous friction force moment on the lateral casing wall. The consideration of this moment has improved considerably the accuracy of the approximate analytic solution.
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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... more
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).
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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... more
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 mechanics, and it is used systematically subsequently throughout the book. We then introduce the definition of the tensors of strains and stress, which characterize a continuum, in a reference frame and in an actual frame without any assumptions on the smallness of strains.
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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... more
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 valid at large Reynolds numbers. This theory enables one to calculate the drag force acting on a plate in the viscous fluid flow. We also outline the theory for the transition from laminar viscous fluid flow to turbulent flow and discuss a number of the semiempirical theories of turbulence.
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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... more
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 of characteristics. The onset
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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... more
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 general principles of the construction of the