# Showing papers in "Ussr Computational Mathematics and Mathematical Physics in 1988"

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TL;DR: Effective methods are proposed for calculating a multidimensional discrete Fourier transform based on a new representation of it and their application to discrete number theory is proposed.

Abstract: Effective methods are proposed for calculating a multidimensional discrete Fourier transform based on a new representation of it.

39 citations

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TL;DR: In this paper, the steady-state harmonic oscillations of a vertically inhomogeneous elastic half-space, caused by oscillations caused by a load applied to its surface, are considered.

Abstract: The steady-state harmonic oscillations of a vertically inhomogeneous elastic half-space, caused by oscillations of a load applied to its surface, is considered. Methods are given for the numerical construction of Green's matrix for a continuously inhomogeneous and layered halfspace, whose stability is ensured by separation of the exponential components in explicit form.

19 citations

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TL;DR: In this paper, the uniqueness in the problem of finding the coefficients of dielectric permittivity and electrical conductivity in a Maxwellian system is considered, and the question of uniqueness is answered.

Abstract: The question of uniqueness in the problem of finding the coefficients of dielectric permittivity and electrical conductivity in a Maxwellian system is considered.

17 citations

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TL;DR: A non-commutative algorithm is proposed for multiplying square 5 × 5 matrices that uses one hundred multiplications and its application to 7 × 7, 10 × 10 and 15 × 15 matrices allows algorithms to be constructed that require 273, 700 and 2300 multiplications respectively.

Abstract: A non-commutative algorithm is proposed for multiplying square 5 × 5 matrices that uses one hundred multiplications. Its application to 7 × 7, 10 × 10 and 15 × 15 matrices allows algorithms to be constructed that require 273, 700 and 2300 multiplications respectively.

17 citations

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TL;DR: In this article, a method for calculating supersonic flows around blunt bodies is proposed based on the numerical solution of the complete two-dimensional equations of a viscous shock layer.

Abstract: A method is proposed for calculating supersonic flows around blunt bodies. This is based on the numerical solution of the complete two-dimensional equations of a viscous shock layer. Allowance for the upward transfer of disturbances through the stream into the subsonic regions is ensured by carrying out global iterations along the whole of the segment of the shock layer considered. Each global iteration is calculated by the marching method. Questions regarding the stability of iterative process are considered as well as the correctness of the formulation of the mixed problem on each global iteration. The results of the calculations are in good agreement with experimental data.

12 citations

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TL;DR: In this article, an algorithm for constructing a limiting problem in a section of the cylinder whose solution is an asymptotic approximation (as h → 0) to the initial problem is presented.

Abstract: An elliptic set of second-order equations in a cylinder of small height h is considered. Dirichlet conditions are specified on the lateral surface of the cylinder, and natural boundary conditions are specified at the ends. An algorithm for constructing a limiting problem in a section of the cylinder whose solution is an asymptotic approximation (as h → 0) to the solution of the initial problem is presented. In the case when the problem in the section is elliptic without a parameter, estimates of the rate of convergence are obtained. A program is compiled which, making the necessary calculations, constructs equations of the limiting problem.

11 citations

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TL;DR: A scheme of second order of accuracy is constructed which leads to a much higher algorithm efficiency and is based on the current state of the art in local linearization.

Abstract: The method of local linearization in its simplest form has first order of accuracy and requires a considerable amount of computer time. A scheme of second order of accuracy is constructed which leads to a much higher algorithm efficiency.

11 citations

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TL;DR: Sufficiently accurate constructive upper estimates are obtained for the length of the shortest and the complexity of the minimum DNFs of almost all Boolean functions with a small number of zeros.

Abstract: An algorithm is proposed for constructing reduced disjunctive normal forms (DNF) of Boolean functions given with zero sets wthat is most effective for a small number of zeros. Sufficiently accurate constructive upper estimates are obtained for the length of the shortest DNFs and the complexity of the minimum DNFs of almost all Boolean functions with a small number of zeros.

10 citations

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TL;DR: In this paper, a difference scheme of the solution in a rectangle of the Dirichlet problem for a differential equation of elliptic type with a small parameter for the leading derivatives is considered.

Abstract: A difference scheme of the solution in a rectangle of the Dirichlet problem for a differential equation of elliptic type with a small parameter for the leading derivatives is considered. The characteristics of the degenerate equation are parallel to the lateral sides of the rectangle. Unlike schemes investigated previously, this difference scheme converges uniformly with respect to the parameter in the whole domain. An approximation of the first-order derivatives is investigated.

10 citations

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TL;DR: In this paper, an algorithm is given for computing the points of solution of a non-linear program, to which correspond Lipschitz (multi-extremal) left-hand sides of the constraints and an objective function.

Abstract: An algorithm is given for computing the points of solution of a non-linear program, to which correspond Lipschitz (multi-extremal) left-hand sides of the constraints and an objective function. The multidimensional problem is reduced to a one-dimensional one (or a system of such problems). The algorithm decision rule uses the results of past iterations, reduced to the form of the corresponding one-dimensional problem. The method allows extra account to be taken of a priori information about the disposition of the required optimum points, as given by a distribution function.

9 citations

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TL;DR: With this method, completely conservative schemes can be obtained of two-dimensional computations on modelling a Rayleigh-Taylor instability in a closed rectangular vessel.

Abstract: A method of computing complex two-dimensional gas-dynamic flows on variable-structure meshes is proposed. The medium is represented as a set of point particles and of domains surrounding the particles. These domains are so-called Dirichlet domains. Discretization of the equations is realized on a pattern formed from “Dirichlet neighbours”. To construct the difference scheme, the method of support operators is used; with this method, completely conservative schemes can be obtained. Results are given of two-dimensional computations on modelling a Rayleigh-Taylor instability in a closed rectangular vessel.

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TL;DR: In this paper, the authors considered the propagation of non-stationary waves in a semi-restricted channel with solid walls and derived the existence and uniqueness of the solution of the problem.

Abstract: The problem of the propagation of non-stationary waves in a semi-restricted channel with solid walls is considered. Theorems concerning the existence and uniqueness of the solution of this problem are obtained and questions concerning the stabilization of the solution as t →+∞ and the existence of a regime of steady-state oscillations are considered.

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TL;DR: In this article, a numerical method for finding the global maximum of a function of several variables in a set specified by constraints of the inequality type, on the assumption that the functions satisfies a Lipschitz condition in the set, with known Lipschnitz constant, is given.

Abstract: A numerical method is given for finding the global maximum of a function of several variables in a set specified by constraints of the inequality type, on the assumption that the functions satisfies a Lipschitz condition in the set, with known Lipschitz constant.

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TL;DR: An analogue of the classical Sedov problem is considered in the case of the gravitational-gyroscopic wave equation and the behaviour of the solution at large times is investigated in this article, where the existence of a limiting amplitude is proved.

Abstract: An analogue of the classical Sedov problem is considered in the case of the gravitational-gyroscopic wave equation and the behaviour of the solution at large times is investigated. In particular, the existence of a limiting amplitude is proved.

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TL;DR: In this article, an algorithm is devised for solving numerically a problem with a power boundary layer, and the uniform convergence of the numerical solution to the exact solution is proved for certain assumptions regarding the coefficients of the equation, for which the usual difference schemes for solving boundary value problems give a poor approximation on uniform meshes of the initial equation.

Abstract: An algorithm is devised for solving numerically a problem with a power boundary layer. For certain assumptions regarding the coefficients of the equation, the uniform convergence of the numerical solution to the exact solution is proved. The usual difference schemes for solving boundary value problems give a poor approximation on uniform meshes of the initial equation, in the zones where the required function has singularities. The approximation can be improved if we introduce new independent variables, with respect to which the function-solution has either no singularities, or their order is reduced. The problem can then be solved with respect to the new variables on a uniform mesh, or in the old variables, on a non-uniform mesh, constructed by transformation of the independent variables. This approach is used below for problems with a power boundary layer.

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TL;DR: In this paper, a method of solving vanational inequalities using a penalty function which can measure the level of feasibility of each point is proposed, and a sequence of approximate solutions such that they are all feasible and have limits which solve the problem is obtained.

Abstract: A method of solving vanational inequalities using a penalty function which can measure the level of feasibility of each point is proposed. The initial point may be arbitrary. A sequence of approximate solutions such that they are all feasible and have limits which solve the problem is obtained.

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TL;DR: In this article, the authors used statistical modelling to calculate the structure of a shock wave in binary and ternary gaseous mixtures in which the concentration of one or two of the components is small.

Abstract: The method of statistical modelling is used to calculate the structure of a shock wave in binary and ternary gaseous mixtures in which the concentration of one or two of the components is small. Weighting algorithms are developed with a scheme of modelling the collisions by Bernoulli trials or with a ballot-box modelling scheme. It is shown that these algorithms lead to a kinetic equation which approximates the Boltzmann equation when there is molecular chaos with an accuracy up to the first order of smallness with respect to the time interval, Δt , into which the evolution of the model system is split and with respect to the characteristic size of a spatial cell. The concentration and temperature profiles, and the distribution functions of pairs of particles of different kinds with respect to their relative velocities, obtained in the calculations, are presented.

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TL;DR: The main characteristics of a universal program package which has been developed on the basis of this multigrid method are presented together with examples of real problems in the theory of elasticity and heat conduction which have been solved.

Abstract: A version of the multigrid method for solving three-dimensional boundary value problems is described. An operator formulation of the method is given. The main characteristics of a universal program package which has been developed on the basis of this method are presented together with examples of real problems in the theory of elasticity and heat conduction which have been solved.

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TL;DR: In this paper, a difference method was proposed for the numerical solution of a fourth-order m-dimensional parabolic equation, which, compared with the usual difference method, enables the number of necessary arithmetic operations to be reduced by a factor m 1 2 in the case of an ordinary computer, and by 2m 1 2 for a two-processor computer.

Abstract: A difference method is proposed for the numerical solution of a fourth-order m-dimensional parabolic equation which, compared with the usual difference method, enables the number of necessary arithmetic operations to be reduced by a factor m 1 2 in the case of an ordinary computer, and by a factor of 2m 1 2 in the case of a two-processor computer.

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TL;DR: In this article, a family of two-layer completely conservative difference schemes is constructed for the spatially one-dimensional equations of gas dynamics, and it is possible to obtain complete conservativeness of the difference scheme by profiling the time weights in a space matched to the solution.

Abstract: A family of two-layer completely conservative difference schemes is constructed for the spatially one-dimensional equations of gas dynamics. It is possible to obtain complete conservativeness of the difference scheme by profiling the time weights in a space matched to the solution of the problem.

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TL;DR: In this paper, a system of equations and inequalities describing the formation of filling in the case of discharge and pouring of an unconnected ideally free-flowing medium is obtained, which is shown to be equivalent to a quasivariational inequality of evolutionary type.

Abstract: The system of equations and inequalities describing the formation of filling in the case of the discharge and pouring of an unconnected ideally free-flowing medium is obtained. The system is shown to be equivalent to a quasivariational inequality of evolutionary type, and a method of solving it numerically is given.

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TL;DR: In this paper, an algorithm for constructing disjunctive normal forms of Boolean functions, specified by enumeration of zero sets, is given and the complexity of the construction is estimated.

Abstract: An algorithm is given for constructing disjunctive normal forms of Boolean functions, specified by enumeration of zero sets. The complexity of the construction is estimated. Examples are given of using the algorithm to construct the shortest disjunctive normal form for some special classes of functions.

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TL;DR: In this paper, an extension of the well-known methods is proposed, suitable for calculating the reaction speeds of strongly excited molecules in the case when the gas motion has a finite effect on the reaction kinetics.

Abstract: Methods and results of solving the kinetic equations are analysed, with the aim of justifying the macroscopic equations of gas dynamics and studying the kinetics of reactions with high-energy thresholds. An extension of the well-known methods is proposed, suitable for calculating the reaction speeds of strongly excited molecules in the case when the gas motion has a finite effect on the reaction kinetics. Then, even in the principal approximation, the solution can describe the effect of the motion and the reaction on the energy distribution of the molecules, and hence on the macroscopic reaction rate.

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TL;DR: In this article, an unbiased and simple implementation of the Monte-Carlo method for calculating the linear functionals of the solution of a complete equation of the Boltzmann type is proposed, and it is shown that the statistical error of the method is proportional to the quantity (nN) − 1 2, where n is the number of modelled particles of the ensemble and N is the independent samples of the ensembles of n particles.

Abstract: An unbiased and simple to implement Monte-Carlo method for calculating the linear functionals of the solution of a complete equation of the Boltzmann type is proposed. It is shown that the statistical error of the method is proportional to the quantity (nN) − 1 2 , where n is the number of modelled particles of the ensemble, and N is the number of independent samples of the ensembles of n particles.

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TL;DR: In this paper, the possibility of applying a model of a non-viscous gas to the numerical modelling of large-scale vortex structures is discussed, and the results of the model are compared with those obtained by other workers during an experimental investigation using visualization of the flow.

Abstract: The possibility of applying a model of a non-viscous gas to the numerical modelling of large-scale vortex structures is discussed. Such structures and observed in real flows at high values of the Reynolds number in the near wake of bodies around which the flow is poor. Both the stationary and non-stationary (of the vortex track type) numerical solutions are considered which exist for the same values of the parameters being determined. Pictures of the flow behind a circular cylinder are presented and they are compared with those obtained by other workers during an experimental investigation using visualization of the flow. An example is given of the modelling of a vortex track which is implemented on the basis of the complete Navier-Stokes equations at a moderate value of the Reynolds number.

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TL;DR: In this paper, a generalization of the Braginskii equation to the case of a plasmochemical system is presented. But this generalization is restricted to a specific case of the system.

Abstract: Hydrodynamic equations of higher accuracy are obtained as well as hydrodynamic equations which are a generalization of the equations in the form of S.I. Braginskii to the case of a plasmochemical system.

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TL;DR: The concepts of universal and local constraints are defined, and the most commonly encountered universal constraints are studied in the context of an algebraic approach to the synthesis of well-posed classification algorithms.

Abstract: In the context of an algebraic approach to the synthesis of well-posed classification algorithms, the concepts of universal and local constraints are defined, and the most commonly encountered universal constraints are studied

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TL;DR: In this paper, an effective numerical algorithm is suggested for calculating the eigenvalues of Mathieu's differential equation when the parameter of the equation takes complex values from a fairly wide range of variation.

Abstract: An effective numerical algorithm is suggested for calculating the eigenvalues of Mathieu's differential equation when the parameter of the equation takes complex values from a fairly wide range of variation. The algorithm is based on using the theory of continued fractions. The efficiency of the algorithm is verified by a series of numerical experiments and by comparing them with known numerical data. Some of the calculated values are presented in the form of tables.

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TL;DR: Several schemes for constructing high-speed e-approximate algorithms for solving discrete extremal problems and examples of specific problems for which solution algorithms are constructed on the basis of these schemes are given.

Abstract: Several schemes for constructing high-speed e-approximate algorithms for solving discrete extremal problems are described. Examples of specific problems for which solution algorithms are constructed on the basis of these schemes are given. These problems basically relate to the theory of scheduling.

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TL;DR: In this article, a modified Hartree-Fock-slater model is used to determine the thermal conductivity and spectral light absorption coefficients of a mixture of chemical elements, where the mixture is considered.

Abstract: Quantum mechanical models of a substance and, in particular, a modified Hartree-Fock-Slater model are used to determine the thermal conductivity and spectral light absorption coefficients. The case when the substance is a mixture of chemical elements is considered.