Siberian Journal of Numerical Mathematics

Volume 6, 2003

Contents

Number 1, pp.1-100
Number 2, pp.101-208
Number 3, pp.209-312
Number 4, pp.313-439


Number 1, pp.1-100

Akhysh (Akishev) A.Sh.
The stability in of some difference schemes for heat conductivity equation 
(in Russian), pp.1-16

In the present paper, the stability in space for a wide class of difference schemes corresponding to the heat conductivity equation with variable coefficients is proved. This paper is the sequel of the author's previous works. The attraction of this methodology consists in that a prior estimations in for difference problems are obtained in the same way as in the case of initial differential problems in space.

Bulgak A.S.
An algorithm for testing the practical regularity of interval matrices
(in Russian), pp.17-23

The problem of testing the practical regularity of the interval matrices is considered. In the present paper, the author proposes an algorithm for testing the practical regularity of the interval matrices.

Funaro D.
Superconsistent discretizations with application to hyperbolic equation
(in English), pp.89-99

A family of finite difference methods for the linear hyperbolic equations, constructed on a six-point stencil, is presented. The family depends on 3 parameters and includes many of the classical linear schemes. The approximation method is based on the use of two different grids. One grid is used to represent the approximated solution, the other (the collocation grid) is where the equation is to be satisfied. The two grids are related in such a way that the exact and the discrete operators have a common space which is as large as possible.

Gheit V.E.
On polynomials, the least deviating from zero in L[-1,1] metric (third part)
(in Russian), pp.37-57

The present paper is the sequel of the results of the second part [2]. The theorems stated in [6] have been proved. These theorems contain the characterization of points of the sets , , from [2, Theorem 2.2] and present a final classification of polynomials, which are the least deviating from zero in the metric L[-1,1] with four prescribed leading coefficients.

Smelov V.V.
On generalized solution of two-dimensional elliptic problem with piecewise constant coefficients based on splitting a differential operator and using specific basis functions
(in Russian), pp.59-72

An alternative method with respect to difference and variational-difference algorithms is offered. It is intended for solving a boundary value problem with the second order elliptic operator in a two-dimensional domain combined of rectangles. Coefficients of a differential operator are assumed to be piecewise constant, i.e., are constant inside each rectangle. An approximate solution of the problem is realized in a generalized version. The proposed method is based on the splitting of the differential operator, using a specific system of the basic functions which ensures approximation of the solution by means of their small number. The final objective is to reduce the problem to a solution of one-dimensional problems with the algorithm oriented to a sufficiently small dimension of algebraic systems of equations and, respectively, to the fast convergence rate of the iterative process as well as to the essentially decreased computer memory.

Sushkevich T.A., Vladimirova E.V.
On a model of radiation transport problems in the spherical shell with allowance for the reflecting boundary
(in Russian), pp.73-88

The radiation transport in the Earth's atmosphere is being investigated to scale over the whole planet. The method of the numerical solution of the general boundary value problem in the radiative transfer theory for a spherical shell with a reflecting underlying surface for the mathematical modeling of the Earth radiation field is proposed. The optical transfer operator of the spherical atmosphere-Earth system has been constructed. The models of the influence functions for the transfer theory spherical boundary value problem are formulated.

Vshivkov V.A., Malyshkin V.E., Snytnikov A.V., Snytnikov V.N.
Numerical simulation of N-body gravitational dynamics by PIC method: a parallel implementation
(in Russian), pp.25-39

The evolution of self-gravitating systems such as the accretion discs is of great interest to astrophysics. The aim of this work is to create a parallel program for the accretion disc simulation on high-performance multiprocessor computers. The disc structure formation is N-body problem in a self-consistent gravity field. A good approximation to the problem is the Vlasov-Liouville kinetic equation. In the present work, the equation is solved by the PIC method. The main difficulty here is the evaluation of gravitational potential which is given by the 3D Poisson equation. The parallel scheme of the algorithm was designed for the MIMD computers in an assembly technology. This means that the program is assembled of minimal fragments, each being a ready-made program containing potential values and the particles from one or more grid layers. The values of a grid potential are uniformly distributed among the processor elements uniformly in the radial direction. As the potential evaluation takes the main time, the distribution of particles is of minor importance here. Test computations conducted on the ICT cluster of Pentium-III workstations showed the linear acceleration as compared to the sequential version.


Number 2, pp.101-208

Amelkin V.A.
Enumeration, coding, and generation of sequences with constraints on lengths of minimum series
(in Russian), pp.101-111

The sets of binary and n-valued serial sequences of the length m with the given values of lengths of minimum series are considered. Exact formulas for the determination of the powers of such sets are obtained. The algorithms of coding and generation for the binary sequences are found.

Andreev A.B., Maximov J.T., Racheva M.R.
Finite element method for calculation of dynamic stresses in the continuous beam on elastic supports
(in English), pp.113-124

We study a damping beam construction on elastic supports. For this kind of a construction a general mathematical model is deduced. The external load is a time-depending harmonic function. The corresponding spectral problem contains the eigenvalue parameter in the boundary conditions. The variational formulations of the considered boundary value problems are obtained. The dynamic stresses of the constructions are determined using the finite element method and the method of normal shapes. Finally, the numerical results related to the problem with practical applications are presented.

Bubyakin A.A., Laevsky Yu.M.
A compact projecting-grid scheme for a class of two-dimensional diffusive equations
(in Russian), pp.125-138

A finite-element compact scheme of fourth order accuracy for the class of elliptic problems is proposed. Namely, the case of coefficients with splitting arguments is considered. The space of grid functions is designed, in which the coercive bilinear form is defined. The mesh energy norm of the error is estimated.

Gorunescu F., Gorunescu M.
Optimization of costs policy in a geriatric queuing model with extra beds provision
(in English), pp.139-147

The planning of a hospital geriatric department is a complex task, concerning both health care itself and the corresponding financial issues. The long-term geriatric services frequently occupy hospital beds for long periods of time incurring high costs. On the other hand, there are a lot of elderly people who are unable to obtain a hospital bed, because they are all occupied. The aim of this paper is to analyze the influence of the admission policy and bed allocation upon the hospital costs. Such methodology can be used by health service managers to optimize the geriatric department activity.

Kuznetsov Yu.I.
The problem of moments on a finite set of points
(in Russian), pp.149-157

The influence of the last diagonal entry bn of the Jacobi matrix on its eigenvalues, which at the same time are the nodes of orthogonality of respective polynomials as well as on the squares of the first components of the normalized eigenvectors - the weights of the orthogonality, is considered. The weights of orthogonality are the distribution masses whose moments are known and given by the positive definite Hankel matrix independent of bn. Using the solutions to the equations with special matrices the first derivatives of bn of the nodes and the weights of orthogonality of the polynomials are calculated. Their asymptotic behavior with bn =>  is discussed.

Leus V.A.
On the condition number of matrices occurring in problems of generation of functions of many variables
(in Russian), pp.159-169

The solvability of the problem of the differentially conditioned generation of a function of many variables in Rm is examined. With dimension m 2, one can say about the probabilistic solution only. It is shown that the probability to have an unambiguous solution is close to unit in the case of the analytical basis functions.

Nikolaeva N.N., Titarenko V.N., Yagola A.G.
Error estimation for solution of Abel equation on sets of monotonic and convex functions
(in Russian), pp.171-180

We consider Abel equation under condition that the exact solution belongs to a compact set. The error of finite-dimensional approximation of the problem is estimated. For the error obtained we construct an area to which the exact and the approximate solutions belong using the method of cutting convex polyhedrons.

Savchenko A.O.
A high order numerical method for the integral Volterra equations with weak singularity
(in Russian), pp.181-195

A new method of numerical solution of the linear integral Volterra equations with high accuracy, based on approximation of integrals by quadratures independent of the kernel values is proposed. This approach does allow to numerically solve special integral equations, for example, the equations with slightly singular kernels. The main idea of the method is to expand a sought for function by the Taylor formula and to use the kernel moments at the subgrid points to find the matrix of the quadrature coefficients.
The dependence of the approximation constant on the number of the subgrid points is analyzed. It is shown that the constant exponentially decreases. An estimate of the solution error for a problem with perturbations of the kernel and the right-hand side is found. The theorem of convergence for the second kind Volterra equations is proved.

Uchaikin V.V., Saenko V.V.
Stochastic solution to partial differential equations of fractional orders
(in English), pp.197-203

Partial differential equations containing the fractional derivatives β ! f/t β (0 < β ≤ 1) and (m ) α/2 (0 < α < 2).are considered. These equations generalize the ordinary diffusion equation to an anomalous one and can be solved by m-dimensional isotropic random walk with delay. In contrast to the ordinary case, a free path distribution should have a heavy tail of the inverse power type with the exponent α, and the delay time distribution should have a similar tail with the exponent β.

Notes
Tanana V.P., Sevast'yanov Ya.M.
On optimal methods of solution to linear equations of the first kind with an approximately specified operator
(in Russian), pp.205-208


Number 3, pp.209-312

Aleksandrov V.M.
An iterative method for computation of time-optimal control of quasilinear systems
(in Russian), pp.227-247

An iterative method of finding the time-optimal control for quasi-linear systems is considered. A system of linear algebraic equations is obtained, which relates deviations of the initial conditions of the normalized conjugate system and the deviation of the finite time to the deviations of phase coordinates resulted from nonlinearity. A numerical algorithm and its modifications are described. The convergence of the iterative procedure has been proved. Some examples are presented.

Axelsson O., Larin M.
An element-by-element version of variable-step multilevel preconditioning methods
(in English), pp.209-226

In this paper, an element-by-element implementation of the recently proposed variable-step multilevel preconditioning method for solving second-order elliptic boundary value problems is considered. A special technique based on the internal properties of the preconditioning are used for analysis of the rate of convergence. Performance results on standard test problems are presented and discussed.

Bezhaev A.Yu.
Spline interpolation of huge multivariate data
(in English), pp.249-261

The paper deals with the ``true" multi-dimensional interpolation problem at scattered meshes with a huge number of interpolating points. For its solution we suggest here a new numerical technology consisting in partitioning of the problem on a number of subproblems and in a successive glueing of solutions to the subproblems. The basis of the partitioning method is the algorithm of optimal hyperplane, dividing a mesh in two intersected ones.

Nikol'skii E.V.
Comparative descriptions for two methods: the classical radiation method and the method of equivalent systems
(in Russian), pp.279-290

In this paper, the author considers two alternative methods of solution of multi-dimensional second order partial differential equations. These methods are called: the classical radiation method and the method of equivalent systems (MES). It is shown that in a sense one can consider the MES as generalization and extension of the classical radiation method. As an example the author makes use of the three-dimensional wave equation.

Rozhenko A.I.
On orthogonal decomposition of space in spline-fitting problem
(in Russian), pp.291-297

A special orthogonal decomposition of a basic space for an abstract quasi-spline-fitting problem is proposed. Using this decomposition, a theorem on representation of smoothing quasi-spline σa. is proved. Exact in order convergence estimates of σa. to the limit quasi-splines  σ0 and σ are obtained. The monotony and the upper convexity of the function ψ-1(β), used in the algorithm of selection of the smoothing parameter αby the residual criterion, are proved.

Sorokin S.B.
The conjugate-operator model for a dynamic problem of the plate theory
(in Russian), pp.299-311

In this paper, according to hypotheses of a technical theory of their plates the equations, making up the mathematical model of the dynamic problem of elasticity theory, are averaged on the operator level. As a result, the conjugate-operator model of the dynamic problem of the theory of plates has been obtained, its possible statements being formulated, and approaches to their numerical realization being discussed. The efficient difference schemes (local-one-dimensional) for the statement ``velocities-moments" are substantiated.

Volkov Yu.S.
On estimation of entries of a matrix inverse to a cyclic band matrix
(in Russian), pp.263-267

Estimates for entries of matrix inverse to the diagonally dominant cyclic band matrix are obtained. These results are used for the evaluation of a norm of the inverse matrix in the case of the column diagonal dominance.

Znak V.I.
Phase-synchronized-weighted median filter and some questions of estimation of quality of its response to a frequency-modulated signal
(in Russian), pp.269-278

We discuss the specific features of the phase-synchronized-weighted median filters when processing the frequency-modulated (sweep) signals. The estimation of the response quality of such a filter is found, and a functional, enabling us to reduce the problem of the choice of the filter structure to the problem of minimization the functional, is introduced. The conclusions obtained are confirmed by the results of numerical statistical modelling.


Number 4, pp.313-439

Marchuk G.I.
Forthy years in the Computer center
(in Russian), pp.313-321

Dymnikov V.P., Volodin E.M., Galin V.Ya., Glazunov A.V., Gritsoun A.S., Dianskii N.A., Lykosov V.N.
Climate and climate change: mathematical theory and numerical modeling
(in Russian), pp.347-379

In the paper, a climate system model constructed on the basis of coupling models of general circulation of the atmosphere and ocean is presented. This model does not use the correction of turbulent heat fluxes on the sea surface. The method to calculate the response operator of climatic models and real climatic system to small external perturbations of forcing is described. The method is based on the use of dissipation-fluctuation relations for systems with a large number of positive Lyapunov's exponents. To illustrate the effectiveness of the proposed method, some results of constructing an approximate response operator for the atmospheric general circulation model are discussed. Numerical experiments to reproduce the present-day climate have been carried out. Climatic characteristics simulated by the coupled model are compared to the characteristics obtained from the community of models participating in the project SMIP. The coupled atmosphere and ocean model response to an increase of the atmospheric CO_2 concentration is analyzed. It is found that the maximal warming about 2-3.5 К takes place in the centre of Euroasia. During the cold season this warming is expressed stronger (3-5 К) than during the warm season (1-1.5 К). Approximately one third of the cold season warming in Euroasia (1-2 К) is explained by a change of the atmospheric dynamics, namely, by an increase of the arctic oscillation index.

Goldin S.V.
Geometrical fundamentals of seismic imaging: realization of contact mappings
(in English), pp.323-345

In this paper, a general formulation of the seismic imaging process on the base of ray theory is proposed. Media and wavefields are considered within geometrical seismics: a medium consists of reflectors and refractors, a useful component of the wave field coinciding with basic terms of a ray series. As connections of reflectors and seismic traveltime surfaces belong to a class of contact mappings, the problem of migration is posed as realization of a given contact mapping in a class of wavefield transforms (including the class of pseudo-differential operators).

Il'in V.P.
On the numerical solution of direct and inverse problems of geoelectromagnetic exploration
(in Russian), pp.381-394

Numerical methods for the solution of direct and inverse problems of above-ground, surface and underground electromagnetic exploration geophysics are considered. Approximations of two-dimensional and three-dimensional boundary value problems are made by finite volume methods of high accuracy. The systems of linear algebraic equations with high order sparse matrices are solved by fast incomplete factorization algorithms with acceleration by conjugate gradients. The questions of computing apparent resistivity and sensitivity of characteristics fields to variations of geometric and material properties of media are described. The solution of inverse problems are implemented by optimization approaches including the interior point methods, the Lagrange multipliers and application of sequential quadratic programming. 

Kargin B.A.
Statistical modelling in stochastic problems of the atmosphere and ocean optics
(in Russian), pp.395-410

The paper deals with some aspects of application of statistical modelling for the solution of two stochastical problems of the atmosphere and ocean optics: the optical radiation transfer in the stochastical cloudiness and in the ocean-atmosphere system with allowance for the oceanic surface perturbations. The statements of problems are presented. Some actual applications are discussed.

Lavrentiev M.M.
Models of a medium and problems of geophysical data interpretation
(in Russian), pp.411-413

In this paper we consider the problems connected with different of the geophysical medium. In particular, we observe the models used in geophysical prospecting and the related results in the theory of inverse problems. The medium model based on the works by academician M.A. Sadovsky is proposed.

Mikhailenko B.G.
Simulation of seismic wave propagation in heterogeneous media
(in English), pp.415-429

The paper presents a review of numerical methods for calculation of seismic wave fields in heterogeneous elastic media. In addition, a special attention is being given to the method for the calculation of transient wave fields for viscoelastic media models. The method is based on a combination of the integral Laguerre transform (with respect to time) with the Fourier-Bessel transform along the radial coordinate and a finite difference technique with respect to the vertical coordinate. Some examples of the calculation of seismic wave fields are given.

Romanov V.G.
On one approach to an inverse problem solution for a hyperbolic equation
(in Russian), pp.431-439

The problem of determining a coefficient under the lower term of a hyperbolic equation is considered. A suitable residual functional is proposed to use for a numerical solution to this problem. A given algorithm allows to estimate an accuracy of an approximate solution on every step of the minimization procedure.