Siberian Journal of Numerical Mathematics

Volume 12, 2009

Contents

Number 1, pp. 1-119
Number 2, pp. 121-241
Number 3, pp. 243-359
Number 4, pp. 361-463


Number 1, pp. 1-119

Gaidomak S.V.
On numerical solution of one quasilinear algebraic-differential system by the method of spline-collocation
(in Russian), pp. 17-27

    In this paper, a quasilinear algebraic-differential system is considered. For its numerical solution of it the spline-collocation method is used. The theorem on convergence of numerical processes is proved. The results of numerical experiments are presented.
Key words:  quasilinear algebraic-differential system, spline-collocation method.

Gheit V.E., Gheit V.V.

On polynomials, the least deviating from zero in L[-1,1] metric, with five prescribed coefficients (in Russian), pp. 29-40

    The properties of polynomials
Rn+5(x), the least deviating from zero in L[-1,1] metric with five given leading coefficients, whose forms were calculated earlier, are studied.Theorems 1, 2 with Theorem  A contain a final classification of polynomials Rn+5(x), whose number of sign changes in (-1,1) is exactly equal to (n+1).
Key words:  non-negative, non-positive polynomials, polynomials, the least deviating from zero in integral metric.

Kamont Z., Czernous W.
Implicit difference methods for Hamilton Jacobi functional differential equations
(in Russian), pp. 57-70

    Classical solutions of initial boundary value problems are approximated in this paper by solutions of associated implicit difference functional equations. The stability is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used for numerical solving of nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent and the corresponding explicit difference methods are not convergent. The results can be applied to differential integral problems and differential equations with deviated variables.
Key words:  initial boundary value problem, functional differential equation, implicit difference method, Newton method.

Korobeynikov S.N., Reverdatto V.V., Polyansky O.P., Sverdlova V.G., Babichev A.V.
Computer simulation of underthrust and subduction at collision of plates
(in Russian), pp. 71-90

    Mathematical simulation of a collision of lithospheric slabs at which one slab is sank into the mantle under another one is carried out. Problems of the crust and the mantle deformation are numerically solved, so that the finite element method is used for spatial discretization of the equations of deformable solid mechanics, and for evolution of the collision process, the step-by-step integration of the quasistatic deformation equations is applied. Problems of the slabs movement are solved at geometric nonlinear statement in view of large deformations of bodies and contact interactions of slabs and the mantle. The solution is numerically carried out by using the MSC.Marc 2005 code, in which the formulations of equations with required types of nonlinearites are implemented. That part of the Earth's crust which has no tendency to sinking in the mantle is simulated by the prescribed movement of a rigid body. Another part of the Earth's crust, which by virtue of properties of the initial geometry should sink, is simulated by a deformable body with an elastic-plastic strain hardening material. The mantle is simulated by an ideal elastic-plastic material with a small value of yield stress. Parts of the Earth's crust with different geometric parameters are considered. From the computer simulation of plates collision it follows that in standard conditions, the underthrust of one slab under another one is realized, and at some initial thickening of a plate in a contact zone the subduction (deep sinking) of this plate is possible. It is shown that in the latter case it is necessary to take into account the known experimental fact of material condensation of a sunk piece of the plate.
Key words:  tectonic processes, subduction, computer simulation, finite element method.

Nabongo D., Boni T.K.
An adaptive scheme to treat the phenomenon of quenching for a heat equation with nonlinear boundary conditions
(in Russian), pp. 107-119

    This paper concerns the study of numerical approximation for the following boundary value problem



where p>0, u0C2([0,1]), u0 (0)=1 and u'0 (1)=-u-p0
(1). We find some conditions under which the solution of a discrete form of the above problem quenches in a finite time and estimate its numerical quenching time. We also prove that the numerical quenching time converges to the real one when the mesh size goes to zero. Finally, we give some numerical experiments to illustrate our analysis.
Key words:  discretizations, heat equation, quenching, numerical quenching time, convergence, nonlinear boundary conditions.

Znak V.I., Grachev O.V.
Some issues of improving the quality of noisy periodic signals and numerical estimation of their parameters and characteristics; cluster approach-statement of problem
(in Russian), pp. 41-55

    In this paper, the technique of processing and analysis of noisy periodic signals are offered, where signals are recordedat  discrete moments of time. The technique includes the two stages: signal quality improvement (processing) with the use of the weighted order statistics filters and the cluster analysis of results of processing. The basic definitions and features of the named type of filtration are listed, as well as formulations and definitions of the cluster analysis are introduced. The above said enables us to state problems of periodic signals analysis. Conclusions concerning the efficiency of the cluster analysis with conditions of using the weighted order statistics filters are proved to be true with the use of results of numerical modeling, where a noisy frequency-modulated signal is employed as a model.
Key words:  weighted order statistics, processing and analysis of periodic signals, cluster analysis.

Lu Z., Chen Y.
L-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations (in Russian), pp. 91-105

    In this paper, we investigate
L-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations using mixed finite element methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive L-error estimates of optimal order for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, we present numerical tests which confirm our theoretical results.
Key words:  
L-error estimates, optimal control problem, semilinear elliptic equation, mixed finite element methods.

Voevodin A.F.
The factorization method for linear and quasilinear singularly perturbed boundary problems for ordinary differential equations
(in Russian), pp. 1-15

    For linear singularly perturbed boundary value problems we offer the method that reduces solving a differential problem to a discrete (difference) problem. The difference equations are constructed by the factorization method and are an exact analogy of differential equations. The coefficients of difference equations are calculated by solving the Cauchy problems for first order differential equations. In this case, the nonlinear Ricatti equations with a small parameter are solved by the asymptotic method, and linear equations are solved by the numerical methods. Solution to the quasilinear singularly perturbed equations is obtained by the implicit relaxation method. The solution to a linearized problem is calculated by analogy with a linear problem at each iterative step. The method is tested with the known Lagestrome-Cole problem.
Key words:  factorization method, asymptotic method, relaxation method.

 


Number 2, pp. 121-241

AverinaT.A.,  Yakunin M.A.

Parameters estimates of a price series model as  solution to linear SDE with a Poisson component (in Russian), pp. 121-129.

 

    The model of a series of price increments with jumps is constructed based on a linear stochastic differential equation with a Poisson component. The estimates of unknown parameters of the model and SDE are obtained with the help of the method of moments. The algorithm for statistical simulation of the solution to SDE with a Poisson component in a general form is proposed. Some results of the numerical experiments are given.

Key words: stochastic differential equations (SDE), Poisson component, price series, estimates of parameters.

 

Balandin A.L.

Vector spherical harmonics in 3-D vector tomography (in Russian), pp. 131-143.

    A method of series expansion with the aid of vector spherical harmonics intended for inverting line integrated experimental (Doppler) data is proposed to investigate the 3-D vector fields in laboratory plasmas in spherical tokamak devices. A number of numerical computations demonstrating the 3-D reconstruction of the model vector fields have been performed to assess the inversion method proposed.

Key words: tomography, X-ray transform, vector spherical harmonics.

 

Borovko I.V.,  Krupchatnikoff V.N. 

The influence  of  the  stratosphere  polar  vortex  dynamics upon the low troposphere circulation (in Russian),  pp. 145-160.

 

    In  this paper,  the response of the extratropical troposphere to the polar  stratosphere temperature decrease, accompanied by the polar vortex strengthening, is investigated. For this purpose, we use a spectral general circulation model with zonally-symmetric boundary conditions on the surface and a heat source given analytically.

Key words: stratosphere,  polar vortex, annular mode.

 

Khatuntseva O.N.

A theoretical definition of dimension of simply connected fractal objects in problems of the viscous “fingers” formation and the dendrites growth (in Russian), pp. 231-241.

 

    The earlier developed method for a description of  discontinuous functions is applied to the determination of parameters specifying fractal objects - dimension and geometrical coefficients for two classes of problems: the viscous ``fingers'' formation and the dendrites growth.

Key words: fractal, fractional dimension, discontinuity of the first kind.

 

Kremer I.A.,  Urev M.V.

A regularization method for the stationary Maxwell equations in an inhomogeneous conducting medium (in Russian), pp. 161-170.

 

    This paper considers a problem of defining the vector potential of a magnetic field with a non-standard calibration in an inhomogeneous conducting medium. The problem in question is the one with constraints on the right-hand side and on the solution itself. The generalized and regularized statement of this problem without constraints is proposed and substantiated. This statement of the problem is equivalent to the original generalized problem with constraints.

Key words:  stationary Maxwell's equations, vector potential, saddle point problem, regularization, discontinuous coefficients.

 

Laevsky Yu.M., Yausheva  L.V.

Simulation of the filtrating gas combustion processes in non-homogeneous porous media (in Russian), pp. 171-187.

 

    A one-dimensional two-temperature model of a combustion front moving with filtration of a fuel gaseous mixture in chemically inert porous media with discontinuous thermo-physical parameters is numerically investigated. From the algorithmic stand point we deal with the new applications of two-level explicit and semi-implicit difference schemes with moving adaptive meshes. From the stand point of physical features of the processes under consideration, the main attention is given to aspects of stabilization of the combustion front, which is important in some technical applications. 

Key words: combustion, porous media, discontinuous parameters, difference scheme, adaptive mesh, stabilization.

 

Moskalensky E.D.

Finding exact solutions to the two-dimensional eikonal equation (in Russian), pp. 201-209.

 

    In this paper, the two-dimensional eikonal equation fx2 + fy2 = φ2, where φ= 1/υ, and υ (x,y), is a waves propagation velocity, is discussed. This non-linear equation is reduced to a quasilinear equation for a new dependent variable u. For some kinds of the functions φ, solutions to the quasilinear equations are found. This means that it is possible to solve the original equation for such φ. This paper also offers an approach to finding a new solution based on a known one.

Key words: wave propagation, inhomogeneous medium, eikonal equation, harmonic functions.

 

Moughrabi I.A.R.

New implicit multi-step quasi-Newton methods (in Russian), pp. 189-200.

 

    Multi-step quasi-Newton methods for optimization use data from more than one previous step to construct the current Hessian approximation. These methods were introduced by Ford and Moughrabi in [3,4], where they showed how to construct such methods by means of  nterpolating curves. To produce a better parametrization of the interpolation, Ford [2] developed the idea of “implicit” methods. In this paper, we describe the derivation of new implicit updates which are similar to the methods I4 and I5 developed in [7]. The experimental results we present here show that both of the new methods produce better performance than the existing methods, particularly as the dimension of the test problem grows. 

Key words:  unconstrained optimization, quasi-Newton methods, multi-step methods.

 

Philipoff Ph., Tchobanov V., Grammatikopoulos M., Michaylov  Ph.

An indefinite boundary soil-structure interaction mathematical model (in Russian), pp. 221-230.

 

    A model of the indefinite boundary soil-structure interaction problem is presented in the paper. The structure is described by finite elements, the soil is described by a partial hyperbolic equation, and the contact between the soil and the structure is described by a matrix integral equation. The structure and soil damping are examined and a theorem for obtaining a structural finite element damping description is demonstrated.

Key words: indefinite wave boundary problems, structural methods, dynamical condensation, structure and soil damping.

 

Romanov L.N.

On minimization of risk for restoration of atmospheric data (in Russian), pp. 211-219.

 

    This paper deals with restoration of gaps in meteorological fields aimed at the weather simulation. The universal approach, based on the local approximation which allows us to restore gaps in the meteorological fields in various conditions, and at the same time to realize the step-by-step weather forecasting, is described. The results of  the unit gap restorations for some meteorological elements are presented. These results are compared to those obtained with the local averaging and by the inertia methods. The possibilities of the data restoration as well as the step-by-step prediction in the case when the global meteorological data are available are discussed.

Key words:  average risk, approximation, small sample, multidimensional fields, process.

 


Number 3, pp. 243-359

Aleksandrov V.M.

A numerical method of solving a linear problem on a minimum consumption of resources (in Russian), pp. 247-267

 

    A simple algorithm of developing a quasi-optimal control relative to the consumption of resources is considered. The control is used as an initial approach to an iterative procedure of computing the optimal control. A system of linear algebraic equations is obtained that approximately relays the increments of the initial conditions of the adjoint system to the increments of the amplitudes of the quasi-optimal control over ultimate values. A local convergence of the computing process with a quadratic rate is proved, a radius of the local convergence being found. The condition of global convergence of the method is determined.

Key words: optimal control, quasi-optimal control, finite control, consumption of resources, linear system, phase trajectory, switching time, adjoint system, variation, iteration, convergence.

 

Kel'manov A.V., Khamidullin  S.A.

On one recognition problem of vector alphabet generating a sequence with a quasi-periodical structure (in Russian), pp. 275-287

 

    In this paper, we analyze one version of the off-line recognition problem of the vector alphabet in the case when this alphabet is a generator of sequences having quasi-periodical vector-fragments, these fragments coinciding with alphabet vectors. It is shown that the solution of this problem is reduced to that of a special optimization problem. We have proven that this problem is solvable in a polynomial time. An algorithm for an exact solution to this problem is justified. This algorithm ensures the maximum-likelihood recognition of the vector alphabet under condition when the noise is additive and is a Gaussian sequence of independent random values having an identical distribution.

Key words: discrete optimization problem, efficient algorithm, alphabet of vectors, off-line recognition, Gaussian noise, maximum-likelihood, numerical sequence, quasiperiodical fragments.

 

Kuzin V.I., Krupchatnikov V.N., Fomenko A.A., olubeva E.N., Martynova J.V., Platov  G.A.

A study of the dynamics of the Northern Eurasia and the Arctic basin climatic system (in Russian), pp. 289-295

 

    This paper studies the dynamics of the Northern Eurasia climate under conditions of the global climate change based on a coupled atmosphere and ocean general circulation models. The estimation and the analysis of a feedback for some parameters of the atmosphere are done. The role of the biosphere in the climate dynamics of the 21-st century is investigated, with allowance for the structure of the surface layer, the vegetation layer, soil and the hydrology dynamics. The main features of the dynamics of the North Atlantic and the Arctic ocean in the periods appropriate to various phases of an index of the North-Atlantic Oscillations (NAO) are investigated.

Key words: climate dynamics, mathematical modeling.

 

Maslovskaya L.V., Maslovskaya  O.M.

The penalty method of grids matching in the mixed Herrmann-Miyoshi scheme (in Russian), pp. 297-312

 

    The penalty method for mixed finite element methods is formulated and studied. The Herrmann-Miyoshi scheme for the biharmonic equation is considered. The main idea is to construct a perturbation problem with two parameters that play the role of penalties. The perturbation problem is constructed by substitution of the main conditions in the mixed variational formulation on the interface by natural conditions that contain parameters. The discretization of the perturbation problem by the finite element method is done. Estimates of the norm of a difference between the solution of a discrete perturbation problem and that of a given problem are obtained. Recommendations for choosing penalties depending on a mesh size and penalties are given.

Key words: mixed scheme, grids matching, the penalty method, loss in convergence rate.

 

Rukavishnikov V.A., Kuznetsova E.V.

A scheme of a finite element method for boundary value problems with non-coordinated degeneration of input data (in Russian), pp. 313-324

 

    We construct a scheme of a finite element method for boundary value problems with non-coordinated degeneration of input data and singularity of solution. The rate of convergence of an approximate solution of the proposed finite element method to the exact Rυ-generalized solution in the weight set W12,υ*+β/2+1(Ω,δ) is investigated, the estimation of finite element approximations is established.

Key words: non-coordinated degeneration of input data, Rυ-generalized solution, singularity of a solution, finite element method.

Shary S.P.

On comparison between Apostolatos-Kulisch and Mayer-Warnke theorems in interval analysis (in Russian), pp. 351-359

 

    This paper deals with comparing Apostolatos-Kulisch theorem and Mayer-Warnke theorem that form a basis of the so-called formal (aka algebraic) approach to the outer interval estimation of the solution sets for interval linear systems of equations. We show that despite a greater generality of Mayer-Warnke theorem, it extends the applicability scope of the formal approach to a very small extent, and a practical significance of such an extension is inessential.

Key words: interval linear equations, solution set, outer estimation, format approach, Apostolatos-Kulisch theorem, Mayer-Warnke theorem.

 

Tsetsokho V.A., Belonosova A.V., Belonosov  A.S.

Calculation formulas of linear geometrical spreading at ray tracing in a 3D block-inhomogeneous gradient medium (in Russian), pp. 325-339

 

    In this paper, recurrent formulas, suitable for direct programming, to calculate a linear geometrical spreading of the central field of seismic rays in a 3D block-gradient medium needed for organizing shooting to area observation systems have been obtained.

For recalculation formulas through the interface, a new representation using a special operator of non-orthogonal projection allowing an additive separation of terms depending only on the ray curvature, the boundary curvature, and the variable character of the velocities ratio along the boundary, has been found.

Formulas for partial derivatives of the eikonal via linear and angular geometrical spreading are presented.

Key words: ray method, geometrical spreading, ray tracing, 3D shooting.

 

Tsibulchik G.M.

Continuation of elastic waves in reverse time (in Russian), pp. 341-350

 

    Methods based on the inverse continuation of the oscillation field have received a wide use in the processing of multi-channel seismic prospecting data. Physically, the idea of this approach is clear: a wave field observed on some surface is continued into the medium and backward in time. Mathematically, all continuation algorithms that are used are based on a scalar model of the wave equation describing sufficiently well the wave nature of oscillations of individual types, but not taking into account the vector nature of these oscillations. It is well known that a system of equations of the dynamic elasticity theory (Lame equations) is a more adequate model for the description of seismic oscillations. In this paper, continuation of the field of elastic oscillations in an inhomogeneous isotropic medium is considered.

Key words: seismic waves, inverse problem, field continuation, Lame equations.

 

Voytishek A.V., Rogasinsky  S.V.

Minimal variance of an integer stochastic value (in Russian), pp. 269-273

 

    In this paper, the statement about a variance minimum of an integer stochastic value with fixed mathematical expectation is proved.

Key words: splitting of collision estimator, discrete integer stochastic value, fixed mathematical expectation, variance minimum.

 


Number 4, pp. 361-463  

Averina T.A.

Statistical simulation methods for a non-homogeneous Poisson ensemble (in Russian), pp. 361-374

 

    In this paper, some Monte-Carlo methods for modeling homogeneous and non-homogeneous Poisson ensembles are offered. Generalization of the Maximum Cross-section Method is constructed and proved for modeling non-homogeneous Poisson ensembles of points.

Key words: Poisson random process, Poisson ensemble, stochastic differential equations, Monte-Carlo methods.

 

Alekseev A.K., Makhnev I.N.

On using the Lagrange coefficients for a posteriori error estimation (in Russian), pp. 375-388

 

    A posteriori error estimation of the goal functional is considered using a differential presentation of a finite difference scheme and adjoint equations. The local approximation error is presented as a Tailor series remainder in the Lagrange form. The field of the Lagrange coefficients is determined by a high accuracy finite difference stencil affecting results of computation. The feasibility of using the Lagrange coefficients for the refining solution and estimation of its uncertainty are considered.

Key words: a posteriori error estimation, postprocessor, adjoint equations.

 

Amelkin V.A.

Numeration of non-decreasing and non-increasing n-valued serial sequences (in Russian), pp. 389-401

 

    We have examined finite sets of n-valued serial sequences, whose structure is not only limited to the number of series and their lengths, but also is limited to the series heights, by whose limitations the order of following series of various heights is given.

    We have obtained solutions of numeration and generation problems for sets of sequences: non-decreasing and non-increasing, in which a difference in heights of the neighboring series is either no less or no greater than a certain quantity. We have developed the algorithms, which assign smaller numbers to lexicographically lower sequences and which ascribe smaller numbers to lexicographically higher sequences.

Key words: series, serial sequences, series length, series height, limitations.

 

Kuznetsov Yu.I.

The eigenvalue of the symmetric Toeplitz matrix (in Russian), pp. 403-407

 

    The algorithm for determination of eigenvalues of the symmetric Toeplitz matrix is worked out in this paper. To this end, common features of problems of eigenvalues for the symmetric Toeplitz matrix and the persymmetric Hankel one are substantiated. The latter is reduced to the problem of eigenvalues for the Jacobi persymmetric matrix.

    In case of the even order, the problem reduces to the Jacobi matrix of the half order.

Key words: symmetric Toeplitz matrix, Hankel structure, Jacobi matrix, persymmetric, tranzitivibilty, Sturm theorem, algorithm, polynomials, roots, eigenvalue.

 

Kushniruk N.N., Namm  R.V.

The Lagrange multipliers method for solving a semicoercive model problem with friction (in Russian), pp. 409-420

 

    Unconditional minimization of a non-differentiable functional, arising in a model friction problem is reduced to conditional minimization of a differentiable functional. A dual scheme, based on a modified Lagrangian functional, is used for the solution of the obtained semicoercive problem.

Key words: semicoercive model friction problem, variational inequality, Lagrange multipliers method.

 

Penenko V.V.

Variational methods of data assimilation and inverse problems for studying the atmosphere, ocean, and environment (in Russian), pp. 421-434

 

    Methods for the combined use of mathematical models and observational data for studying and forecasting the evolution of the natural processes in the atmosphere, ocean and environment are presented. Variational principles for estimation of the functionals defined on a set of the functions of state, parameters and sources of the models of processes are the theoretical background. Mathematical models with allowance for uncertainties are considered as constraints to the class of functions. The main attention is paid to methods of successive data assimilation and to the inverse problems.

Key words: variational principles, data assimilation, adjoint problems, sensitivity analysis, uncertainty assessment, inverse problems, models of atmospheric dynamics and chemistry.

 

Prigarin S.M., Hahn K., and Winkler  G.

Variational dimension of random sequences and its application (in Russian), pp. 435-448

 

    A concept of variational dimension for a random sequence with stationary increments is introduced. In the Gaussian case, the variational dimension in the limit coincides with the Hausdorff dimension of a proper random process. Applications of the concept are illustrated by examples of the neurology data and the network traffic analysis.

Key words: random sequences with stationary increments, variational dimension, Hausdorff dimension, fractal, self-similarity, data analysis.

 

Smirnov S.V.

On barotropic trapped wave solutions with no-slip boundary conditions (in Russian), pp. 449-463

 

    Barotropic trapped wave solutions of a linearized system of the ocean dynamics equations are described, for semi-infinite, f-plane model basin of a constant depth bordering a straight, vertical coast, for some «typical» values of the model parameters. No-slip boundary conditions are considered. When the wave length is shorter than the Rossby deformation radius, the main features of the wave solutions are as follows: the Kelvin wave exponential offshore decay scale essentially decreases as the wave length decreases, an additional wave solution propagating in the opposite direction appears.

Key words: ocean dynamics, trapped waves, Kelvin wave.