Siberian Journal of Numerical Mathematics

Volume 17, 2014

Contents

Number 1, pp. 1-99
Number 2, pp. 101-216
Number 3, pp. 217-313
Number 4, pp. 315-427


Number 1, pp. 1-99

 

Aida-zade Kamil Rajab, Abdullaev Vagif Maarif

On the numerical solution to loaded systems of ordinary differential equations with non-separated multipoint and integral conditions (in Russian), pp. 1-16

 

We propose a numerical method of solving systems of linear non-autonomous ordinary loaded differential equations with non-separated multipoint and integral conditions. This method is based on the operation of convolution of integral conditions to local conditions. This approach allows reducing the solution to the original problem to solving the Cauchy problem with respect to a system of ordinary differential equations and to linear algebraic equations. Numerous computational experiments on several test problems with application of the formulas and schemes of the numerical solution have been carried out. The results of the experiments have shown a sufficiently high efficiency of the approach described.

Key words: ordinary loaded differential equations system, non-separated conditions, integral conditions, non-local multipoint conditions, sequential shift operation.

 

Aleksandrov Vladimir

A method of optimal real-time computation of a linear system with retarded control (in Russian), pp. 17-30 

 

A new method of solving time-optimal control problems in real time has been developed. The method is based on the following: 1) approximating the attainability sets with a family of hyperplanes; 2) subdividing the whole computational process into the computations performed beforehand and those that are carried out while the control takes place; 3) integrating differential equations only over the displacement intervals of the control completion moment and the switching moments. The labor-intensive characteristic of the method is evaluated. Characteristics of calculating the optimal control of a linear system with retarded control in real time are considered. The results of modeling and numerical estimations are presented. 

Key words: optimal control, speed, switching moment, retardation, adjoint system, phase trajectory.

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Behl Ramandeep, Kanwar V., Sharma Kapil K.

New modified optimal families of King's and Traub-Ostrowski's method (in Russian), pp. 31-42 

         Based on quadratically convergent Schröder's method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King's family of fourth-order methods and Traub-Ostrowski's method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King's method and Traub-Ostrowski's method are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King's family, Traub-Ostrowski's method, and Jarratt's method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature. 

Key words: nonlinear equations, Newton's method, King's family, Traub-Ostrowski's method, Jarratt's method, optimal order of convergence, efficiency index.

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Vikhtenko Ellina Mihailovna, Maksimova Nadezhda Nikolaevna, Namm Robert Viktorovich

A sensitivity functionals in variational inequalities of mechanics and their application to duality schemes (in Russian), pp. 43-52 

 

Characteristic properties of a sensitivity functional in the variational inequalities mechanics on an example of a scalar Signorini problem are investigated. Applications of sensitivity functionals in duality schemes are considered. 

Key words: scalar Signorini problem, duality scheme, modified Lagrangian functional, sensitivity functional.

Kokurin Mikhail Yurjevich, Kozlov Alexander Ivanovich

On a posteriori approximation of a set of solutions to a system of quadratic equations with the use of the Newton method (in Russian), pp. 53-65

 

For quadratic systems of algebraic equations we propose an algorithm generating a posteriori estimates of a convex hull of a set of solutions using the results of a step of the Newton method. Results of numerical tests are given. 

Key words: quadratic operator, the Newton method, a posteriori estimate, numerical range, convex hull.

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Okuonghae R.I., Ikhile M.N.O.

A family of highly stable second derivative block methods for stiff IVPs in ODEs (in Russian), pp. 67-81

 

This paper considers a class of highly stable block methods for the numerical solution of initial value problems (IVPs) in ordinary differential equations (ODEs). The boundary locus of the proposed parallel one-block, r-output point algorithms shows that the new schemes are A-stable for output points r=2(2)8 and A(α)-stable for output points r=10(2)20, where r is the number of processors in a particular block method in the family. Numerical results of the block methods are compared with a second derivative linear multistep method in [8]. 

Key words: block methods, continuous methods, collocation and interpolation, boundary locus, A(α)-stability, stiff IVPs.

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Prashanth M., Gupta D.K., Singh S.

Semilocal convergence for the Super-Halley's method (in Russian), pp. 83-99

 

The semilocal convergence of Super-Halley's method for solving nonlinear equations in Banach spaces is established under the assumption that the second Frёchet derivative satisfies the ω-continuity condition. This condition is milder than the well known Lipschitz and Hölder continuity conditions. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where the Lipschitz and Hölder continuity conditions fail. Difficult computation of the second Frёchet derivative is also avoided by replacing it with a divided difference containing only the first Frёchet derivatives. A number of recurrence relations based on two parameters are derived. A convergence theorem is established to estimate a priori error bounds along with the domains of existence and uniqueness of the solutions. The R-order of convergence of the method is shown to be at least three. Two numerical examples are worked out to demonstrate the efficiency of our method. It is observed that in both examples the existence and uniqueness regions of solution are improved when compared with those obtained in [7]. 

Key words: nonlinear operator equations, ω-continuity condition, recurrence relations, R-order of convergence, a priori error bounds.


Number 2, pp. 101-216 

To the 7Oth Birthday of Academician B.G. Mikhailenko (in Russian), pp. 101-103

 

To the 80th Anniversary of Gennadii Alekseevich Mikhailov (in Russian), pp. 105-109

 

Abdikalykov A.K., Ikramov Kh.D., Chugunov V.N. 

On eigenvalues of (T+H)-circulants and (T+H)-skew-circulants (in Russian), pp. 111-124

 

Explicit formulas for calculating eigenvalues of the Hankel circulants, Hankel skew-circulants, (T+H)-circulants, and (T+H)-skew-circulants are obtained. It is shown that if ϕ ≠ ±1, then the set of matrices that can be represented as sums of a Toeplitz ϕ-circulant and a Hankel ϕ-circulant is not an algebra.

Key words: Toeplitz matrix, Hankel matrix, circulant, skew-circulant, eigenvalues.

 

Burmistrov A.V., Korotchenko M.A.

Weight Monte Carlo algorithms for estimation and parametric analysis of the solution to the kinetic coagulation equation (in Russian), pp. 125-138 

 

The Smoluchowski equation with linear coagulation coefficients depending on two parameters is considered. We construct weight algorithms for estimating various linear functionals in an ensemble, which is governed by the equation under study. The algorithms constructed allow us to estimate the functionals for various parameters as well as parametric derivatives using the same set of trajectories. Moreover, we construct the value algorithms and analyze their efficiency for estimating the total monomer concentration as well as the total monomer and dimer concentration in the ensemble. A considerable gain in computational costs is achieved via the approximate value simulation of the time between interactions combined with the value simulation of the interacting pair number. 

Key words: statistical modeling, evolution of many-particle system, Smoluchowski equation, value function, parametric derivative, multiplicative weight, computational cost.

Imomnazarov Kh.Kh., Mikhailov A.A. 

Application of a spectral method for numerical modeling of propagation of seismic waves in porous media for dissipative case (in Russian), pp. 139-147 

 

This paper presents the algorithm, based on the application of the spectral Laguerre method for approximation of temporal derivatives as applied to the problem of seismic wave propagation in the porous media with dissipation of energy. The initial system of equations is written down as the first order hyperbolic system in terms of velocities, stresses and pore pressure. For the numerical solution of the problem in question, the method of a combination of the analytical Laguerre transformation and a finite difference method is used. The proposed method of the solution can be considered to be an analog to the known spectral method based on the Fourier transform. However, unlike the Fourier transform, application of the integral Laguerre transform with respect to time allows us to reduce the initial problem to solving a system of equations in which the parameter of division is present only in the right-hand side of equations and has a recurrent dependence. As compared to the time-domain method, with the help of an analytical transformation in the spectral method it is possible to reduce an original problem to solving a system of differential equations, in which there are only derivatives with respect to spatial coordinates. This allows us to apply a known stable difference scheme for recurrent solutions to similar systems. Such an approach is effective when solving dynamic problems for porous media. Thus, because of the presence of the second longitudinal wave with a low velocity, the use of difference schemes in all coordinates for stable solutions requires a consistent small step both with respect to time and space, which inevitably results in an increase in computer costs. 

Key words: Laguerre transform, porous media, numerical modeling, wave field, difference scheme.

 

Mikhailenko B.G., Mikhailov A.A. 

Numerical modeling of acoustic-gravity waves propagation in a heterogeneous «Earth-Atmosphere» model with a wind in the atmosphere (in Russian), pp. 149-162

 

A numerical-analytical solution for seismic and acoustic-gravitational waves propagation is applied to a heterogeneous «Earth-Atmosphere» model. Seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. Propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations with the a wind. The proposed algorithm is based on the integral Laguerre transform with respect to time, the finite integral Fourier transform along the spatial coordinate with the finite difference solution of the reduced problem.

Key words: seismic waves, acoustic-gravitational waves, Navier-Stokes equations, Laguerre transform, finite difference method.

 

Mikhailenko B.G., Fatyanov A.G. 

Numerical-analytical modeling of wave fields for complex subsurface geometries and structures (in Russian), pp. 163-176

 

In this paper we propose an analytical method of modeling seismic wave fields for a wide range of geophysical media: elastic, non-elastic, anisotropic, anisotropic-non-elastic, porous, random-inhomogeneous, etc. for super-remote (far) distances. As finite difference approximations are not used, there is no grid dispersion when computing wave fields for arbitrary media models and observation points. The analytical solution representation in the spectral domain makes possible to carry out the analysis of a wave field in parts, specifically, to obtain the primary waves. Based on the developed program of computing the wave fields, we have carried out the simulation of water waves and seismic «ringing» on the Moon. The monotone displacement resonant to the lower frequency area with increasing the recording distance has been explained. Such a displacement was detected in experiments with a seismic vibrator. 

Key words: mathematical modeling, analytical solution, full wave fields, primary waves, elastic, non-elastic, anisotropic-non-elastic, porous, random-inhomogeneous media.

 

Mikhailov G.A.

About efficient algorithms of numerically-statistical simulation (in Russian), pp. 177-190 

 

A set of numerical algorithms for the simulation of random variables and functions as well for the parametrical numerically-statistical analysis are considered. Important specifications and explanations of the algorithms formulations and substantiation, which are effective from the standpoint of practice, are given. 

Key words: base random number, probability density function, discrete superposition method, branching of trajectories, similar trajectory method, random field, histogram.

Pchelintsev A.N.

Numerical and physical modeling of the Lorenz system dynamics  (in Russian), pp. 191-201 

 

This paper describes a modification of a power series for the construction of approximate solutions of the Lorenz system. The results of the computer-aided simulation are presented. Also, the physical modeling of the dynamics of the Lorenz system of the processes occurring in the circuit are considered. 

Key words: Lorenz system, analog multiplier, integrator, method of power series, radius of convergence, free convection, Lorenz attractor.

Smirnov S.V.

On calculation of seiche oscillations of the middle part of Peter the Great Gulf (in Russian), pp. 203-216 

 

         Characteristics of barotropic seiche oscillations of the middle part of Peter the Great Gulf are considered with the use of spectral-finite difference model. The model is based on the linearized system of shallow water equations. Difference approximation is carried out on an irregular triangular spatial mesh. The numerical method involves the solution of the eigenvalue problem and is able to directly obtain a set of frequencies and the corresponding forms of seiche oscillations. The grid computational domain covers the Amurskiy gulf and the Ussuriyskiy gulf. The Zolotoy Rog bay and the Alekseev bay are described in more detail on the grid. The calculated and presented spatial-temporal characteristics of a number of seiche oscillations corresponding to well-defined peaks of the energy spectrum of the sea level data from the station «Vladivostok» of the Russian of tsunami warning service. The results of calculations for the Alekseev bay are compared to the data of natural measurements and the solutions of the Cauchy problem. 

Key words: seiches, harbour oscillations.


Number 3, pp. 217-313  

 

Afanasyev I.V.

A cellular automata model of three organisms populations in lake Baikal (in Russian), pp. 217-227

 

A cellular automata model of population dynamics of three organisms in Lake Baikal is proposed and investigated. Each species is divided into age groups. There are eight groups all together. The model allows one to take into account a spatial organisms distribution, a seasonal dependency of birth rates, a possible habitat pollution and water streams. A computational experiment was carried out for the case of pollution that is in the south area of lake Baikal. It demonstrates that the population dynamics tends to the oscillating process with a period equal to 1 year. The assessment of the critical pollution intensity which leads to the total extinction is presented. The model was verified within production-to-biomass and frequency of occurrence ratios. 

Key words: cellular automata, discrete modeling, populations dynamics, lake Baikal, prey-predator systems.

 

Lutsenko N.A., Tarasov G.V., Gyrnik K.A.

An OpenMP version of the parallel algorithm for calculation of unsteady gas flow through porous objects with energy sources: Analysis and Application (in Russian), pp. 229-244

 

The gas flows in the gravity field through the porous objects with energy sources, which may originate from the natural or man-caused disasters, have been investigated. An OpenMP version of the parallel algorithm has been developed for the calculation of unsteady 2D gas flows through porous self-heating media of complex subsurface geometries. The structure of the sequential algorithm and the transition from it to the OpenMP version have been described, the performance and efficiency of parallelization have been analyzed. The unsteady gas flows through axisymmetric porous self-heating objects with a partial closure of the object outlet (with a top cover) have been investigated by means of the developed parallel algorithm. The influence of the partial closure of the object outlet on the cooling process of the porous objects with a non-uniform distribution of heat sources has been analyzed. 

Key words: parallel algorithms, numerical modeling, porous objects, gas cooling, heat release.

 

Orlov A.V., Malyshev A.V.

The test problem generation for quadratic-linear pessimistic bilevel optimization (in Russian), pp. 245-257

 

The generation method of quadratic-linear bilevel optimization test problems in a pessimistic formulation is proposed and justified. The propositions about the exact form and the number of local and global pessimistic solutions in generated problems are proved. 

Key words: test problem generation, bilevel optimization, guaranteed (pessimistic) solution, kernel problems.

 

Romankov A.S., Romenski E.I.

The Runge-Kutta/WENO method for solving equations for small-amplitude wave propagation in a saturated porous medium (in Russian), pp. 259-271

 

A high-accuracy Runge-Kutta/WENO method up to fourth order with respect to time and fifth order with respect to space is developed for the numerical modeling of the small-amplitude wave propagation in a steady fluid-saturated porous medium. The system of governing equations is derived from the general thermodynamically compatible model of a compressible fluid flow through a saturated elastic porous medium, which is described by the hyperbolic system of conservation laws with allowance for finite deformations of the medium. The results of numerical solution of one- and two-dimensional wavefields demonstrate efficiency of the method developed. 

Key words: high-accuracy methods, hyperbolic system of conservation laws, saturated elastic porous media, wave propagation.

 

Tripathy M., Sinha Rajen Kumar

Convergence of H1-Galerkin mixed finite element method for parabolic problems with reduced regularity of initial datas  (in Russian), pp. 273-288

 

We study the convergence of an H 1-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming reduced regularity of the initial data. More precisely, for a spatially discrete scheme error estimates of order \mathcal{O}( h2t -1/2) for positive time are established assuming the initial function p0 ϶ H 2(Ω) ∩ H01(Ω). Further, we use an energy technique together with a parabolic duality argument to derive error estimates of order \mathcal{O}( h2t -1) when p0 is only in H01(Ω). A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established. 

Key words: parabolic problems, H1-Galerkin mixed finite element method, semi-discrete scheme, backward Euler method, error estimates.

 

Shary S.P. 

On the full rank interval matrices  (in Russian), pp. 289-304

 

For interval matrices, the paper considers the problem of determining whether a matrix has a full rank. We propose the full rank criterion that relies on the search for diagonal dominance as well as criteria based on pseudoinversion of the midpoint matrix and comparison of the midpoint and the radius matrices for the interval matrix under study. 

Key words: interval matrix, full rank, full rank criteria.

 

 Shlychkov V.A., Krylova A.I. 

A numerical model of density currents in estuaries of the Siberian rivers (in Russian), pp. 305-313

 

A numerical model for studying the dynamic mixing of the sea and the river waters in the estuarial area is proposed. Computations are based on the two-dimensional longitudinal vertical stratified fluid mechanics equations and the equation of transport of salt. The model focuses on the reproduction of the local density currents at the mouth of arms branched deltas of the rivers of Siberia. The results of numerical experiments are given, the dynamic structure of the flow and salinity profiles are compared to the observational data. 

Key words: numerical simulation, turbulence, gradient-density flows, river flow, sea area, transport of salt.


Number 4, pp. 315-427

 

Vitvitsky A.A.

Cellular automata with a dynamic structure for simulating the biological tissues growth (in Russian), pp. 315-327

 

         The concept of cellular automata with the dynamic structure of a cellular space (DCA) is proposed. The DCA extends the capabilities of classical cellular automata (CA), and allows using the cellular automata approach to the problems of simulating the biological tissues growth. A DCA differs from a classical CA in that its cells are not located on a regular lattice, and intercellular connections are explicitly described by the neighborhood matrix. In addition, insertion and partition operators are introduced for the DCA. These operators allow one to dynamically change the cell space-structure. Based on this extension, the DCA-model of the apical meristem escape of Arabidopsis Thaliana growth is constructed, being a parallel composition of the two DCA: the asynchronous two-dimensional DCA simulating self-regulation in biological cells, and the synchronous one-dimensional DCA simulating growth and division of biological cells. The results of computer simulations have shown that the behavior of the proposed DCA-model matches the behavior of the existing model based on the composition of differential equations and the method of L-system (Lindenmayer system). Furthermore, the proposed DCA-model allows one to simulate growth of individual biological cells and to visualize the substances dynamics in these cells (decay, synthesis and diffusion). 

Key words: computer simulation, cellular automata, cellular automata with dynamic structure, orphogenesis, apical meristem escape, Arabidopsis Thaliana.

 

Kotel'nikov E.A. 

Minimization of a quadratic function on the sphere (in Russian), pp. 329-338 

 

In this paper, a sequential algorithm for solving the problem of minimization of a quadratic function on a sphere is proposed. At each iteration of the scheme, a two-dimensional problem of minimization is solved. Numerical comparisons with other methods are presented. 

Key words: quadratic optimization on sphere, Cholesky decomposition, trust region, step trajectory, quadratic model.

 

Leonov A.S.

Which of inverse problems can have a priori approximate solution accuracy estimates comparable in order with the data accuracy (in Russian), pp. 339-348 

 

It is proved that a priori global accuracy estimate for approximate solutions to linear inverse problems with perturbed data can be of the same order as approximate data errors for well-posed in the sense of Tikhonov problems only. A method for assessing the quality of selected sets of correctness is proposed. The use of the generalized residual method on a set of correctness allows us to solve the inverse problem and to obtain a posteriori accuracy estimate for approximate solutions, which is comparable with the accuracy of the problem data. The approach proposed is illustrated by a numerical example. 

Key words: linear inverse problems, correctness in the sense of Tikhonov, a priori and a posteriori accuracy estimate.

 

Litvenko K.V., Prigarin S.M. 

Numerical stochastic models of the sea surface undulation and extreme ocean waves (in Russian), pp. 349-361

 

The paper deals with simulation of the time-space stochastic structure of the sea surface undulation and extreme ocean waves. Numerical algorithms are constructed on the basis of conditional spectral models and models of time series adapting data of observations. Estimates for frequencies of extreme waves appearance are studied on the basis of the theory of random processes and fields.

Key words: simulation of random fields, conditional spectral models, time series, sea surface undulation, extreme ocean waves (freak-waves, rogue waves).

 

Moskalensky E.D.

On finding exact solutions of the two-dimensional eikonal equation when the front of the wave propagating in a medium is a circle  (in Russian), pp. 363-372 

 

Wave propagation in a two-dimensional medium is considered in the case when the front of the wave is a circle with the center (a(t),0) and radius r(t). A question is posed: what is the distribution of velocity in the medium? Common characteristics and examples of such media are given.

Key words: wave propagation, front of wave, eikonal equation.

 

Okuonghae R.I., Ikhile M.N.O.

L(α)-stable variable order implicit second derivative Runge Kutta methods  (in Russian), pp. 373-387 

 

This paper considers the extension of the popular Runge Kutta methods (RKMs) to second derivative Runge Kutta methods (SDRKMs) for the direct solution of stiff initial value problems (IVPs) of ordinary differential equations (ODEs). The methods are based on using collocation and interpolation techniques. The last stage of the input approximation is identical to the output method. The SDRKMs are L(α)-stable for the methods examined. Numerical experiments are given comparing one of these methods with a two derivative Runge Kutta method (TDRKM) and a second derivative linear multistep method (SDLMM). 

Key words: second derivative, Runge Kutta method, collocation, interpolation.

 

Salov G.I.

A new three-sample non-parametric statistical test with its special case equivalent to the Whitney test  (in Russian), pp. 389-397

 

In this paper, we propose a new non-parametric statistical test for the problem of homogeneity of three samples. We consider an alternative for which one sample values tend to be stochastically larger than every one from the two other samples values. The Whitney test is equivalent to special (linear) case of this test. Some comparisons are made for the cases with samples from exponential and uniform distributions. 

Key words: three samples, homogeneity test, nonparametric statistical test.

 

Smelov V.V.

A network version of the non-standard trigonometric basis and its advantages with respect to a similar polynomial basis  (in Russian), pp. 399-409 

 

A trigonometry-based functional basis as a network version is proposed. It is aimed at the approximation with high orders of accuracy of smooth and piecewise-smooth functions. A comparative analysis of the features of the basis proposed and a polynomial one is made. The trigonometric version offers considerable advantages over the polynomial bases. 

Key words: functional basis, elliptic operator, energy scalar product, functional, generalized solution, conjugation condition.

 

 Tarakanov V.I., Lysenkova S.A., Nesterenko M.V.

Iterative scheme of finding a spectrum of the product of two non-commutative operators  (in Russian), pp. 411-427

 

We consider spectral features and an iterative scheme of finding a spectrum of the product of two non-commutative partially symmetric operators in the Hilbert space H. In this case it is assumed that one of operators is compact, the second not necessarily being compact and even restricted in H. Numerical implementation of the iterative scheme for finding the operator spectrum of the problem of eigen-oscillations of the Rayleigh beam is presented. 

Key words: operator, spectrum, iterative algorithm.