Siberian Journal of Numerical Mathematics

Volume 3, 2000

Contents

Number 1, pp.1-88
Number 2, pp.89-190
Number 3, pp.191-294
Number 4, pp.295-403


Numerb 1, pp.1-88

S.S. Artemiev, A.A. Nosikova, and S.V. Soloboev
Monte Carlo method for share's price modeling (in Russian), pp.1-10

The questions of share's price modeling by Monte Carlo method are discussed. Share's pricing model which is considered as possible alternative to the classical model is obtained. Some new characteristics of risk and profit for obtained model are derived. The results of the option premium calculation with the new basis model are considered.

I.A. Blatov
On the asymptotically exact estimates of preconditioners of the incomplete factorization type (in Russian), pp.11-42

The methods of incomplete block factorization for the model elliptic boundary value problem are considered. The filling-in dependent two-side asymptotically exact estimates of the precondition number are proved.

S.N. Korobeinikov
Numerical solution of nonlinear problems on deformation of elastic shells of revolution at eigenstates (in Russian), pp.43-56

The quasi-static problems on axisymmetric deformation of shells of revolution made from an elastic material with consideration for its geometric nonlinearity are numerically solved. The special attention is given to the determination of eigenstates of shells appropriate to the non-trivial solution of a homogeneous rate problem. The Riks–Wempner approach is used wherein the parameter of the external load intensity is entered as unknown. The basic nonlinear problem in determination of eigenvalues and appropriate eigenvectors reduces to the linearized (generalized) problem in eigenvalues. The developed algorithm of solving problems on deformation of shells in the vicinity of eigenstates is approved for a problem on deformation of the longitudinally compressed simply supported circular cylindrical shell. Numerical solutions are compared with analytical ones. It has been found that the behaviour of solution of the problem in the vicinity of an eigenstate is sensitive to perturbations of geometric parameters of the shell due to the dense spectrum of eigenvalues.

Sh. Smagulov, N.M. Temirbekov, and K.S. Kamaubaev
Modeling of boundary conditions for pressure by fictitious domain method in the incompressible flow problems (in Russian), pp.57-72

A new version of fictitious domain method, as applied to viscous incompressible flow problems in velocity–pressure formulation, is considered. The Dirichlet boundary condition for pressure at the boundary of auxiliary domain is proposed. It allows to unify the computation models and to construct the computational economic algorithms. We consider a singularly perturbed elliptic boundary value problem

S.B. Sorokin
Error bounds of two-sided approximations for the Sturm–Liouville problem (in Russian), pp.73-88

In the paper, the investigation of two-side approximations method for eigenvalue problem based on the well-known fictitious method is pursued. The expansion of eigenvalues of auxiliary problem in power series by small parameter of continuation irrespective of its sign is proved. Unimprovable error bounds of two-sided approximations are obtained. The conjugate-factorized structure of the problem operator plays the decisive role in obtaining the result. The research is realized on descrete level.


Number 2, pp.89-190

Gurii Ivanovich Marchuk (on the occasion of his 75th birthday) (in Russian), pp.89-92

Gurii Ivanovich Marchuk (on the occasion of his 75th birthday) (in English), pp.93-95

E.N. Akimova, T.I. Seregnikova
The parallel algorithms for solving the three-dimensional problems of elasticity by the boundary integral equations method
(in Russian), pp.97-107

The article is devoted to parallelizing the algorithm for solving the three-dimensional problem of elasticity by the boundary integral equations method in bounded axially symmetric domains with symmetric or asymmetric boundary conditions.

V.B. Barakhnin, N.V. Borodkin
The second order approximation TVD scheme on moving adaptive grids for hyperbolic systems
(in Russian), pp.109-121

The second order approximation finite difference scheme for moving adaptive grids which is the generalization of the well-known Harten scheme is presented. The conditions sufficient to ensure that the scheme is the TVD are obtained. The numerical tests of the scheme for the system of equation of shallow water when solutions contain shocks and rarefaction waves are conducted.

L.V. Gilyova, V.V. Shaidurov
Justification of asymptotic stability of the triangulation algorithm for a three-dimensional domain
(in Russian), pp.123-136

An algorithm of triangulation construction (the subdivision into tetrahedrons) for a three-dimensional bounded domain with a smooth curvilinear boundary is considered. The algorithm starts on a given coarsest triangulation. The consequent finer triangulations are recurrently constructed by the subdivision of tetrahedrons of the previous level into 8 parts with correction of the location of vertices near the boundary to approximate the boundary. To evaluate the quality of a triangulation a certain quantitative criterion is used. It is proved that a successful (in the sense of this criterion) initial triangulation moderately detailed guarantees good quality of the consequent finer triangulations under arbitrary number of recurrent implementations of subdivision algorithm.

E.V. Goryunov, Kh.Kh. Imomnazarov
Numerical solution of combined one-dimensional inverse problems for Maxwell's equation and equations of porous media
(in Russian), pp.137-149

Combined one-dimensional inverse problems for Maxwell's equation and equations of porous media are solved numerically using the optimization approach. Representative series of numerical calculations for various models of media are given.

V.I. Drobyshevich
Implicit scheme on different time meshes for semilinear parabolic equations
(in Russian), pp.151-158

A method of the construction of difference schemes with different time-step in the subdomains is suggested. It is associated with the interpolation of a solution on a boundary of subdomains. In the case of a semilinear parabolic equation, it is proved that the solution of difference problem converges to that of a differential problem with the order O(t).

Yu.I.Kuznetsov
Hamilton form of the Jacobi matrices
(in Russian), pp.159-164

The algorithm of congruent transformation of the positive definite (semi-definite) Jacobi matrix to the form, in which the sum of inner row elements equals to zero, is presented. The definition domain of the parameter of such (not unique) transformation is defined.

Yu.M. Laevsky, P.V. Banushkina
The compound explicit schemes
(in Russian), pp.165-180

In the paper, a new approach to the design of the explicit schemes for the solution to the boundary value parabolic problems is proposed. Its basis is a block presentation of grid operator with essentially different spectral properties of the blocks. It has been shown that for corresponding choice of integration steps a stability is provided by independent conditions for every block. With the help of the Chebyshev type schemes the method to relax one of conditions is suggested.

A.I. Tyatyushkin
Parallel computations in Optimal control problems
(in Russian), pp.181-190

The multimethod's technology for the solution of the optimal control problems is implemented under the form of parallel optimization processes with the choice of a best approximation. Corresponding to this technology the solution is found by a multimethods' algorithm consisting of a sequence of steps of different methods applied to the optimization process in order to accelerate it. Such a technology allows to consider some particularities of the problem of interest at all stages of its solution and to improve the efficiency of optimal control search.


Number 3, pp.191-294

A.S. Alekseev, B.G. Mikhailenko
Numerical-analytical algorithms of solution to the forward and the inverse problems in seismology
(in English), pp.191-214

The paper deals with numerical-analytical algorithms of solution to the forward and the inverse seismological problems based on a combination of the finite integral Fourier transforms with the finite difference methods. Such an approach allows the splitting of the 3D problems to a series of the 1D problems and their parallelized solution on a multiprocessor computer.

A.B. Andreev, T.D. Todorov
Lumped mass error estimates for an isoparametric finite element eigenvalue problem
(in English), pp.215-228

The error estimate for eigenfunctions and eigenvalues of the second order elliptic operator is analyzed and justified for a class of curved isoparametric triangular finite elements. The quadrature formula giving the lump of the mass matrix is considered. The use of the same nodes for an isoparametric triangle finite element of more than one degree and a quadrature formula is the phenomenon investigated in the paper. At the end of the paper, the numerical results are presented.

P.W. Hemker, G.I. Shishkin, L.P. Shishkina
Distributing the numerical solution of parabolic singularly perturbed problems with defect correction over independent processes
(in English), pp.229-258

For a singularly perturbed parabolic equation on an interval, the first boundary value problem of reaction-diffusion type is studied. For the approximation of the boundary value problem we use previously developed finite difference schemes, of high e-uniform order of accuracy in time, based on defect correction. The new approach developed in this paper is the introduction of a partitioning of the domain for these e -uniform schemes. We determine conditions under which the difference schemes applied independently on the subdomains can accelerate (e-uniformly) the solution of the boundary value problem without losing the accuracy of the original schemes. Hence, the simultaneous solution on the subdomains can in principle be used for parallelization of the computational method.

I.V. Konnov
Properties of gap functions for mixed variational inequalities
(in English), pp.259-270

Various gap functions for a class of mixed variational inequalities containing a P-mapping function and a convex separable function which are not necessarily differentiable are considered. Such problems have a number of applications in mathematical physics, economics, and operations research. The initial problem is shown to be equivalent to a constrained optimization problem for a conventional gap function which may be non-differentiable. At the same time, the D-gap function allows one to reduce the initial problem to the problem of finding stationary points of a continuously differentiable function. This latter problem can be solved by standard unconstrained differentiable optimization methods.

V.M. Kovenya
Splitting methods for the numerical solution of multi-dimensional problems of gas dynamics
(in English), pp.271-280

On the basis of the method of splitting into physical processes and spatial directions, schemes of minimum dissipation for a numerical solution of the Euler equations are proposed written in various gas dynamic variables. Their justification for a multi-dimensional case is given and their properties are investigated.

V.A. Ogorodnikov, A.V. Protasov
Variational methods of data assimilation in the problem of stochastic modelling of complexes of hydrometeorological fields
(in English), pp.281-294

A new method of dynamic probabilistic numerical modelling of ensembles of independent realizations of complexes of space-time fields of hydrometeorological elements based on the variational principle is proposed. An ensemble of realizations satisfies statistical climatic characteristics in the atmosphere, and each realization of this ensemble satisfies the hydrothermodynamic numerical model.


Number 4, pp.295-403

 100th Anniversary of M.A. Lavrentyev (in Russian), pp.295-296

100th Anniversary of M.A. Lavrentyev (in English), pp.297-298

V.P. Il'in, A.G. Marchuk, Ya.I. Fet
M.A. Lavrentyev: the historical role in the computerization of our country
(in Russian), pp.299-304

A.F. Voevodin, V.V. Ostapenko
On calculation of hydraulic bore in open channels
(in Russian), pp.305-321

In this paper, the review of the results devoted to construction of finite-difference methods of numerical calculation of hydraulic bore (shock) in open channels, arising, in particular, at the dam destruction is presented. The problem of construction of such methods was set up to the scientists of Laboratory of Applied Hydrodynamics of the Institute of Hydrodynamics SB RAS by M.A. Lavrentiev at the beginning of 60th. In the present paper, the solution to this problem obtained for the model of one-layer ``shallow water" is considered. At the same time, the questions of conservativity of difference schemes, weak finite-difference approximation of systems of conservation laws, and construction of high accuracy shock capturing difference schemes connected with the problem mentioned above are discussed here.

T.V. Asarnova, I.A. Kolesnikov
Estimates of the elements of inverse matrices for operators with banded matrices
(in Russian), pp.323-331

In this paper, we have received the concrete estimates of the elements of inverse matrices for the bounded operators with banded (finite-diagonal) matrices acting in abstract Banach space. The method is based on the analysis of the Fourier series of some strongly continuous periodic operator-valued functions.

A.V. Kel'manov
The recognition error probability bounds for quasi-periodic sequence formed from given number of identical subsequences
(in Russian), pp.333-344

The upper and lower bounds of the recognition error probability of the quasi-periodic sequence formed from the given number of identical subsequences with unknowns (determined) instants of their beginning is obtained. The case is considered, when the unobservable quasi-periodic sequence is distorted by uncorrelated Gaussian interference with the known dispersion, and the instants of the beginning and ending of observations above the distorted sequence do not break first and last of a subsequence of hidden quasi-periodic sequence into two parts. The theoretical outcomes are illustrated by the data of numerical modeling.

A.V. Kopylov
Finite difference method for numerical solution to a boundary value problem for a mixed difference differential equation
(in Russian), pp.345-355

A boundary value problem for a mixed linear difference differential equation in a bounded rectangular domain is considered. A difference scheme for the problem is constructed. Stability, existence, and uniqueness of the solution of arising system of difference equations is proved. Weak convergence of approximations to the generalized solution of the considered boundary value problem is also proved.

A.D. Lyashko, S.E. Zhelezovskii
Correctness of an operator-differential scheme and substantiation of the Galerkin method for hyperbolic equations
(in Russian), pp.357-368

A theorem on the conditional correctness of an operator-differential scheme is proved. Using this theorem, the Galerkin method for an abstract quasilinear hyperbolic equation is substantiated in the case when the coercive solvability conditions are absent and the existence of the sufficiently smooth exact solution is supposed. The unique solvability of the approximate problems is stated and the error estimate exact by the order of approximation is obtained. The use of these results is illustrated by an example of finite element schemes applied to the first initial boundary value problem for a second-order hyperbolic equation.

M.I. Nechepurenko
Rational operators in commutative algebras
(in Russian), pp.369-376

In this paper, a certain axiomatics of the operational calculus in the associative-commutative algebras is presented. The algebra of rational operators with respect to a fixed zero undivisor is considered.

A.V. Selikhov
Autowave emergence conditions for a Cellular Neural Network
(in Russian), pp.377-394

In this paper, a formal substantiation of a parameter choice for the Cellular Neural Network, which generates autowave processes like traveling round front and traveling round pulse, is presented on the base of the investigation of neuron pair phase plane properties. Required conditions determining the value of point-like autowave source are given for two types of the CNN – with one and two stable equilibrium points on the phase plane of neuron pair. Simulating results of the autowaves in the CNN with the parameters determined correspondingly to the substantiation are also presented.

V.P. Tanana, A.A. Shtarkman
On convergence of finite dimensional approximations of L-regularized solutions
(in Russian), pp.395-403

Necessary and sufficient conditions for convergence of finite dimensional approximations of L-regularized solutions are obtained.