Siberian Journal of Numerical Mathematics

Volume 8, 2005

Contents

Number 1, pp.1-88
Number 2, pp.89-176
Number 3, pp.177-271
Number 4, pp.273-362


Number 1, pp.1-88

Andramonov M.Yu.
Solving systems of nonlinear equations by the parametric approach with an arbitrary initial point
(in Russian), pp. 11-22

We propose a number of algorithms for solving systems of nonlinear equations, when a good approximation to solution is unknown, and the Newton method is not efficient. These methods are based on the choice of weights for an auxiliary function and on the descent in the space of weights. The convergence depends on relations between the measures of regions of attraction of the solutions. In order to improve the performance, we consider perturbation methods.
Key words:
nonlinear equations, arbitrary initial point, random weights.

Andreev A.B., Maximov J.T., Racheva M.R.
Finite element modelling for a beam on the Winckler type basis with variable rigidity
(in English), pp. 23-30

We study constructing a beam on the Winckler basis that is under the influence of a cross-force. This force rotates around the axis of the beam. The rigidity of this basis depends on the time variable. A general mathematical model is deduced for this type of constructions. Variational formulations of the boundary value problems in question are obtained. The finite element method is used to determine the stresses of a beam. We discuss the corresponding eigenvalue problems in order to apply the method of normal shapes. Finally, a numerical result with practical application is presented.
Key words:
finite element method, the Winckler type basis, dynamic stresses.

Averina T.A., Artemiev S.S.
Numerical solution to stochastic differential equations with growing variance
(in Russian), pp. 1-10

The paper considers a new method for transition from an initial unstable in the mean square SDE system to the SDE system with a solution close to a stationary process. The SDE systems for a stochastic component are obtained with the use of the Ito formula both in the case of linear and nonlinear initial SDE systems.
Key words:
stochastic differential equations (SDEs), unstable SDEs, numerical methods for solution of SDEs, Monte Carlo methods.

Kashuba E.V., Rukavishnikov V.A.
On the p-version of the finite element method for the boundary value problem with singularity
(in English), pp. 31-42

The one-dimensional first-type boundary value problem for the second order differential equation with strong singularity of a solution caused by coordinated degeneration of input data at the origin is considered. For this problem we define the solution as -generalized one. It has been proved that solution belongs to the weighted Sobolev space under proper assumptions for coefficients and the right-hand side of the differential equation. The scheme of the finite element method is constructed on a fixed mesh using polynomials of an arbitrary degree p (the p-version of the finite element method). The finite element space contains singular polynomials. Using the regularity of -generalized solution, the estimate for the rate of convergence of the second order with respect to the degree p of polynomials is proved in the norm of the weighted Sobolev space.  
Key words:
the p-version of the finite element method, boundary value problems with singularity, the weighted Sobolev spaces, an orthonormalized singular polynomials set.

Kretinin A.V.
Forming a neuronet database for perceptrons structure optimizatio
(in Russian), pp. 43-55

Results of the solution of a variety of neuronet approximations of different topology functions are used for formation of a training set on which the neuronet database is constructed for the perceptrons structure optimization.
Key words:
perceptron, structure optimization.

Nemirovskii Yu.V., Yankovskii A.P.
Generalization of the Runge-Kutta methods and their application to integration of initial-boundary value problems of mathematical physics
(in Russian), pp. 57-76

An idea is proposed and tested to generalize the Runge-Kutta methods to a bidimensional case for the approximate integration of the initial-boundary value problems corresponding to the partial differential equations. It is shown that some classical finite difference schemes of integration of the equation of transport and non-stationary one-dimensional heat conductivity can be obtained as consequence of such generalization. New schemes of high orders of accuracy for various problems of mathematical physics are obtained. Stability of these schemes is proved, and results of calculations for problems with large gradients of the solution are presented. On concrete examples it is shown that classical schemes of low orders of accuracy unsatisfactorily describe solutions of such problems, and the schemes of high orders constructed by means of the generalized Runge-Kutta methods presented, give a good approximation to exact solutions.
Key words:
numerical integration, initial-boundary value problems, generalization of the Runge-Kutta methods, large gradients of solution, stability of numerical schemes.

Shevaldin V.T.
Approximation by local parabolic splines with arbitrary knots
(in Russian), pp. 77-88

For the class of the functions with almost bounded second derivatives, a new linear local method of parabolic spline approximation on an arbitrary grid is constructed. This method has some smoothing properties and inherits the monotonicity and the convexity of the initial data (values of a function at the grid points). On this class the error of approximation by the splines constructed is exactly calculated.
Key words:
local method, parabolic spline approximation, the error of approximation.


Number 2, pp. 89-176

Andreev A.B., Todorov T.D.
Superconvergence of the gradient for cubic triangular finite elements (in English),
pp.89-100

Superconvergence of the gradient of approximate solutions to second order elliptic equations is analyzed and justified for the 10-node cubic triangular elements. The existence of superconvergent points is proved. A recovery gradient technique in a subdomain is presented. The superclose property is proved. A rigorous proof of the superconvergent error estimate in a recovered gradient function is obtained. Numerical experiments supporting the theory under study are presented.

Key words:
finite element method, superconvergence, recovered gradient.

Artemiev S.S., Korsun A.E., Yakunin M.A.
Investigation of probability characteristics for a particular trade algorithm
(in Russian), pp. 101-108

We investigate probability characteristics of a random sequence which forms the total profitability for the trade algorithm based on a simple model of a price series. In the case of the model of a price series with a Gaussian distribution of profitabilities, the formulas for calculation of some probability characteristics are obtained.
Key words:
trade algorithm, profitability, probability density.

Kamenskii A.G., Kamenskii G.A.
On convergence of a finite difference scheme to solution of the third boundary value problem for a system of abstract elliptic equations (in Russian), pp. 109-126

There is considered the third boundary value problem for abstract elliptic equations. The problem of stability of solutions to a system of elliptic equations on a restricted domain by non-smooth perturbations of the boundary of this domain is studied. There are proposed a difference scheme for an approximate solution of the considered problem and the conditions for the convergence of solutions of this scheme to the exact solution of the problem.
Key words:
elliptic type equations, difference schemes, boundary value problems.

Larin M.
Using a compensation principle in the algebraic multilevel iteration method for finite element matrices
(in English), pp. 127-142

In the present paper, an improved version of the algebraic multilevel iteration (AMLI) method for finite element matrices, which was offered in [7], is proposed. To improve the quality of the AMLI-preconditioned, or (which is the same) to speed up the rate of convergence, a family of iterative parameters defined on an error compensation principle is proposed and analyzed. The performance results on standard test problems are presented and discussed.
Key words:
algebraic multilevel iteration method, preconditioned conjugate gradient method, finite element approximation.

Popov A.S.
The search for the best cubature formulae invariant under the octahedral group of rotations with inversion for a sphere
(in Russian), pp. 143-148

The definition of the best cubature formula invariant under the octahedral group of rotations with inversion for a sphere is given. The process of searching for the best cubature formulae of the given symmetry type is described. The table which contains the main characteristics of all the best today cubature formulae of the octahedral group of rotations with inversion up to the 53rd algebraic order of accuracy is given. The weights and the coordinates of the new cubature formulae of the 21st, 25th, 27th, 31st and 33rd orders of accuracy are given to 16 significant digits.
Key words: numerical integration, cubature formulae, octahedral group of rotations.

Shapeev A.V.
Investigation of mixed spectral and finite difference approximation on the basis of a viscous flow problem in a diffusor
(in Russian), pp. 149-162

A general approach to derivation of efficient numerical methods based on a mixed spectral and finite difference approximation for problems, whose solutions are smooth in some variables and non-smooth in other variables is considered. The approach considered is applied to the problem of a viscous liquid flow in a plane diffusor. Properties of the numerical method are investigated on the basis of computational experiments.
Key words:  non-stationary self-similar, fluid flow in a diffusor, confusor; mathematical simulation, numerical method, mixed spectral and finite difference approximation, discretization.

Shkarupa E.V.
A functional random walk-on-grid algorithm for the biharmonic equation. The error estimation and optimization
(in Russian), pp. 163-176

We consider a functional algorithm of random walk-on-grid as applied to the global solution of the Dirichlet problem for the biharmonic equation. In the metric space C, a certain upper error bound is constructed, and optimal values (in the sense of the upper error bound) of the algorithm parameters, i.e., the number of grid nodes and the sample size are obtained. We carry out numerical comparison of efficiency of the algorithm in question and the global random walk on spheres algorithm, based on the use of the fundamental solution to the biharmonic equation for the problem of a bending of a thin elastic plate with a simply supported boundary.
Key words: Monte Carlo methods, functional algorithms, random walks, biharmonic equation, error estimation, optimization.


Number 3, pp. 177-271

On the anniversary of Gurii Ivanovich Marchuk (in Russian), pp. 177-178

Alekseev A.S., Glinski B.M.,Kotelevski S.P., Kuchin N.V., Malyskin V.E., Selikhov A.V.
History of creation of the Siberian Super Computer Center, state-of-the-arts and prospects for its development (in Russian), pp. 179-187

The history of creation of computational resources of Siberian Super Computer Center (SSCC) and organization of the large-scale problem solution are considered. The results of the project on the development of multicomputer system SIBERIA in the 80-s, the current structure of the SSCC and prospects of its development are considered and analyzed.
 Key words: supercomputing, supercomputer center, multicomputer, numerical modelling.

Andreev A.B., Petrov M.S., Todorov T.D.
General results for lumped mass approximation of isoparametric eigenvalue problem on triangular meshes
(in English),
pp. 189-205

This paper deals with a FE-numerical quadrature method giving a diagonalization of the mass matrix (lumped mass matrix). The method is applied for a class of second order selfadjoint elliptic operators defined on a bounded domain in the plane. The isoparametric finite element transformations and triangular Lagrange finite elements are used.

The paper concludes with the investigation started by the authors in \cite{and1,and2} for the isoparametric variant of the lumped mass modification for second order planar eigenvalue problems. Thus the relationship between the possible quadrature formulas and the precision of the method is proved. The effect of these numerical integrations on the error in eigenvalues and eigenfunctions is estimated. At the end of the paper, the numerical results confirming the theory are presented.
 Key words: eigenvalue problem, isoparametric FEM, lumped mass, numerical integration.

Kwak Do Y., Lee Jun S.
The V-cycle multigrid convergence of some finite difference scheme for the Helmholtz equation (in English), pp. 207-218

In this paper, we analyze the V-cycle multigrid algorithm for a positive definite Helmholtz equation on a hexagonal grid. Specifically, we apply the V-cycle multigrid algorithm to the numerical scheme based on the mean value solutions for the Helmholtz equation on hexagonal grids introduced in \cite{An-Do}, and show its convergence. The theory for the V-cycle multigrid convergence is carried out in the framework in \cite{Br-Xu91} by estimating the energy norm of the prolongation operator and proving the approximation and regularity conditions. In numerical experiments, we report the eigenvalues, condition number and contraction number.
Key words:
multigrid method, mean value solution, finite difference methods.

Lisitsa V.V.
Optimal grids for solution to the wave equation with variable coefficients
(in Russian), pp. 219-229

This paper represents investigation of the method of constructing optimal grids. Extension of this method to the wave equation with variable coefficients and estimation of numerical solution obtained on optimal grids are also considered. The experiments presented illustrate a decrease of the time required for calculations.
Key words:
optimal grids, Pade-Chebyshev approximation, Gaussian quadrature rules, Lanczos method.

Lugovkin S.E.
Verification of data in certain problems of linear programming and the generalized Bernshtein problem (in Russian), pp. 231-244

We study the problem of finding the necessary and sufficient conditions of the fact that a given set of numbers could be a set of probabilities of certain events and probabilities of the pairwise interselection of such events. The algorithm of building a system of inequalities for solving this problem is obtained. The application of this algorithm to more general problems is discussed.
Key words:
event, probability, the exception method, reliability, compatibility of parameters.

Marchenko M.A.
Monte Carlo simulation of spatially inhomogeneous coagulation of particles altogether with their diffusion (in Russian), pp. 245-258

Monte Carlo algorithm for simulation of coagulation of particles altogether with their diffusion is developed. The problem to solve is the boundary-value problem for the 1D Smoluchowski equation containing convection and diffusion terms. The stochastic particles method is underlying the algorithm. The principal features of the algorithm are the use of special Markov process and a splitting scheme according to physical processes. A special technique to reduce the estimator variance is developed. The method of tentative estimation of the algorithm parameters is given.
Key words: Monte Carlo, Smoluchowski's equation, coagulation, diffusion, nucleation.

Tabarintseva E.V.
On error estimation for the quasi-inversion method for solving a semi-linear ill-posed problem
(in Russian), pp. 259-271

In this paper, the approximate solutions error estimates are obtained for an ill-posed semi-linear Cauchy problem. The continuity module of the inverse operator is used as a standard estimator for obtaining the error estimates. The value of the continuity module is calculated for two classes of uniform regularization of the original problem. The quasi-inversion method is used to construct stable approximate solutions.
Key words: differential-operator equation, Cauchy problem, ill-posed problem, method of approximate solution, error estimate.


Number 4, pp. 273-362

Akhysh A.Sh.
The \ellp stability of some difference schemes for one system of nonlinear parabolic equations (in Russian), pp. 273-280

In the present paper, the stability in the space \ellp for certain difference schemes for one system of nonlinear parabolic equations is proved. This paper is a sequel to the author's previous works.
Key words: system of nonlinear parabolic equations, stability in the space \ellp of difference schemes,
explicit scheme, implicit scheme, splitting scheme.

Artemiev S.S., Voynov A.N., Korsun A.E., Serdtseva N.A.
Parametrical analysis of trade algorithms by Monte Carlo method (in Russian), pp. 281-287

A parametric analysis of fundamental characteristics of profitability and risk of the two trade algorithms is realized by
Monte Carlo method. Numerical experiments are executed on the model prices of stocks, which are a discrete analogue to stochastic differential equations. The description of a modeling program is presented.
Key words: parametric analysis, trade algorithm, Monte Carlo method, profitability, risk.

Bukenov M.M.
Dynamic problem of linear viscoelasticity in velocity-stress statement
(in Russian), pp. 289-295

A conjugate-operator viscoelastic model in the velocity-stress statement is studied. To implement it numerically, a class of implicit difference schemes based on the spatial variables splitting is constructed.
Key words: model, velocity-stresses, finite difference schemes.

Gusev S.A.
Monte Carlo estimates of derivatives with respect to parameters of the solution of the parabolic equation based on numerical SDE solution
(in Russian), pp. 297-306

In this paper, a statistical method of estimation of the solution of the parabolic equation and its derivatives with respect to parameters is proposed. This method is based on the numerical solution of stochastic differential equations (SDE's) by the Euler method. The order of convergence of using functionals of the SDE's is determined. Some numerical results are given.
Key words: parabolic equation, derivatives with respect to parameters, stochastic differential equations, Euler method.

Kalinkin A.A., Laevsky Yu.M.
On extrapolation with respect to a parameter in the perturbed mixed variational problem
(in Russian), pp. 307-323

In this paper, the extrapolation with respect to a regularization parameter in the mixed variational problem is investigated. The estimates obtained are applied to a few examples of boundary value problems. The results of numerical experiments are given.
Key words: mixed finite element method, extrapolation, Stokes problem.

Krupchatnikoff V.N., Borovko I.V.
Some features of the polar vortex dynamics on the isentropic surfaces (in Russian), pp. 325-335

In this paper, some features of the polar vortex dynamics are investigated. We use a mathematical model, in which a stream with a linear shift with overlapped stationary waves is taken as basic state. The interaction between the basic stream and the
non-stationary Rossby waves is examined. The stability of trajectories is studied. The numerical estimation of some characteristic parameters is made, and the phenomenon of chaotic advection is discussed.
Key words: dynamics of the stratosphere, polar vortex, stability of the trajectories.

Nurmoldin Y.Y.
Restoration of functions, integrals, and solutions to the heat conductivity
equation from the Ulyanov
U2-classes (in Russian), pp. 337-351

The paper dealt with a problem of numerical integration, and approximate restoration of functions and solutions to the heat conductivity equation with functions of distribution of starting temperatures from the classes U2(β,θ,α) defined by the rate of decreasing the trigonometric Fourier coefficients. Optimal orders of errors of the quadrature formulas, restoration, and discretization by the trigonometric Fourier coefficients in L2 and L metrics are obtained.
Key words: the optimal quadrature formulas, the optimal approximate restoration of functions and decisions of the heat conduction equation.

Reddy M.V.
Average discrepancy for periodic integrands
(in English), pp. 353-362

In the numerical integration of periodic integrands over the s-dimensional unit cube, various performance criteria such as Pα and R have previously been used. In this paper, we use a criterion called L2 discrepancy. An analogue of this quantity has previously been used to study the error in the case of non-periodic integrands. For this quantity we obtain expressions for the average in the case of number-theoretic and 2
s copy rules. The values of these averages are then compared for roughly the same number of points.
Key words: average L2-discrepancy, number-theoretic rule, 2
s copy rule.