Siberian Journal of Numerical Mathematics

Volume 10, 2007

Contents

Number 1, pp.1-122
Number 2, pp. 123-228
Number 3, pp. 229-324
Number 4, pp. 325-428


Number 1, pp.1-122

Aleksandrov V.M.
Iterative method for computing time optimal control in real time mode
(in Russian), pp.1-28

We propose a simple method for forming a piecewise constant finite control in the real-time mode, which transfers a linear system from any initial state to the origin in a fixed time. The relations for a sequence of finite controls to be transformed into the fast time optimal control are obtained. Computations are carried out while the system is monitored. The iterative process of computing the optimal control reduces to a sequence of solutions to linear algebraic equations and the Cauchy problems. Effective techniques for setting an initial approximation are proposed, which significantly decrease the number of iterations. A sequence of finite controls is proved to converge to the time optimal control. Results of modeling and computing are given.
Key words:
optimal control, finite control, linear system, phase trajectory, speed, switching moments, adjoint system, variation, iteration.

Bakirov N.K.
Optimal error of numerical integration with regard to function values at integration points
(in Russian), pp.29-42

In the paper, we consider a corrected definition for the numerical integration error norm with regard to function values at integration points. Optimal and suboptimal integration formulas are obtained for different functional spaces. Key words: optimal integration error, trapezoids formula.

Borisova N.M.
On modeling of hydraulic bore propagation at incline bank
(in Russian), pp.43-60

In the present paper, the numerical algorithm for the hydraulic bore propagation onto a dry channel on the basis of the shallow water equations is proposed. This algorithm is based on a modified total momentum conservation law. The results of numerical simulation of generation, propagation and run-up onto the inclined shore of the hydraulic bore, arising after the total or partial (in the two-dimensional case) dam-break, and of the wave like tsunami, arising after a quick local bottom rise.
Key words: the theory of shallow water equations, hydraulic bore, dry channel, inclined shore.

Larin M.R.
On a modification of algebraic multilevel iteration method for finite element matrices
(in English), pp.61-76

Today, multigrids and multilevel methods for solving a sparse linear system of equations are well known. They are both robust and efficient. In \citeAL1997, the algebraic multilevel iteration (AMLI) method for finite element matrices has been proposed. However, this method has two restrictions on the properties of the original matrix, which can fail in practice. To avoid them and to improve the quality of the AMLI-preconditioner, a family of relaxation parameters is suggested and analyzed.
Key words: algebraic multilevel iterative method, preconditioned conjugate gradient method, finite element matrices.

Philipoff Ph., Michaylov Ph.
"Belene'' Nuclear Power Plant: numerical and experimental free field signals
(in English), pp.105-122

Investigation of "BELENE" Nuclear Power Plant (NPP) free field signals is presented. The SH wave propagation through multilayer geological media in the region is considered. The original structural model of the geological column is developed. The layers are isotropic, with a constant depth and parallel skyline. The SH rays are with an arbitrary angle regarding the layers. The seismic SH waves are generated by a special detonation device. The main results of the study are graphically illustrated. The comparison between the original "BELENE" NPP experimental and the numerical surface (free field) signals (obtained by the direct problem, formulated in Section 4) for the investigated geological column is made.
Key words: NPP, structural model, FEM, digital seismic signals, power spectral density.

Senashova M.Yu.
Error estimation of computing a multivariable function and its gradient
(in Russian), pp.77-88

Graphs for the calculation of composite functions of multiple variables, the duality principle for obtaining the composite function gradient are described. Algorithms for the estimation of the computation error of the composite function and its gradient are presented.
Key words: error estimations, graph computation.

Smelov V.V.
Approximation of piecewise smooth functions by a small binary basis from eigenfunctions of the two Sturm-Liouville problems under mutually symmetric boundary conditions
(in Russian), pp.89-104

A method for construction of specific basis functions is formulated. This method is based on eigenfunctions of the two general Sturm-Liouville problems under two different mutually symmetric versions of boundary conditions. The expansion of smooth and piecewise smooth functions leads to rapidly convergent series. This result is the basis for approximation of the above-mentioned functions by means of a small number of terms.
Key words: piecewise smooth function, approximation, the Sturm-Liouville problem, eigenfunctions, rapidly convergent series.


Number 2, pp. 123-228

OBITUARY Anatoly Alekseev
(in Russian), pp.123-125

Averina T.A., Alifirenko A.A.
The analysis of stability of a linear oscillator with multiplicative noise
(in Russian), pp.127-145

In this paper, we investigate a linear SDE of second order in the Ito sense with a multiplicative noise with real parameters. This equation was reduced to a two-dimensional linear SDEs system of first order with the help of replacement of variables. This linear SDEs system is linearization of an arbitrary two-dimensional nonlinear system. We investigate the stability of a trivial solution to a linear system SDE. We obtain conditions for parameters of the system for various modes of stability. We compare the known numerical methods on the solution of an oscillatory system.
Key words: stochastic differential equations (SDEs), linear oscillator, stability of trivial solution, numerical methods for SDEs.

Baranovsky N.V.
Landscape parallelization and forest fire danger prediction
(in Russian), pp.147-158

In this paper, the new landscape parallelization approach to solving the problem of the forest fire danger prediction is considered. One of the versions is discussed, and evaluations of calculation speedup using this approach are given.
Key words:  parallel realization, forest fire danger prediction, mathematical simulation, multiprocessor systems.


Kel'manov A.V., Khamidullin S.A.
Optimal detection of a given number of unknown quasiperiodic fragments in a numerical sequence
(in Russian), pp.159-175

The a posteriori\/  approach to the problem of the noise-proof detection of unknown quasiperiodic fragments in a numerical sequence is studied. It is assumed that the number of elements in the fragments is given. The case is analyzed, where (1) the number of fragments is known; (2) the index of a sequence term corresponding to the beginning of a fragment is a deterministic (not random) value; (3) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is established that the problem under consideration is reduced to testing a set of hypotheses about the mean of a random Gaussian vector. It is shown that the search for a maximum-likelihood hypothesis is equivalent to finding the arguments which yield a maximum for auxiliary object function. It is proven that maximizing this auxiliary object function is a polynomial-solvable discrete optimization problem. An exact algorithm for solving this auxiliary problem is substantiated. We derive and prove an algorithm for the optimal (maximum-likelihood) detection of fragments. The kernel of this algorithm is the algorithm for solution to an auxiliary problem. The results of numerical simulation are presented.
Key words:  numerical sequence, a posteriori processing, quasiperiodic fragment, optimal detecting, effective algorithm.

Knaub L.V., Laevsky Yu.M., Novikov E.A.
Variable order and step integrating algorithm based on the explicit two-stage Runge-Kutta method
(in Russian), pp.177-185

The inequality for a stability control of the explicit two-stage Runge-Kutta like method is obtained.With the usage of stages of this scheme, the methods of first and second order are developed.The method of first order has a maximal length of the stability interval equal to 8. The algorithm of variable order and step is created, for which the most efficient computational scheme is chosen from the stability criterion. Numerical results with an additional stability control and variable order demonstrate an increase in efficiency.
Key words:  ordinary differential equations, stiff systems, error control, stability control.

Mamatov A.R.
On theory of the duality linear maximin problems with connected variables
(in Russian), pp.187-193

The theorems are proved for linear maximin problems with connecting restrictions, for which conditions follows the local optimality of plans of the first players of these problems. This local optimality is due to the concurrence of values of object functions of this problem and the one dual to it.
Key words:
  linear maximin problem with connected variables, players, support, dual problem.


Novikov M.A.
On precise edges of polynomials
(in Russian), pp.195-208

This paper discusses definitions of precise edges of polynomial functions at infinitely distant points (x0, y0). It has been found that the limit equalities at these points are necessary conditions: limx→x_0, y→y_0  f'x (x,y) = 0limx→x_0, y→y_0   f'y (x,y) = 0limx→x_0, y→y_0   (x f'x (x,y)+yf'y (x,y)) = 0. This allows one to obtain both finite and limit solutions of the system of necessary extremum conditions. The most typical properties of the polynomials, which have their precise edges, as well as the largest and the smallest values of polynomials at infinitely distant points have been revealed. An algorithm of finding the precise edges, which is based on constructing a parametric solution for a system of nonlinear equations, has been developed. The problems to be solved are reduced to some simpler, analysis by applying the aids of computer algebra aimed at determination of the largest and the smallest values of polynomials. The corresponding examples are given.
Key words:  polynomial, form, infinitely distant point, local extremum, precise edge, the smallest value of polynomial, parametric solution of a system of algebraic equations.

Simonov N.A.
Random walk-on-spheres algorithms for solving mixed and Neumann boundary-value problems
(in Russian), pp.209-220

We propose a new approach to constructing Monte Carlo methods for solving mixed boundary value problems for elliptic equations with constant coefficients. We derived a mean-value relation for point values of the solution. As a consequence, the walk-on-spheres algorithm can still be used even after a trajectory hits the reflecting boundary. Such an approach is
significantly more efficient than the standard one.
Key words:  Monte Carlo, random walk, walk-on-spheres, mixed boundary-value problem, Poisson equation, mean value theorem.


Tanana V.P., Tabarintseva E.V.
On a method to approximate discontinuous solutions of nonlinear inverse problems
(in Russian), pp.221-228

A method to approximate discontinuous solutions of nonlinear inverse problems is suggested. An inverse problem for a nonlinear parabolic equation is considered as an example. A sharp error estimation for the constructed approximate
solution is obtained.
Key words:  inverse problem, nonlinear operator equation, approximate solution.


Number 3, pp. 229-324

Evstigneev V.A., Arapbaev R.N., Osmonov R.A.
The dependence analysis: basic tests for data dependence
(in Russian), pp.247-265

In this paper, a comparative review of tests for data dependence applied in parallelized compilers is presented. Comparisons of advantages and disadvantages of such tests using both examples and estimated characteristics of individual criteria are given. A comparative table of all considered tests is presented.
Key words: parallelizing compilers, data dependence, loop parallelization, optimization, linear diophantine equation.

Gorunescu M., Gorunescu F.
Modeling the kinetics behind the patients flow
(in English), pp.229-235

In many practical applications, such as modeling the patients flow through a hospital, the dynamical system under consideration is described by a compartmental network system. A law of mass conservation governs this kinetic system, the instantaneous flow balances around the compartments are expressed by first order differential equations, and the state variables are constrained to remain non-negative along the system trajectories. The aim of this paper is to develop a compartmental kinetic model of the patients flow, providing a reliable picture of the dynamics behind the movement of patients. The snapshot of the modeled health care system on short or even medium-term will enable the hospital staff to simulate in vitro different scenarios and help them to make an optimum decision.
Key words: compartmental network system, patients flow, numerical integration.

Gusev S.A.
Solving SDE's numerically to estimate parametric derivatives of the solution to a parabolic boundary value problem with a Neumann boundary condition
(in Russian), pp.237-246

In this paper, a parabolic boundary value problem with a Neumann boundary condition is considered. The diffusion process with reflection from the boundary corresponds to the boundary problem. A statistical method to estimate the solution and parametric derivatives of the considered problem is proposed. This method is based on solving SDE's by the Euler method.
The order of convergence of the obtained estimates is established. The results of numerical computations are presented.
Key words: parabolic boundary value problem, reflected diffusion, parametric derivatives, stochastic differential equations, Euler method.

Korkmasov F.M.
On a two-dimensional analogue of the orthogonal Jacobi polynomials of a discrete variable
(in Russian), pp.277-284

It is shown that if Piα,β (x) (α,β > -1, i=0,1,2,...) are classical Jacobi polynomials, the system of polynomials of two variables Ψmnα,β (x,y)rm,n=0 = Pmα,β (x) Pnα,β (y)rm,n=0  (r = m + n  N 1) is an orthogonal system on the grid ΩNN=(xi, yj)Ni,j=0  [-1,1]2, where xi and yj are zeros of the Jacobi polynomial PNα,β (x). Given an arbitrary continuous function f(x,y) on the square [-1,1]2, we construct two-dimensional discrete partial Fourier-Jacobi sums of the rectangular type Sm,n,Nα,β (f;x,y) over the orthonormal system ̂Ψmnα,β (x,y)rm,n=0. Estimates of the Lebesgue function Lm,n,Nα,β (f;x,y) for the discrete Fourier-Jacobi sums Sm,n,Nα,β (f;x,y) depending on the position of a point (x,y) on the square [-1,1]2 are obtained.Besides, an application of the orthogonal Jacobi polynomials of a discrete variable Ψmnα,β (x,y) to some applied problems of geophysics is considered.
Key words: continuous function, Jacobi polynomials, Lebesgue function, grid, best approximation, orthogonal system,
discrete partial Fourier-Jacobi sums, Christoffel numbers.


Lisitsa V.V.
Unsplit Perfectly Matched Layer for a system of equations of dynamic elasticity theory
(in Russian), pp.285-297

This paper presents an original approach to the construction of a Perfectly Matched Layer based on the Optimal Grids technique. This PML allows one to reach a suitable reduction of the reflections for all incident angles. The use of the Optimal Grids approach makes it possible to considerably decrease the computational time, because high accuracy of the solution can be reached using a small number of grid points.
Key words: optimal grids, Perfectly Matched Layer, artificial boundary conditions, equations of elasticity.

Nechepurenko M.I., Okol'nishnikov V.V., Pishchik B.N.
Simulation of complex engineering systems
(in Russian), pp.299-305

In this paper, complex technical systems that are objects of automation are considered. A structure, a set of models, and functions of the simulation environment to determine an optimal strategy for the control of such systems are proposed.
Key words: engineering system, process control system, simulation, simulator.


Novikov E.A., Tuzov A.O.
Six-stages method of order 3 for the solution of additive stiff systems
(in Russian), pp.307-316

In this paper, we construct a method of the third order of accuracy to solve additive autonomous stiff systems of ordinary differential equations. Inequalities for accuracy control are obtained. The results of calculations are presented.
Key words: stiff systems, additive systems, one-step methods, Runge-Kutta methods, (m,k)-methods, error estimation.


Savchenko A.O., Savchenko O.Ya.
Calculation of currents on the surface of a superconducting axially symmetric body screening an external coaxial magnetic field
(in Russian), pp.317-324

A one-dimensional integral equation for the finding of currents on the surface of a superconducting axially symmetric body is given. For the case of an ellipsoid of rotation in a homogeneous magnetic field and for a sphere in a magnetic field with polynomial values on the axis of symmetry, an exact solution is obtained. The axis of symmetry of the body and the axis of the external magnetic field coincide. A numerical algorithm based on a combination of a projective method and an iterative regularization method to solve first kind Fredholm equations is proposed. B-splines were chosen as projectors. The results of a numerical reconstruction of the sought-for functions for some particular cases with the use of the method proposed are presented.
Key words: current, magnetic field, superconductor, axially symmetric body, screening, first kind Fredholm equations, B-splines.


Zadorin A.I.
Method of interpolation for a boundary layer problem
(in Russian), pp.267-275

A singularly perturbed boundary value problem for a second order ordinary differential equation is considered. It is assumed that the solution is found at the nodes of a uniform or nonuniform mesh. An interpolation method taking into account the boundary layer part of the solution is proposed. Using the constructed interpolation function, we find the derivative of the solution with an accuracy uniform with respect to a parameter at any point of the interval.
Key words: ordinary differential equation, boundary layer, mesh solution, linear interpolation, exponential interpolation, numerical differentiation.
 


Number 4, pp. 325-428

Alekseev A.K.
On the error transfer calculation via adjoint equations
(in Russian), pp.325-334

The calculation of a flow parameter uncertainty depending on the error in input data: initial conditions, boundary conditions, coefficients may be conducted using adjoint equations. For the pointwise error estimation, this approach is
advantageous from the computational standpoint since it needs solving only one (adjoint) system of equations in addition to the system that simulates a flowfield. The fields of"adjoint temperature'',"adjoint density'', etc. enable the calculation of an impact of any input data error on the uncertainty of a reference pointwise parameter. The considered approach can be applied to the estimation of a functional variation under the action of a small random error or a variation of input data away from a stationary point. In the vicinity of such a stationary point, the error can also be computed using adjoint equations but with much higher computational costs.
Key words: adjoint equations, error transfer.

Bandman O.L.
Parallel implementation of cellular automata algorithms for simulation of spatial dynamics
(in Russian), pp.335-348

Cellular Automaton (CA) is a mathematical model for the spatial dynamics which is mainly used to simulate
phenomena with a strong nonlinearity and discontinuity. Since the CA simulation problems size is usually very large, highly efficient methods, algorithms, and software for coarse grained parallelization are urgently needed. The engrained opinion that the fine-grained parallelism of CA eliminates the problem of coarse-grained parallelization is shown to be incorrect. The problems need to be solved. So, a general approach to the CA coarse-grained parallelization based on the CA-correctness conditions is presented. First, the formal model used for the CA representation (Parallel Substitution Algorithm) and the CA correctness conditions are given. Then parallelization methods are considered for synchronous and asynchronous CA. To achieve an acceptable efficiency for asynchronous CA, a method of its approximation with a block-synchronous CA is proposed. All the methods presented are illustrated by computer simulation results.
Key words: cellular automaton, synchronous mode, asynchronous mode, block-synchronous mode, fine-grained parallelism, coarse-grained parallelism, efficiency of parallelization, parallel substitution algorithm, correctness conditions.

Kuznetsov Yu.I.
A nonlinear oscillation model with separation of variables
(in Russian), pp.349-360

This paper deals with one-dimensional and two-dimensional oscillatory systems. The theorem about separation of spatial and time variables for this problem is proved. The ODE system for the Fourier coefficients of the solution was found. Numerical experiments with a one-dimensional oscillatory system point to the existence of other oscillations -- the energy oscillations inside some cluster of harmonics.
Key words:  nonlinear, the oscillation systems, one-dimensional, two-dimensional, Kronecker product, ODE system, separation of variables, clusters of harmonics, energy oscillations, approximant.


Moskalensky E.D.
On one approach to solving the eikonal equation fx2+fy2+fz2=φ2
(in Russian), pp.361-370

The paper offers a new approach to finding a solution to the eikonal equation fx2+fy2+fz2=φ2 with a variable velocity of the waves propagation V(x,y,z) (φ=1/V). It is based on a certain change of a dependent variable and reduction of the equation to a system of three quasilinear equations. It is shown that for certain cases of the function V(x,y,z), exact solutions to this equation can be found with the help of the approach proposed.
Key words: wave propagation, inhomogeneous medium, eikonal equation.

Palymskiy I.B., Fomin P.A., Hieronymus H.
The Rayleigh-Benard convection in gas with chemical reactions
(in English), pp.371-383

The problem of the Rayleigh-Benard convection for a chemical equilibrium gas is solved numerically. The gas is assumed to be incompressible, and the layer boundaries are assumed to be flat, isothermal, and free from the shear stress. The Boussinesq model with the coefficient at the buoyancy term depending on the transverse coordinate is used. The resultant nonlinear system of equations is solved by a previously developed numerical method based on the spectral representation of vorticity and temperature fields. Convection in incompressible gas is impossible. But, as is shown here, in an incompressible gas with chemical reactions, convection is possible owing to the anomalous dependence of the thermal expansion coefficient on temperature. Linear analysis shows that the critical Rayleigh number is essentially decreasing
at a low pressure. The instability domain spreads toward higher temperatures as the pressure increases. By the numerical method, various convection nonlinear modes are obtained: stationary, periodic, quasi-periodic, and stochastic convection.
The proposed model of convection of a chemical equilibrium gas can be useful for the understanding of the transition of a cellular combustion of surface systems into an explosion (initiation of the surface detonation) and for the calculation of operating modes of chemical reactors.
Key words: combustion, numerical analysis, simulation, hydrodynamics, convection, heat transfer, gases.

Shlychkov V.A.
Numerical study of the Kelvin-Helmholtz instability resolution in a floodplain channel flow
(in Russian), pp.417-428

Models of water flows with different velocities in the river channel are studied. The tangential velocity discontinuity causes
development of the so-called Kelvin and Helmholtz instability. The mathematical statement of the problem of studying instability based on the two-dimensional planimetric equations of the turbulent flow hydrodynamics was formulated. When constructing a numerical model, a finite difference scheme of a high spatial resolution is used, providing a possibility of the direct description of large eddies generated in the turbulence field on the line of separation. Analysis of energy transformations in the system "channel-floodplain" is made, the possibility of realization of the antigradient transport, known as negative viscosity phenomenon, is shown.
Key words: numerical modeling, turbulence, Kelvin--Helmholtz, monotone scheme, Kelvin-Helmholtz instability, river flow.


Silva J.DA, Carrasquilla A. and Priimenko V.
The two-dimensional GPR modeling for near-surface investigation using the Dirichlet-Neumann boundary condition combination
(in English), pp.385-399

We have developed an algorithm to simulate a Ground Penetrating Radar (GPR) survey responses in the two-dimensional (2D) geological media using a finite element numerical method (FEM). The scalar transverse electric mode of Maxwell's wave equation was simulated utilizing a combination of the Dirichlet and the Neumann boundary conditions. Immediately, the program designed was used to analyze various survey situations, observing such effects as antenna frequencies selection, pipes and buried tanks locations and karst cavities detection in limestone. Several pipes configurations were studied, mainly those filled with fresh water, salt water, oil and air. Thus, all these tests permitted us to conclude that the target size and conductivity change the hyperbolic pattern of the GPR response, and, the shape of the tails gives a measure of velocity and depth. In this form, we have shown how efficient GPR is to map the underground conditions and their benefits to environmental and hydrogeological studies. The results obtained allow us to perform all kinds of the 2D models using smaller meshes, which traduce in faster calculations, and, in this form, to select optimal parameters and conditions to
provide more information, which can potentially help us to develop better field surveys and, consequently, to obtain better
interpretations.
Key words: 2D GPR survey, the Dirichlet and the Neumann conditions, the finite element method, numerical simulations.

Tarnavsky G.A., Aliev A.V., and Tarnavsky A.G.
Mathematical modeling of formation of doping nanostructures in basic material (nanotechnologies for microelectronics)
(in Russian), pp.401-416

Physical-chemical processes, which constitute the basis of one of segments of a technological cycle for designing new semiconductor materials for nanoelectronics, were numerically simulated. This production stage the burning of basic material (Si, Ti or Ge) in oxygen is intended for the formation of special nanostructures of donor (P, As or Sb) and acceptor (B, Ga or Al) dopings regularly distributed in the basic material before starting the burning. In this paper, investigation of the growth of dynamics of an oxide film and the study of redistribution of dopings by physical-chemical processes of segregation on "oxide/material" wave front is carried out for some version of employed configurations of the base surface ("trench") partly closed by protecting masks, which preserve some segments of the surface from
oxidation. The distributions of doping concentration, with generation of different domains, including specific nanostructures   short-located zones (60--80 nm) of elevated concentrations of the donor and the acceptor dopings, are obtained and analyzed. These nanostructures of the donor and the acceptor dopings in the base provide the required semiconductor electrophysical properties of material.
Key words: nanotechnologies, design of new materials, mathematical modeling, oxidation of crystal silicon, doping segregation.