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

**
Aleksandrov V.M.
Iterative method for computing time optimal control in real time mode **
(

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 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 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** (

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** (

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** (

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 **(

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.

**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 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.

Landscape parallelization and forest fire danger prediction

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.

Optimal detection of a given number of unknown quasiperiodic fragments in a numerical sequence

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.

Variable order and step integrating algorithm based on the explicit two-stage Runge-Kutta method

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.

On theory of the duality linear maximin problems with connected variables

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:

On precise edges of polynomials

This paper discusses definitions of precise edges of polynomial functions at infinitely distant points (

Random walk-on-spheres algorithms for solving mixed and Neumann boundary-value problems

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.

On a method to approximate discontinuous solutions of nonlinear inverse problems

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.

**Evstigneev V.A., Arapbaev R.N., Osmonov R.A. The dependence analysis: basic tests for data dependence **(

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.

Modeling the kinetics behind the patients flow

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 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.

On a two-dimensional analogue of the orthogonal Jacobi polynomials of a discrete variable

It is shown that if

discrete partial Fourier-Jacobi sums, Christoffel numbers.

Unsplit Perfectly Matched Layer for a system of equations of dynamic elasticity theory

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.

Simulation of complex engineering systems

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.

Six-stages method of order 3 for the solution of additive stiff systems

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.

Calculation of currents on the surface of a superconducting axially symmetric body screening an external coaxial magnetic field

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.

Method of interpolation for a boundary layer problem

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.

**Alekseev A.K. On the error transfer calculation via adjoint equations **(

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.

Parallel implementation of cellular automata algorithms for simulation of spatial dynamics

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.

A nonlinear oscillation model with separation of variables

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.

On one approach to solving the eikonal equation

The paper offers a new approach to finding a solution to the eikonal equation

The Rayleigh-Benard convection in gas with chemical reactions

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.

Numerical study of the Kelvin-Helmholtz instability resolution in a floodplain channel flow

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.

The two-dimensional GPR modeling for near-surface investigation using the Dirichlet-Neumann boundary condition combination

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.

Mathematical modeling of formation of doping nanostructures in basic material (nanotechnologies for microelectronics)

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.