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2017 year, number 3

1.
Optimal resource consumption control of perturbed systems

V.M. Aleksandrov
Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090
Keywords: оптимальное управление, расход ресурса, возмущение, время перевода, быстродействие, моменты переключений, итерационный процесс, сопряженная система, фазовая траектория, optimal control, resource consumption, perturbation, moving time, switching moments, iterative process, conjugate system, phase trajectory

Abstract >>
A method for calculating the optimal consumption of the resource control of perturbed dynamic systems. This method includes both normal and singular solutions. According to the method proposed the problem is subdivided into three independent tasks: 1) a consideration of the effects of perturbations on the system; 2) computation of the optimal control structure; 3) computation of the switching moments of optimal control. A consideration of the effects of perturbations on the system and transfer to a non-zero final state are reduced to the transformation of the initial and final states of the systems. The structure calculation is based on the relation between deviations in the initial conditions of the conjugate systems and deviations of the phase trajectory at the completion instant. An iterative algorithm has been developed, its characteristics being considered. The results of modeling and numerical calculations are given.



2.
The choice of the equation of state in mathematical models of pipeline transportation of natural gas

E.A. Bondarev1, A.F. Voevodin2, K.K. Argunova1, I.I. Rozhin1
1Institute of Oil and Gas Problems of Siberian Branch of Russian Academy of Sciences, Oktyabrskaya, 1, Yakutsk, Russia, 677980
2Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Lavrentiev st., 15, Novosibirsk, 630090
Keywords: уравнение состояния, природный газ, гиперболические уравнения, equation of state, natural gas, hyperbolic equations

Abstract >>
By comparison with reliable experimental data in a wide range of pressure and temperature, it has been shown that the Redlich-Kwong equation of state appropriately reflects all the characteristics of the coefficient of compressibility, the throttling factor and the normalized difference of specific isobaric and isochoric heat capacities. It has been found that this equation corresponds to the inequalities required to ensure a set of equations of gas flow in pipelines to be hyperbolic.



3.
The plane wave refraction on convex and concave obtuse angles in geometric acoustics approximation

A.N. Kremlev
Research Institute of Applied Informatics and Mathematical Geophysics, 14 A. Nevskogo ul., Kaliningrad, 236041
Keywords: уравнение эйконала, уравнение Гамильтона-Якоби, лучевой параметр, преломление на выпуклом и вогнутом углах, время первых вступлений, аналитическое вязкое решение, головная волна, конечно-разностная схема Годунова, eikonal equation, Hamilton-Jacobi equation, ray parameter, refraction on convex and concave obtuse angle, first arrival times, analytical viscosity solution, head wave, Godunov finite difference scheme

Abstract >>
The strict analytical solution to the eikonal equation for the plane wave refracted on convex and concave obtuse angles has been built. It has a shock line for the ray vector field and the first arrival times at the convex angle and a rarefaction cone with diffracted waves at the concave angle. This cone corresponds to the Keller diffraction cone in the geometric diffraction theory. The comparison of the first arrival times, the Hamilton-Jacoby equation times for downward waves and the conservation ray parameter equation times was made. It is shown that these times are equal only for pre-critical incident angles and are different for sub-critical angles. It is shown that the most energetic wave arrival times, which have dominant practical importance, are equal to the times calculated for the conservation ray parameter equation. The numerical algorithm proposed for these times calculation may be used for arbitrary velocity models.



4.
A priori error estimates of finite volume method for nonlinear optimal control problem

Z. Lu1,2, L. Li1, L. Cao1, Ch. Hou3
1Chongqing Three Gorges University, Chongqing, 404000, P.R. China
2Tianjin University of Finance and Economics, Tianjin, 300222, P.R. China
3Guangdong University of Finance, Guangzhou, 511300, P.R. China
Keywords: априорные оценки ошибки, нелинейная задача оптимального управления, метод конечных объемов, вариационная дискретизация, a priori error estimates, nonlinear optimal control problem, finite volume method, variational discretization

Abstract >>
In this paper, we study a priori error estimates for a finite volume element approximation of a nonlinear optimal control problem. The schemes use discretizations base on a finite volume method. For the variational inequality, we use a method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate, and control variables is Oh2) or O(h2√|lnh|) in the sense of L2-norm or L-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.



5.
The outer layer method for solving boundary value problems of the elasticity theory

V.I. Mashukov
Siberian Transport University, D. Kovalchuk 191, Novosibirsk, 630049, Russia
Keywords: теория, упругость, граничные интегральные уравнения, внешний слой, двумерные, задачи, метод сопряжённых градиентов, theory, elasticity, boundary integral equations, external layer, two-dimensional, objectives, conjugate gradients method

Abstract >>
This paper presents an algorithm for solving boundary value problems of the elasticity theory, suitable to solve contact problems and those whose scope of deformation contains thin layers of a medium. The solution is represented as a linear combination of subsidiary solutions and fundamental solutions to the Lame equations. Singular points of fundamental solutions of the Lame equations are located as an external layer of the deformation around the perimeter. Coefficients of the linear combination are determined by minimizing deviations of a linear combination from the boundary conditions. To minimize deviations, the conjugate gradient method is applied. Examples of calculations for mixed boundary conditions are presented.



6.
A difference scheme for a conjugate-operator model of the heat conduction problem in the polar coordinates

S.B. Sorokin1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
2Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
Keywords: задача теплопроводности, математическая модель, дискретный аналог, полярные координаты, сходимость, разностная схема, problem of heat conductivity, mathematical model, discrete analog, polar coordinates, convergence, difference scheme

Abstract >>
In the polar coordinates, a discrete analog of the conjugate-operator model of the heat conduction problem preserves the structure of the original model. The difference scheme converges with the second order of accuracy for the cases of discontinuous parameters of the medium in the Fourier law and irregular grids. An efficient algorithm for solving the discrete conjugate-operator model in the case when the thermal conductivity tensor is a single operator.



7.
Solution to a stochastic Darcy equation by the polynomial chaos expansion

I.A. Shalimova, K.K. Sabelfeld
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: полиномиальный хаос, метод стохастических коллокаций, стационарное уравнение Дарси, метод Монте-Карло, разложение Кархунена-Лоэва, polynomial chaos, probabilistic collocation method, Darcy equation, Monte Carlo method, Karhunen-Loeve expansion

Abstract >>
This paper deals with the solution of a boundary value problem for the Darcy equation with a random hydraulic conductivity field. We use an approach based on the polynomial chaos expansion in the probability space of input data. We use the probabilistic collocation method to calculate the coefficients of the polynomial chaos expansion. A computational complexity of this algorithm is defined by the order of a polynomial chaos expansion and the number of terms in the Karhunen-Loève expansion. We calculate different Eulerian and Lagrangian statistical characteristics of the flow by the Monte Carlo and probabilistic collocation methods. Our calculations show a significant advantage of the probabilistic collocation method in comparison with the conventional direct Monte Carlo algorithm.



8.
A multi-point numerical integrator with trigonometric coefficients for initial value problems with periodic solutions

J.O. Ehigie1,2, S.N. Jator3, S.A. Okunuga4
1Nanjing Agricultural University, Nanjing 210095, China
2University of Lagos, Lagos 23401, Nigeria
3Austin Peay State University, Clarksville, TN, USA
4Department of Mathematics, Lagos 23401, Nigeria
Keywords: блочный метод, периодическое решение, тригонометрические коэффициенты, метод коллокации, block method, periodic solution, trigonometric coefficients, collocation technique

Abstract >>
Based on a collocation technique, we introduce a unifying approach for deriving a family of multi-point numerical integrators with trigonometric coefficients for the numerical solution of periodic initial value problems. A practical 3-point numerical integrator is presented, whose coefficients are generalizations of classical linear multistep methods such that the coefficients are functions of an estimate of the angular frequency ω . The collocation technique yields a continuous method, from which the main and complementary methods are recovered and expressed as a block matrix finite difference formula which integrates a second order differential equation over non-overlapping intervals without predictors. Some properties of the numerical integrator are investigated and presented. Numerical examples are given to illustrate the accuracy of the method.