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Siberian Journal of Numerical Mathematics

2017 year, number 4

The flux predictor-corrector scheme for solving a 3D heat transfer problem

K.V. Voronin1, Yu.M. Laevsky1,2
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: смешанный метод конечных элементов, тепловой поток, схема расщепления, схема предиктор-корректор, mixed finite element method, heat flux, splitting scheme, predictor-corrector scheme

Abstract >>
In this paper we propose and study the flux predictor-corrector scheme in the three-dimensional case. This scheme is depleted of drawbacks of that constructed on the basis of the Douglas-Gunn prototype-scheme. The scheme proposed demonstrates the second order of accuracy.

About the power law of the PageRank vector distribution. Part 1. Numerical methods for finding the PageRank vector

A. Gasnikov1,2, E. Gasnikova1, P. Dvurechensky2,3, A. Mohammed1, E. Chernousova1
1Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Russia, 141700
2Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, build. 1, Moscow, Russia, 127051
3Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin, Germany, 10117
Keywords: марковская цепь, эргодическая теорема, мультиномиальное распределение, концентрация меры, оценка максимального правдоподобия, Google problem, градиентный спуск, автоматическое дифференцирование, степенной закон распределения, Markov chain, ergodic theorem, multinomial distribution, measure concentration, maximum likelihood estimate, Google problem, gradient descent, automatic differentiation, power law distribution

Abstract >>
In Part 1 of this paper, we consider the web-pages ranking problem also known as the problem of finding the PageRank vector or Google problem. We discuss the connection of this problem with the ergodic theorem and describe different numerical methods to solve this problem together with their theoretical background, such as Markov Chain Monte Carlo and equilibrium in a macrosystem.

An approximation scheme for a problem of finding a subsequence

A.V. Kelmanov1,2, S.M. Romanchenko1, S.A. Khamidullin1
1Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090
2Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
Keywords: последовательность, евклидово пространство, минимум суммы квадратов расстояний, -трудность, FPTAS, euclidean space, sequence, minimum sum of squared distances, -hardness, FPTAS

Abstract >>
We consider a strongly -hard Euclidean problem of finding a subsequence in a finite sequence. The criterion of the solution is a minimum sum of squared distances from the elements of a sought subsequence to its geometric center (centroid). It is assumed that a sought subsequence contains a given number of elements. In addition, a sought subsequence should satisfy the following condition: the difference between the indices of each previous and subsequent points is bounded with given lower and upper constants. We present an approximation algorithm of solving the problem and prove that it is a fully polynomial-time approximation scheme (FPTAS) when the space dimension is bounded by a constant.

Correctness of the problem of propagation of nonlinear acoustic-gravity waves in the atmosphere from pressure variations on the lower boundary

Yu. Kurdyaeva1, S. Kshevetskii1, N. Gavrilov2, E. Golikova3
1Immanuel Kant Baltic Federal University IKBFU, Building 14 A, Nevskogo str., Kaliningrad
2Saint-Petersburg University, Ulyanovskaya str., Peterhof, Saint-Petersburg, Russia
3A.M. Obukhov Institute of Atmospheric Physics RAS, Pyzhevskii per., 3, Moskou, Russia, 119017
Keywords: численное моделирование, модель атмосферы, акустико-гравитационные волны, нелинейность, корректность, граничная задача, суперкомпьютерная программа, numerical simulation, atmospheric model, acoustic-gravity waves, nonlinearity, correctness, boundary problem, supercomputer program

Abstract >>
Currently, there are international microbarograph networks, with high resolution recording the wave pressure variations on the Earth's surface. This increases the interest in the problems of wave propagation in the atmosphere from variations in the atmospheric pressure. A complete system of nonlinear hydrodynamic equations for an atmospheric gas with lower boundary conditions in the form of wavelike variations on the Earth's surface is considered. Since the wave amplitudes near the Earth's surface are small, linearized equations are used in the analysis of the problem correctness. With the help of the wave energy functional method, it is shown that in the non-dissipative case, the solution of the boundary value problem is uniquely determined by the variable pressure field on the Earth's surface. The corresponding dissipative problem is correct if, in addition to the pressure field, suitable conditions on the velocity and temperature on the Earth's surface are given. In the case of an isothermal atmosphere, the problem admits analytical solutions that are harmonic in the variables x and t . A good agreement between numerical solutions and analytical ones is shown. The study has shown that in the boundary value problem, the temperature and density can rapidly vary near the lower boundary. An example of the solution of a three-dimensional problem with variable pressure on the Earth's surface, taken from experimental observations, is given. The developed algorithms and computer programs can be used to simulate the atmospheric waves from pressure variations on the Earth's surface.

The cubature formulas on a sphere invariant to the icosahedral group of rotations with inversion

A.S. Popov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: численное интегрирование, инвариантные кубатурные формулы, инвариантные многочлены, группа вращений икосаэдра, numerical integration, invariant cubature formulas, invariant polynomials, icosahedral group of rotations

Abstract >>
An algorithm of the search for the best (in a sense) cubature formulas on a sphere that are invariant with respect to the transformations of the icosahedral group of rotations with inversion is described. This algorithm is applied to finding the parameters of all the best cubature formulas of this symmetry type up to the 79th order of accuracy. The parameters of the new cubature formulas of the 21st, 25th and 29th orders of accuracy to 16 significant digits are given.

A boundary value problem for one overdetermined stationary system emerging in the two-velocity hydrodynamics

M.V. Urev1,2,3, Kh.Kh. Imomnazarov1, Jian-Gang Tang4
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
3Siberian Institute of Management, ul. Nizhegorodskya, 6, Novosibirsk, 630102
4YiLi Normal University, 448, Jiefang Road, Yinning Xinjiang, P.R. of China
Keywords: переопределенная стационарная система двухскоростной гидродинамики, множитель Лагранжа, метод конечных элементов, overdetermined two-velocity stationary hydrodynamics system, Lagrange multiplier, finite element method

Abstract >>
In this paper we investigate the two-velocity stationary hydrodynamics system with a single pressure and inhomogeneous divergent and boundary conditions for the two velocities. This system is overdetermined. By replacing the unknown functions, the problem is reduced to a homogeneous one. The solution of the resulting system is reduced to the consecutive solutions of the two boundary value problems: the Stokes problem for a single velocity and pressure, and overdetermined system for the other velocity. We present the generalized statements of these problems and their discrete approximation using the finite element method. To solve the overdetermined problem we apply a version of the regularization methods.

A description of pairs of the quasi-commuting Toeplitz and Hankel matrices

V.N. Chugunov1, Kh.D. Ikramov2
1Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina st., 8, Moscow, Russia, 119333
2Lomonosov Moscow State University, Moscow, Leninskie Gory, Russia, 119991
Keywords: теплицева матрица, ганкелева матрица, П•-циркулянт, квазикоммутирующие матрицы, Toeplitz matrix, Hankel matrix, П•-circulant, quasi-commuting matrices

Abstract >>
We say that the square matrices A and B are of the same order quasi-commute if AB = σ BA for some scalar σ. Classical relations of commutation and anti-commutation are particular cases of this definition. We give a complete description of pairs of the quasi-commuting Toeplitz and Hankel matrices for σ ≠ ± 1.

The Lebesgue constant of local cubic splines with equally-spaced knots

V.T. Shevaldin1, O.Ya. Shevaldina2
1Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskaja st., 16, Ekaterinburg, Russia, 620990
2Ural Federal University, 19 Mira street, Ekaterinburg, Russia, 620002
Keywords: константа Лебега, локальные кубические сплайны, равноотстоящие узлы, Lebesgue constants, local cubic splines, equally-spaced knots

Abstract >>
It is proved that the uniform Lebesgue constant (the norm of a linear operator from C to C) of local cubic splines with equally-spaced knots, which preserve cubic polynomials, is equal to 11/9.