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2018 year, number 3

The analysis of numerical differentiation formulas on the Shishkin mesh with of a boundary layer

A.I. Zadorin
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090
Keywords: функция одной переменной, пограничный слой, формула численного дифференцирования, сетка Шишкина, оценка погрешности, one-variable function, boundary layer, numerical differentiation formula, Shishkin mesh, error estimate

Abstract >>
The problem of numerical differentiation of functions with large gradients in the boundary layer is investigated. The problem is that in the case of functions with large gradients and a uniform grid, the relative error of the classical difference formulas for derivatives can be significant. It is proposed to use the Shishkin mesh to obtain a relative error of the formulas independent of a small parameter. Error estimates that depend on the number of nodes of the difference formulas for a derivative of a given order are obtained. It is proved that the error estimate is uniform in terms of a small parameter. In the case of the uniform grid, the region of the boundary layer is allocated, outside of which the numerical differentiation formulas have an error that is uniform in terms of a small parameter. The results of the numerical experiments are presented.

On congruent selection of the Jordan blocks from a singular square matrix

Kh.D. Ikramov
Lomonosov Moscow State University, Moscow, Leninskie Gory, 1, Russia, 119899
Keywords: конгруэнтное преобразование, жорданова клетка, СР-разложение, рациональный алгоритм, congruent transformation, Jordan block, SN-decomposition, rational algorithm

Abstract >>
The concept of a regularizing decomposition was introduced by R. Horn and V. Sergeichuk. This means the representation of a square matrix by a direct sum of the Jordan blocks with zero on the principal diagonal and a non-singular matrix. Such a representation is attained via congruent transformations and differs from the Jordan normal form. For the reasons explained in this paper, we prefer to speak about the SN-decomposition of a matrix (in other words, singular-non-singular decomposition) rather than the regularizing decomposition. Accordingly, the algorithms providing the former decomposition are called SN-algorithms. We propose a rational algorithm that considerably simplifies the SN-algorithms proposed by Horn and Sergeichuk.

The novel class of exact solutions of the two-dimensional eikonal equation when the velocity in a medium depends on one spatial coordinate

E.D. Moskalensky
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: распространение волн, фронт волны, уравнение эйконала, wave propagation, front of wave, eikonal equation

Abstract >>
The method to obtain solutions of the two-dimensional eikonal equation has been developed for the case when the velocity of wave propagation in a medium depends only on one spatial coordinate. We present several examples, where the initial problem is transformed to one or several ordinary differential equations using the substitution of the solution into a suitable general form. The dynamics of the wave propagation for each solution obtained is illustrated.

A comparison of radial basis functions

A.I. Rozhenko
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: сплайн, алгоритм, радиальная базисная функция, воспроизводящее ядро, тренд, внешний дрейф, интерполяция, сглаживание, регрессия, сплайн с натяжением, регуляризованный сплайн, spline, algorithm, radial basis function, reproducing kernel, trend, external drift, interpolation, smoothing, regression, tension spline, regularized spline

Abstract >>
A survey of algorithms for approximation of multivariate functions with radial basis functions (RBF) splines is presented. Algorithms of interpolation, smoothing, selecting the smoothing parameter, and regression with splines are described in detail. These algorithms are based on the properties of conditional positive definiteness of the spline radial basis function. Several families of the radial basis functions generated by means of conditionally complete monotone functions are considered. Recommendations for the selection of the spline basis and on the preparation of the initial data for approximation with the help of the RBF spline are given.

On the estimate of accuracy of the auxiliary boundary conditions method for solving a boundary value inverse problem for a nonlinear equation

E.V. Tabarintseva
South Ural State University, Chelyabinsk, prosp. Lenina, 76, Russia, 454080
Keywords: параболическое уравнение, обратная задача, модуль непрерывности обратного оператора, метод приближенного решения, оценка погрешности, parabolic equation, inverse problem, modulus of continuity of the inverse operator, approximate method, error estimate

Abstract >>
An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-way estimates for the norms of values of a nonlinear operator in terms of the norms of values of the corresponding linear operator are obtained. Consequent by the two-way estimates are established for the modulus of continuity of a nonlinear inverse problem in terms of the modulus of continuity of the corresponding linear problem. The auxiliary boundary conditions method to construct stable approximate solutions to the nonlinear inverse problem is used. An accurate in order error estimate for the auxiliary boundary conditions method on a uniform regularization class has been obtained.

Generating boundary conditions for the tsunami propagation calculation on imbedded grids

K. Hayashi1, An. Marchuk2, A. Vazhenin1
1The University of Aizu, Aizu-Wakamatsu, Fukushima, Japan
2Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: численный расчёт распространения цунами, граничные условия, вложенные сетки, numerical computation of the tsunami propagation, boundary conditions, nested grids

Abstract >>
The boundary conditions that are used for the numerical modeling of the tsunami generation and propagation have been studied. This study focuses on the generating boundary conditions which make it possible to simulate the tsunami wave with desired characteristics (amplitude, time period and, generally speaking, waveform). Taking into account the fact that the water flow velocity in the propagating tsunami wave is uniquely determined by its height and the ocean depth, it is possible to simulate the wave which propagates inward from the boundary into the area of simulation. This can be done by setting the wave height and the water flow velocity over the boundary. By such a way the numerical modeling of the tsunami propagation from a source up to the coast was implemented on a sequence of refining grids. In the conducted numerical experiment, the wave parameters are transmitted from a bigger area into a subarea via boundary conditions. In addition, such a method allows generating a wave that has certain characteristics on a specified line.

Mixed methods for optimal control problems

T. Hou
Beihua University, Jilin 132013, China
Keywords: эллиптические уравнения, задачи оптимального управления, апостериорные оценки ошибки, смешанные методы конечных элементов, elliptic equations, optimal control problems, a posteriori error estimates, a mixed finite element method

Abstract >>
In this paper, we investigate a posteriori error estimates of a mixed finite element method for elliptic optimal control problems with an integral constraint. The gradient for our method belongs to the square integrable space instead of the classical H(div; Q) space. The state and co-state are approximated by the P2-P1 (velocity-pressure) pair, and the control variable is approximated by piecewise constant functions. Using a duality argument method and an energy method, we derive residual a posteriori error estimates for all variables.