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Numerical Analysis and Applications

2020 year, number 2

1.
The numerical solution of the direct Zakharov-Shabat scattering problem

N.I. Gorbenko1,2, V.P. Il’in1,2, A.M. Krylov1, L.L. Frumin2,3
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
3Institute of Automation and Electrometry, Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia
Keywords: прямая задача рассеяния, схема четвертого порядка, тождество Марчука, direct scattering problem, fourth order difference scheme, Marchuk identity

Abstract >>
The numerical solution of the direct scattering problem for a system of the Zakharov-Shabat equations is considered. Based on the Marchuk identity, a fourth order method of approximation accuracy is proposed. The numerical simulation of the scattering problem is carried out using an example of two characteristic boundary value problems with known solutions. The calculations have confirmed high accuracy of the algorithm proposed, which is necessary in a number of practical applications for optical and acoustic sensing of media in optics and geophysics applied.



2.
The minimization problem for the sum of weighted convolution differences: the case of a given number of elements in the sum

A.V. Kel’manov1,2, L.V. Mikhailova1, P.S. Ruzankin1,2, S.A. Khamidullin1
1Sobolev Institute of Mathematics, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: числовые последовательности, разность взвешенных сверток, переменная длина свертки, минимум суммы, точный полиномиальный алгоритм, численное моделирование, ECG-подобный сигнал, PPG-подобный сигнал, numerical sequences, difference of weighted convolutions, variable length convolution, minimum of sum, exact polynomial-time algorithm, numerical modeling, electrocardiogram-like signal, photoplethysmogram-like signal

Abstract >>
We consider an unstudied optimization problem of summing the elements of the two numerical sequences: Y of length N and U of length q ≤ N . The objective of the optimization problem is to minimize the sum of differences of weighted convolutions of sequences of variable lengths (which are not less than q ). In each of the differences, the first convolution is the unweighted autoconvolution of the sequence U nonlinearly expanded in time (by repetitions of its elements), and the second one is the weighted convolution of an expanded sequence with a subsequence of Y . The number of differences is given. We show that the problem is equivalent to that of approximation of the sequence Y by an element of some exponentially sized set of sequences. Such a set consists of all the sequences of length N which include, as subsequences, a given number M of admissible quasiperiodic (fluctuating) repetitions of the sequence U . Each quasiperiodic repetition is generated by the following admissible transformations of the sequence U : (1) shifting U in time, so that the differences between consecutive shifts do not exceed Tmax ≤ N , variable expansion of U in time consisting in repeating each element of U , with variable multiplicities of the repetitions. The optimization objective is minimizing the sum of the squares of element-wise differences. We demonstrate that the optimization problem in combination with the corresponding approximation problem are solvable in polynomial time. Specifically, we show that there exists an algorithm which solves the problems in the time O(T3max M N ). If Tmax is a fixed parameter of the problem, then the algorithm running time is O( M N ). In the examples of numerical modeling, we show the applicability of the algorithm to solving applied problems of noise-robust analyzing electrocardiogram-like and photoplethysmogram-like signals.



3.
The use of the piecewise parabolic method on a local stencil for constructing a low dissipation of a numerical solution scheme for mathematical modeling of special relativistic hydrodynamic flows

I.M. Kulikov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
Keywords: математическое моделирование, кусочно-параболический метод на локальном шаблоне, релятивистская гидродинамика, numerical modeling, numerical methods of high accuracy order, special relativistic hydrodynamics

Abstract >>
In this paper a new numerical method with a low-dissipation of a numerical solution, based on a combination of the Godunov method with the Roe scheme, and a piecewise parabolic method on a local stencil is described. The construction of the numerical method is described in considerable detail, the method is validated on the one-dimensional Riemann problem. The results of the numerical simulation of the collision of two relativistic gas spheres are presented.



4.
A family of solutions of the two-dimensional eikonal equation

An.G. Marchuk, E.D. Moskalensky
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
Keywords: распространение волны, фронт волны, уравнение эйканала, точечный источник, wave propagation, wave front, eikanal equation, point source

Abstract >>
The propagation of a wave from the point source in the case when the velocity in the medium v is expressed as v = 1/√ y is considered. Exact solutions of the corresponding eikonal equation are obtained and numerically verified. The ambiguity of the solution to this question in the case when the point source is situated at the origin is shown.



5.
Finite element method for the Stokes-Darcy problem with a new boundary condition

O. El Moutea1, H. El Amri1, A. El Akkad2
1Ecole Normale Superieure Casablanca, Morocco, African Union
2Centre Regional des MГ©tiers d'Education et de Formation de Fes Meknes, Morocco, African Union
Keywords: задача Стокса-Дарси, смешанный метод конечных элементов, свободный поток, поток пористой среды, стабилизированная схема, Stokes-Darcy problem, mixed п¬Ѓnite element method, free flow, porous media flow, stabilized scheme

Abstract >>
This paper considers numerical methods for approaching and simulate the Stokes-Darcy problem, with a new boundary condition. We study herein a robust stabilized fully mixed discretization technique, this method ensures the stability of the finite element scheme and does not use any Lagrange multipliers to introduce a stabilization term in the temporal Stokes-Darcy problem discretization. The well-posedness of the finite element scheme and its convergence analysis are also derived. Finally, the effciency and accuracy of the numerical methods are illustrated by different numerical tests.



6.
Source indentification for the Smoluchowski equation using an ensemble of the adjoint equation solutions

A.V. Penenko1,2, A.B. Salimova1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: уравнение Смолуховского, обратная задача идентификации источников, метод Ньютона-Канторовича, сопряженные уравнения, оператор чувствительности, Smoluchowsky equation, inverse source problem, Newton-Kantorovich method, adjoint equations, sensitivity operator

Abstract >>
A source identification algorithm for the systems of nonlinear ordinary differential equations of the production-destruction type is applied to the inverse problem for the discretized Smoluchowski equation. An unknown source function is estimated by time series of measurements of the specific size particles concentration. Based on an ensemble of adjoint equations solutions, the sensitivity operator is constructed that links the perturbations of the sought for model parameters with perturbations of the measured values. This reduces the inverse problem to a family of quasilinear operator equations. To solve the equations, an algorithm of the Newton-Kantorovich type is used with r -pseudoinverse matrices. The eficiency and properties of the algorithm are numerically studied.



7.
To calculating heat-conducting vapor-gas-droplet mixture flows

V.S. Surov
South Ural State University, Chelyabinsk, Russia
Keywords: гиперболическая модель односкоростной теплопроводной парогазокапельной смеси, межфракционный теплообмен, метод Годунова, линеаризованный римановский решатель, hyperbolic model of a heat-conducting vapor-gas-drop mixture, inter-fractional heat transfer, Godunov’s method, linearized Riemann solver

Abstract >>
A characteristic analysis of the equations of a single-velocity heat-conducting vapor-gas-drop mixture is carried out, in which the interfraction heat exchange is taken into account and their hyperbolicity is shown. The computational formulas of the Godunov method with a linearized Riemann solver are presented with whose use a number of the mixture flows are calculated.



8.
On reducing the inverse boundary value problem to the synthesis of two ill-posed problems and their solution

V.P. Tanana1,2
1South Ural State University, Chelyabinsk, Russia
2Chelyabinsk State University, Chelyabinsk, Russia
Keywords: оценка погрешности, модуль непрерывности, преобразование Фурье, некорректная задача, error estimation, modulus of continuity, Fourier transform, ill-posed problem

Abstract >>
This paper concerns the solution of the inverse boundary value problem for the equation of thermal conductivity and the estimation of the approximate solution error. The Fourier transform with respect to time, which allows one to obtain an error estimate, is not applicable to the problem to be solved. Therefore, in the equation of thermal conductivity, the variable was replaced, which led to the synthesis of problems and allowed obtaining an estimate.