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Numerical Analysis and Applications

2021 year, number 2

Generalized bivariate Hermite fractal interpolation function

Sangita Jha1, A.K.B. Chand1, M.A. Navascues2
1Indian Institute of Technology Madras, Chennai, India
2Universidad de Zaragoza, Zaragoza, Spain
Keywords: fractals, fractal interpolation, hermite interpolation, fractal surface, convergence

Abstract >>
Fractal interpolation provides an efficient way to describe the smooth or non-smooth structure associated with nature and scientific data. The aim of this paper is to introduce a bivariate Hermite fractal interpolation formula which generalizes the classical Hermite interpolation formula for two variables. It is shown here that the proposed Hermite fractal interpolation function and its derivatives of all orders are good approximations of the original function even if the partial derivatives of the original functions are non-smooth in nature.

The errors investigation in problems for solving simple equations of mathematical physics by iterative methods

V.P. Zhitnikov1, N.M. Sherykhalina1, R.R. Muksimova2
1Ufa State Aviation Technical University, Ufa, Bashkortostan, Russia
2Saint Petersburg State University of Civil Aviation, Saint Petersburg, Russia
Keywords: heat equation, implicit scheme, Laplace equation, biharmonic equation, iteration method, numerical filtration

Abstract >>
The error caused by the inaccuracy of the equation system solution by iterative methods has been investigated. The upper error estimate for the axially symmetric heat equation is found in the accumulation process in several time steps. The upper estimate shows the linear dependence of the error on the threshold value of the limiting criterion for the iterations number, the quadratic error growth from the range partitions number, and its independence of the time partitions number. The computing experiment shows a good correspondence of the obtained estimate to real errors with boundary and initial conditions of various types. The quadratic error growth for the Laplace equation, caused by the accuracy limitation for applying the iteration method, on the number of range partitions n, is empirically found. A similar error growth for the biharmonic equation is found in proportion to n4.

A computational model of fluid filtration in fractured porous media

M.I. Ivanov, I.A. Kremer, Yu.M. Laevsky
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
Keywords: fluid filtration, fractured porous media, double porosity, porous blocks, fractures, conservation laws, mixed finite element method, upwind scheme, maximum principle

Abstract >>
The paper discusses a computational 3D double porosity model of a two-phase incompressible fluid filtration in a fractured-porous medium. Conservation laws are formulated in the integral form, and for their spatial approximation, a combination of the mixed finite element method to determine the total flow and pressure velocities is used and the finite volume method to determine the saturations in porous blocks and in fractures. The approximation of equations for saturations according to an explicit scheme with upwinding to eliminate unphysical oscillations is carried out. The model under consideration includes the injection and production wells with total flow rates. For the total velocities and pressures, the Neumann problem is formulated, for which the condition of unique solvability is indicated and a method for solving it without additional conditions is proposed. For an explicit upwind scheme for solving equations for saturations, a weak maximum principle is established, illustrated by computational experiments.

A rational algorithm for checking the congruence of unitoid matrices

Kh.D. Ikramov1, A.M. Nazari2
1Lomonosov Moscow State University, Moscow, Russia
2University of Arak, Arak, Iran
Keywords: congruence, unitoid matrix (unitoid), cosquare, similarity, Toeplitz decomposition, indices of inertia, Pythagorean triples, Maple, circulants

Abstract >>
A matrix is said to be unitoid if it can be brought to diagonal form by a congruence transformation. We say that an algorithm is rational if it is finite and uses the arithmetic operations only. There exist rational methods designed for checking congruence of particular classes of unitoid matrices, for example, Hermitian, accretive, or dissipative matrices. We propose a rational algorithm for checking congruence of general unitoid matrices. The algorithm is heuristic in the sense that the user is required to set the values of two integral parameters M and N. The choice of these values depends on the available a priori information about the proximity of neighboring canonical angles of the matrices under checking.

Mathematical substantiation of pulsed electromagnetic soundings for new problems of petroleum geophysics

M.N. Nikitenko, V.N. Glinskikh, D.I. Gornostalev
The Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Keywords: electromagnetic soundings, transient method, numerical-analytical solution, computational algorithm, layered homogeneous geoelectric model, Bazhenov formation

Abstract >>
This paper concerns the development of fundamental theoretical foundations and the creation of algorithms and software for pulsed electromagnetic soundings in relation to studying an unconventional source of hydrocarbons with hard-to-recover reserves - the Bazhenov formation. We carry out the mathematical substantiation of a new geophysical technology for the localization of oil-prospective zones, using a spatially distributed system of highly deviated wells. For the fast mathematical simulation, we obtained a solution to the problem of pulsed electromagnetic soundings in layered homogeneous models of media for an arbitrary current pulse in the electromagnetic source, which allows deep parallelization. Based on the created computational algorithm, a parallel one was developed, as well as a fast computer program for numerical simulation of the signals of the new system on multiprocessor devices of the Siberian Supercomputer Center, SB RAS. We carried out a large-scale numerical simulation and analysis of the signals in realistic geoelectric models of the Bazhenov formation to estimate an applicable scope of the new pulsed electromagnetic sounding installation. The calculations show that determining spatial locations of the formation boundaries is possible when logging the wells over a wide range of the sonde spacings. We analyzed the applicability of the diagonal and off-diagonal field components to ensure high sensitivity for mapping the reservoir boundaries and evaluating its internal heterogeneities. The results obtained form a basis for the further design of the optimal configuration of the new electromagnetic sounding system.

Lipschitz-like mapping and its application to convergence analysis of a variant of Newton's method

M.H. Rashid1,2
1Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhongguancun east road, Haidian district, Beijing-100190, China
2University of Rajshahi, Rajshahi-6205, Bangladesh
Keywords: set-valued mappings, lipschitz-like mappings, generalized equations, variant of Newton's method, semilocal convergence

Abstract >>
Let X and Y be Banach spaces. Let f: Ω → Y be a Frèchet differentiable function on an open subset Ω of X and F be a set-valued mapping with closed graph. Consider the following generalized equation problem: 0 in f(x)+ F(x). In the present paper, we study a variant of Newton's method for solving generalized equation (1) and analyze semilocal and local convergence of this method under weaker conditions than those considered by Jean-Alexis and Piètrus [13]. In fact, we show that the variant of Newton's method is superlinearly convergent when the Frèchet derivative of f is (L,p)-Hölder continuous and (f+F)-1 is Lipzchitz-like at a reference point. Moreover, applications of this method to a nonlinear programming problem and a variational inequality are given. Numerical experiments are provided which illustrate the theoretical results.

A family of fifth-order iterative methods for finding multiple roots of nonlinear equations

J.R. Sharma1, H. Arora2
1Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, India
2D.A.V. University, Sarmastpur, Jalandhar, India
Keywords: nonlinear equations, iterative methods, fast algorithms, multiple roots, attraction basins

Abstract >>
We present a family of fifth order iterative methods for finding multiple roots of nonlinear equations. Numerical examples are considered to check the validity of theoretical results. The results show that the new methods are competitive to other methods for multiple roots. Basins of attraction for new methods and some existing methods are drawn to observe the dynamics in the complex plane.