Home – Home – Jornals – Numerical Analysis and Applications 2021 number 2

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 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.

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 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.

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.

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.

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.