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Siberian Journal of Numerical Mathematics

2021 year, number 1

On analytical families of matrices generating bounded semigroups

P.A. Bakhvalov, M.D. Surnachev
Keldysh Institute of Applied Mathematics, Moscow, Russia
Keywords: spectral analysis, difference scheme, Riesz projection, matrix transform, block diagonalization

Abstract >>
We consider linear schemes with several degrees of freedom (DOFs) for the transport equation with a constant coefficient. The Fourier transform decomposes the scheme into a number of finite systems of ODEs, the number of equations in each system being equal to the number of OFs. The matrix of these systems is an analytical function of the wave vector. Generally such a matrix is not diagonalizable and, if it is, the diagonal form can be non-smooth. We show that in a 1D case for L2-stable schemes the matrix can be locally transformed to a block-diagonal form preserving the analytical dependence on the wave number.

On the optimal approximation of geophysical fields

I.V. Boikov, V.A. Ryazantsev
Penza State University, Penza, Russia
Keywords: heat fields, classes of functions, parabolic equations

Abstract >>
In this paper, optimal methods of approximation of some geophysical fields involving gravitational and heat fields are considered. A review of results on this problem is presented. We have developed the algorithm of approximation of multidimensional heat fields which are described by heat equation with constant coefficients. In order to do that, we introduce classes of functions that include solutions of heat equations, and continuous splines uniformly approximating the functions from these classes in the whole domain of definition. We give the upper bounds for the Kolmogorov diameters of the introduced classes of functions. For a wider class of the introduced classes of functions, the Kolmogorov diameters is estimated from below.

Existence results for second-order nonlinear differential inclusion with nonlocal boundary conditions

Noureddine Bouteraa, Slimane Benaicha
University of Oran1, Ahmed Benbella. Algeria
Keywords: second-order, differential inclusion, multi-valued, selection theorem

Abstract >>
In this paper, we investigate the existence of solutions for a second-order differential inclusion with nonlocal boundary conditions. To establish the existence results for the given problem, first we apply Schaefer's fixed point theorem combined with a selection theorem due to Bressan and Colombo. Second, our result is based on the fixed point theorem for multivalued maps due to Covitz and Nadler. An example is given to illustrate the obtained results.

Semilocal convergence of Modified Chebyshev-Halley method for nonlinear operators in case of unbounded third derivative

Neha Gupta, J.P. Jaiswal
Maulana Azad National Institute of Technology, Bhopal, M.P. India
Keywords: Banach space, semilocal convergence, П‰-continuity condition, Chebyshev-Halley method, error bound

Abstract >>
In the present discussion, we analyze the semilocal convergence of a class of modified Chebyshev-Halley methods under two different sets of assumptions. In the first set, we just assumed the bound of the second order Frèchet derivative in lieu of the third order. In the second set of hypotheses, the bound of the norm of the third order Frèchet derivative is assumed at initial iterate preferably supposed it earlier on the domain of the given operator along with fulfillment of the local ω-continuity in order to prove the convergence, existence and uniqueness followed by a priori error bound. Two numerical experiments strongly support the theory included in this paper.

A priori error estimates and superconvergence of P02-P1 mixed finite element methods for elliptic boundary control problems

Ch. Xu
Beihua University, Jilin, China
Keywords: elliptic equations, boundary control problems, a priori error estimates, superconvergence, P-P mixed finite element methods

Abstract >>
In this paper, we discuss a priori error estimates and superconvergence of P 02- P 1 mixed finite element methods for elliptic boundary control problems. The state variables and co-state variables are approximated by a P 02- P 1 (velocity-pressure) pair and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, the state variables and the co-state variables. Then we obtain a superconvergence result for the control variable by using postprocessing projection operator.

Adaptive grids and high-order schemes for solving singularly-perturbed problems

V.D. Liseikin1,2, V.I. Paasonen1,2
1Federal Research Center for Information and Computational Technologies, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: singularly perturbed equations, small parameter, boundary and interior layers, grid generation method

Abstract >>
Layer-resolving grids remain an important element of comprehensive software codes when solving real-life problems with layers of singularities as they can substantially enhance the efficiency of computer-resource utilization. This paper describes an explicit approach to generating layer-resolving grids which is aimed at application of difference schemes of an arbitrary order. The approach proposed is based on estimates of derivatives of solutions to singularly-perturbed problems and is a generalization of the approach developed for the first order schemes. The layer-resolving grids proposed are suitable to tackle problems with exponential-, power-, logarithmic-, and mixed-type boundary and interior layers. Theoretical results have been confirmed by the numerical experiments on a number of test problems with such layers; the results were compared to those obtained with difference schemes of different orders of accuracy.

Restoration of the ocean depth using recorded tsunami arrival times

An.G. Marchuk
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
Keywords: inverse problem, medium restoration, tsunami detectors, wave arrival time, cell of rectangular grid

Abstract >>
Usually, the ocean depth is measured by the echo sounder in the course of a vessel cruise. However, there are a lot of areas in the World Ocean which are free of navigation. This means that by now there are no direct bathymetry measurements. For the numerical modeling of the trans-oceanic tsunami we need the digital bathymetry of the whole water area. The thus arising problem is to restore (at least, approximately) the depth values in the areas without reliable bathymetric data. This can be done in the process of the propagation of a real tsunami with the help of detectors used. Two algorithms for the depth restoration based on the tsunami arrival times are proposed. The algorithm has been tested on the depth restoration problem in the area with a sloping bottom, where the tsunami arrival times are known at the nodes of a rectangular grid.

On the numerical solution of the generalized Burgers-Huxley equation by Lie-Trotter splitting method

Y. Cicek, S.O. Korkut
Izmir Katip Celebi University, Izmir, Turkey
Keywords: convergence analysis, Burgers-Huxley equation, Lie-Trotter splitting method, nonlinear differential equations, Sobolev spaces

Abstract >>
In this paper, the Lie-Trotter splitting method (LSM) is used to solve the generalized Burgers-Huxley equation (GBHE) numerically. We first establish the local error bounds of approximate solutions of the GBHE with the help of the theory of differential operators in a Banach space. Then we prove the global convergence by using a telescoping identity. At the end, the accuracy of the method is provided by numerical results which are compared with earlier studies.