Home – Home – Jornals – Numerical Analysis and Applications 2014 number 2

Explicit formulas for calculating eigenvalues of the Hankel circulants, Hankel skew-circulants, (T+H)-circulants, and (T+H)-skew-circulants are obtained. It is shown that if Φ ≠ ±1, then the set of matrices that can be represented as sums of a Toeplitz Φ-circulant and a Hankel Φ-circulant is not an algebra.

The Smoluchowski equation with linear coagulation coefficients depending on two parameters is considered. We construct weight algorithms for estimating various linear functionals in an ensemble, which is governed by the equation under study. The algorithms constructed allow us to estimate the functionals for various parameters as well as parametric derivatives using the same set of trajectories. Moreover, we construct the value algorithms and analyze their efficiency for estimating the total monomer concentration as well as the total monomer and dimer concentration in the ensemble. A considerable gain in computational costs is achieved via the approximate value simulation of the time between interactions combined with the value simulation of the interacting pair number.

This paper presents the algorithm, based on the application of the spectral Laguerre method for approximation of temporal derivatives as applied to the problem of seismic wave propagation in the porous media with dissipation of energy. The initial system of equations is written down as the first order hyperbolic system in terms of velocities, stresses and pore pressure. For the numerical solution of the problem in question, the method of a combination of the analytical Laguerre transformation and a finite difference method is used. The proposed method of the solution can be considered to be an analog to the known spectral method based on the Fourier transform. However, unlike the Fourier transform, application of the integral Laguerre transform with respect to time allows us to reduce the initial problem to solving a system of equations in which the parameter of division is present only in the right-hand side of equations and has a recurrent dependence. As compared to the time-domain method, with the help of an analytical transformation in the spectral method it is possible to reduce an original problem to solving a system of differential equations, in which there are only derivatives with respect to spatial coordinates. This allows us to apply a known stable difference scheme for recurrent solutions to similar systems. Such an approach is effective when solving dynamic problems for porous media. Thus, because of the presence of the second longitudinal wave with a low velocity, the use of difference schemes in all coordinates for stable solutions requires a consistent small step both with respect to time and space, which inevitably results in an increase in computer costs.

A numerical-analytical solution for seismic and acoustic-gravitational waves propagation is applied to a heterogeneous «Earth-Atmosphere» model. Seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. Propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations with the a wind. The proposed algorithm is based on the integral Laguerre transform with respect to time, the finite integral Fourier transform along the spatial coordinate with the finite difference solution of the reduced problem.

In this paper we propose an analytical method of modeling seismic wave fields for a wide range of geophysical media: elastic, non-elastic, anisotropic, anisotropic-non-elastic, porous, random-inhomogeneous, etc. for super-remote (far) distances. As finite difference approximations are not used, there is no grid dispersion when computing wave fields for arbitrary media models and observation points. The analytical solution representation in the spectral domain makes possible to carry out the analysis of a wave field in parts, specifically, to obtain the primary waves. Based on the developed program of computing the wave fields, we have carried out the simulation of water waves and seismic «ringing» on the Moon. The monotone displacement resonant to the lower frequency area with increasing the recording distance has been explained. Such a displacement was detected in experiments with a seismic vibrator.

A set of numerical algorithms for the simulation of random variables and functions as well for the parametrical numerically-statistical analysis are considered. Important specifications and explanations of the algorithms formulations and substantiation, which are effective from the standpoint of practice, are given.

This paper describes a modification of a power series for the construction of approximate solutions of the Lorenz system. The results of the computer-aided simulation are presented. Also, the physical modeling of the dynamics of the Lorenz system of the processes occurring in the circuit are considered.

Characteristics of barotropic seiche oscillations of the middle part of Peter the Great Gulf are considered with the use of spectral-finite difference model. The model is based on the linearized system of shallow water equations. Difference approximation is carried out on an irregular triangular spatial mesh. The numerical method involves the solution of the eigenvalue problem and is able to directly obtain a set of frequencies and the corresponding forms of seiche oscillations. The grid computational domain covers the Amurskiy gulf and the Ussuriyskiy gulf. The Zolotoy Rog bay and the Alekseev bay are described in more detail on the grid. The calculated and presented spatial-temporal characteristics of a number of seiche oscillations corresponding to well-defined peaks of the energy spectrum of the sea level data from the station «Vladivostok» of the Russian of tsunami warning service. The results of calculations for the Alekseev bay are compared to the data of natural measurements and the solutions of the Cauchy problem.