Home – Home – Jornals – Numerical Analysis and Applications 2016 number 3

In this paper we consider the random structure systems with distributed transitions. A theorem about the form of conditional structure distributions has been proved. To simulatethese distributions a statistical algorithm using a randomized method of maximum cross-section is constructed. Also, a modified version of this algorithm using the simulation of one random number has been constructed. The algorithms developed were used for the simulation of the numerical solution of random structure systems with distributed transitions. The theorem about a weak convergence of the numerical solution, obtained by the algorithms developed has been proved.

A family of interior point algorithms for the linear programming problems is considered. In these algorithms, the entering into the domain of admissible solution of the original problem is represented as optimization process of the extended problem. This extension is realized by adding just one new variable. The main objective of the paper is to give a theoretical justification of the proposed procedure of entering into the feasible domain of the original problem, under the assumption of non-degeneracy of the extended problem. Particularly, we prove that given the constraints of the original problem being consistent, the procedure leads to a relative interior point of the feasible solutions domain.

A class of preconditioners for solving non-Hermitian positive definite systems of linear algebraic equations is proposed and investigated. It is based on the Hermitian and skew-Hermitian splitting of the initial matrix. The generalization for saddle point systems which have semidefinite or singular (1,1) blocks is given. Our approach is based on an augmented Lagrangian formulation. It is shown that such preconditioners are effective for the iterative solution of systems of linear algebraic equations by the GMRES.

In this paper, we consider reaction-diffusion systems arising from two-component predator-prey models with Smith growth functional response. The mathematical approach used here is twofold, since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with a fourth-order difference operator, advance the resulting nonlinear system of ordinary differential equations with a family of two competing exponential time differencing (ETD) schemes, and analyze them for stability. A numerical comparison of these two methods for solving various predator-prey population models with functional responses is also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled.

This paper presents the results of numerical experiments with a model of the coastal trapped waves, which made it possible to identify two features that are important in terms of the regional modeling of the shelf zone interaction with the open ocean. The first feature is the fact that the wave train of this type may be formed as a result of the wind action at a considerable distance from the place where their impact may occur. The propagation of waves along the coastline takes place without significant loss of wave energy, provided that the coastline and topography of the shelf zone contain no features comparable to the Rossby radius. However, the wave loses its energy while passing capes, submarine canyons and in the case when the width of a shelf decreases. For the regional modeling, the possibility of remote wave generation should be well understood and taken into account. The second feature is that a propagating wave is able to spend part of its energy on the formation of density anomalies on a shelf by raising the intermediate waters of the adjoining offshore areas of the open ocean. Thus, the coastal trapped waves carry the wind energy from the areas of the wind impact to other coastal areas, where it can bring about the formation of density anomalies and other types of motion.

The direct simulation Monte Carlo method is now widely used to solve the problems of rarefied gas dynamics. While solving stationary problems a special feature of the method is using dependent sample values of random variables to calculate macroparameters of a gas flow. In this paper, the possibility of using the results of statistical physics to estimate the statistical error of the DSMC method is theoretically analyzed. A simple approach to approximate evaluating the statistical error while calculating components of the velocity and temperature is proposed. The approach is tested on a number of problems.

In this paper, the problem of constructing a spline σ in the Hilbert space satisfying bilateral restrictions z^- ≤ A σ ≤ z⁺ with a linear operator A and minimizing a squared Hilbert seminorm is studied. A solution to this problem could be obtained with the convex programming iterative methods, in particular, with the gradient projection method. A modification of the gradient projection method allowing one to reveal a set of active restrictions in a smaller number of iterations is offered. The efficiency of the modification proposed is shown on the problem of approximation with a pseudo-linear bivariate spline.