Home – Home – Jornals – Numerical Analysis and Applications 2018 number 2

The cluster analysis is used in various fundamental and applied fields and is a current topic of research. Unlike conventional methods, the proposed algorithms are used for clustering objects represented by vectors in space with the non-observance of the axiom of symmetry. In this case, the feature of solving the clustering problem is the use of an asymmetric proximity measures. The first one among the proposed clustering algorithms sequentially forms clusters with a simultaneous generalization to clustered objects from previously created clusters to a current cluster if this reduces the quality criterion. This approach to the formation of clusters allows reducing the computational costs as compared with existing non-hierarchical cluster algorithms. The second algorithm is a modified version of the first algorithm. The second algorithm allows reassigning the main objects of clusters to further reduce the proposed quality criterion.

The paper deals with development of spectrum dichotomy methods for the matrices with a large norm. Such matrices appear as a result of discretization of differential operators. The results of some numerical experiments including the stability investigation for the Poiseuille plane-parallel stream are given.

The paper considers a model of double porosity for a fractured porous medium using a combination of classical and gradient mass transfer functions among cracks and porous blocks in the case of a flow of a weakly compressible single-phase fluid. As compared to well-known models, such a mass transfer function allows one to take into account the anisotropic properties of filtration in a more general form. The results of numerical tests for two-dimensional and three-dimensional model problems are presented. The computational algorithm is based on the use of finite element approximation with respect to space and completely implicit approximations with respect to time.

The optimal q-homotopy analysis method is employed in order to solve partial differential equations (PDEs) featuring a time-fractional derivative. Then, in order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples are given. All numerical calculations in this manuscript have been carried out with Mathematica package.

The splitting method for the CABARET scheme approximating the non-uniform scalar conservation law with convex and monotonically increasing flux function has been proposed. It was shown that at the first step of this method, when the uniform conservation law is approximated, the CABARET scheme is monotonic and its numerical solutions do not have non-physical oscillations in the shock wavefronts. Test computations that illustrate these properties of the CABARET scheme are presented.

A nonlinear distributed second order equation is considered. An algorithm for tracking a prescribed solution based on constructions from the feedback control theory is designed. The algorithm is stable with respect to informational noise and computational errors. It is oriented to a large enough time interval, where the solution is considered.

The study of Painleve's equations has increased during the last years, due to the awareness that these equations and their solutions can accomplish good results both in the field of pure mathematics and theoretical physics. In this paper we introduced an optimal homotopy asymptotic method (OHAM) approach to propose analytic approximate solutions to the second Painlevè equation. The advantage of this method is that it provides a simple algebraic expression that can be used for further developments while maintaining good performance and fitting closely the numerical solution.

A mathematical model of the sea thermodynamics, developed at the Institute of Numerical Mathematics of the Russian Academy of Sciences is considered. The problem of variational assimilation of daily-averaged sea surface temperature (SST) data is formulated and investigated taking into account the observation error covariance matrices. On the basis of variational assimilation of satellite observation data, the inverse problem of restoring a heat flux on the sea surface is solved. The stability of the optimal solution of the problem of variational data assimilation is studied, and the results of numerical experiments for the model of the Baltic Sea dynamics are presented.