Volume 26, Nº 4 (2024)

The present paper is devoted to the properties of semigroup dynamical systems $(G,X)$, where the semigroup $G$ is generated by a finite family of contracting transformations of the complete metric space $X$. It is proved that such dynamical systems $(G,X)$ always have a unique global attractor $\mathcal{A}$, which is a non-empty compact subset in $X$, with $\mathcal{A}$ being unique minimal set of the dynamical system $(G,X)$. It is shown that the dynamical system $(G,X)$ and the dynamical system $(G_{\mathcal{A}},\mathcal{A})$ obtained by restricting the action of $G$ to $\mathcal{A}$ both are not sensitive to the initial conditions. The global attractor $\mathcal{A}$ can have either a simple or a complex structure. The connectivity of the global attractor $\mathcal{A}$ is also studied.  A condition is found under which $\mathcal{A}$ is not a totally disconnected set. In particular, for semigroups $G$ generated by two one-to-one contraction mappings, a connectivity condition for the global attractor $\mathcal{A}$ is indicated. Also, sufficient conditions are obtained under which $\mathcal{A}$ is a Cantor set. Examples of global attractors of dynamical systems from the considered class are presented.

It is proved that for graded non-alternating Hamiltonian Lie algebras over a perfect field of characteristic two corresponding to a flag of the variables’ space  the condition of the embedding theorem of filtered deformations is fulfilled. The group of one-dimensional homology of the first member of the standard filtration for a graded non-alternating Hamiltonian Lie algebra is described. In the case when the number of variables $n\neq 4$, the estimate is obtained for multiplicity of the standard module over an orthogonal Lie algebra in a composition series of the homology group with respect to the natural structure of a module over the null-member of the grading. For $n=4$ the estimate is true if a set of variables coordinated with the flag contains a variable of height greater than 1 which is non-isotropic with respect to Poisson bracket, corresponding to the non-alternating Hamiltonian form. The homology computation employs the normal shape of non-alternating Hamiltonian form, corresponding to its class of equivalence. The monomials of the divided power algebra included into the commutant of the filtration’s first member are found. The multiplicity of the standard module over an orthogonal Lie algebra in a composition series of the first member of grading of the homology group is calculated. This calculation is based on the structure of weights with respect to a special maximal torus of the $p$-closure of the null-member of the standard grading in the Lie algebra of linear operators acting on the negative part of the grading of a non-alternating Hamiltonian Lie algebra.

In iterative algorithms for fully conservative difference schemes (FCDS) for the equations of gas dynamics in Euler variables, new methods for selecting adaptive artificial viscosity (AAV) have been developed, which are used both in explicit iterative processes and in the separate tridiagonal matrix algorithm. Various methods for incorporating AAV are discussed in this paper, including those for effectively suppressing oscillations in velocity profiles. All iterative methods are described in detail and block diagrams are given. A grid embedding method for modeling on spatially irregular sects is proposed. Calculations of the classical arbitrary discontinuity decay problem (the Sod problem) using FCDS and the developed AAV methods in different iterative processes have been performed. Comparative analysis is carried out and the efficiency of the developed improved iterative processes and approaches to the choice of AAV in comparison with the works of other authors is shown. All calculations are illustrated. The figures show variants of solutions of the Sod problem on uniform and non-uniform meshes, as well as a comparison of the methods proposed in the paper for the calculation of the Sod problem on a uniform grid.

This article presents the results of a numerical study of the turbulent flows’ structure in the construction elements under consideration, for which grid models are constructed that correspond to turbulence modeling approaches. More specifically, these approaches invoke Reynolds Averaged Navier-Stokes equations (RANS), equations closed using one or another semi-empirical turbulence model, as well as vortex–resolving approach, in particular, the method of large vortices modeling (Large Eddy Simulation – LES). The flow calculations were performed both in stationary and non-stationary settings using the LOGOS complex on a parallel supercomputer. From the analysis of the results obtained, it is concluded that the averaged flow parameters found within a non-stationary formulation using a zone RANS-LES transition in the turbulence model qualitatively and quantitatively better coincide with experimental data than the results of stationary calculations based on the use of the RANS approach. Verification of the numerical technique was carried out by experimental data obtained on the FT-18 aerodynamic stand on the basis of Nizhny Novgorod State Technical university named after R.E. Alekseev. A quantitative criterion for the effect of structural changes on the uniformity of the flow is the vorticity level.