Currently, the Discontinuous Galerkin Method (DGM) is widely used to solve complex multi-scale problems of mathematical physics that have important applied significance. When implementing it, the question of choosing a discrete approximation of flows for viscous terms of the Navier-Stokes equation is important.

It is necessary to focus on the construction of limiting functions, on the selection of the best discrete approximations of diffusion flows, and on the use of implicit and iterative methods for solving the obtained differential-difference equations for the successful application of DGM on three-dimensional unstructured grids.

First-order numerical schemes and second-order DGM schemes with Godunov, HLLC, Rusanov-Lax-Friedrichs numerical flows and hybrid flows are investigated. For high-order precision methods, it is necessary to use high-order time schemes.

The Runge-Kutta scheme of the third order is used in the work. The equations are written as a system of first-order equations, when solving the Navier-Stokes equation by the discontinuous Galerkin method.

Introduction. Mathematical modeling of hydrodynamic processes in shallow reservoirs of complex geometry in the presence of coastal engineering systems requires an integrated approach in the development of algorithms for constructing computational grids and methods for solving grid equations. The work is devoted to the description of algorithms that allow to reduce the time for solving SLAE by using an algorithm for processing overlapping geometry segments and organizing parallel pipeline calculations. The aim of the work is to compare the acceleration of parallel algorithms for the methods of Seidel, Jacobi, modified alternately triangular method and the method of solving grid equations with tridiagonal preconditioner depending on the number of computational nodes.

Materials and Methods. The numerical implementation of the modified alternating-triangular iterative method for solving grid equations (MATM) of high dimension is based on parallel algorithms based on a conveyor computing process. The decomposition of the computational domain for the organization of the pipeline calculation process has been performed. A graph model is introduced that allows to fix the connections between neighboring fragments of the computational grid. To describe the complex geometry of a reservoir, including coastal structures, an algorithm for overlapping geometry segments is proposed.

Results. It was found that the efficiency of implementing one step of the MATM on the GPU depends only on the number of threads along the Oz axis, and the step execution time is inversely proportional to the number of nodes of the computational grid along the Oz axis. Therefore, it is recommended to decompose the computational domain into parallelepipeds in such a way that the size along the Oz axis is maximum, and the size along the Ox axis is minimal. Thanks to the algorithm for combining geometry segments, it was possible to speed up the calculation by 14–27 %.

Discussion and Conclusions. An algorithm has been developed and numerically implemented for solving a system of large-dimensional grid equations arising during the discretization of the shallow water bodies’ hydrodynamics problem by MATM, adapted for heterogeneous computing systems. The graph model of a parallel-pipeline computing process is proposed. The connection of water body’s geometry segments allowed to reduce the number of computational operations and increase the speed of calculations. The efficiency of parallel algorithms for the methods of Seidel, Jacobi, modified alternately triangular method and the method of solving grid equations for problems of hydrodynamics in flat areas, depending on the number of computational nodes, is compared.

Introduction. Increasing accuracy in the approximation of fractional integrals, as is known, is one of the urgent tasks of computational mathematics. The purpose of this study is to create and apply a second-order difference analog to approximate the fractional Riemann-Liouville integral. Its application is investigated in solving some classes of fractional differential equations. The difference analog is designed to approximate the fractional integral with high accuracy.

Materials and Methods. The paper considers a second-order difference analogue for approximating the fractional Riemann-Liouville integral, as well as a class of fractional differential equations, which contains a fractional Caputo derivative in time of the order belonging to the interval (1, 2).

Results. To solve the above equations, the original fractional differential equations have been transformed into a new model that includes the Riemann-Liouville fractional integral. This transformation makes it possible to solve problems efficiently using appropriate numerical methods. Then the proposed difference analogue of the second order approximation is applied to solve the transformed model problem.

Discussion and Conclusions. The stability of the proposed difference scheme is proved. An a priori estimate is obtained for the problem under consideration, which establishes the uniqueness and continuous dependence of the solution on the input data. To evaluate the accuracy of the scheme and verify the experimental order of convergence, calculations for the test problem were carried out.

Two neurological models of information warfare are considered. For each of them, the optimal control problem is considered, assuming that the Campaign Planner is associated with the governing body of one of the belligerent parties and distributes the volume of propaganda broadcasting in time.

The cost functional reflects the Planner’s desire to maximize the number of their supporters at a given time while minimizing costs during the campaign.

The problem is studied analytically, using the Pontryagin’s maximum principle.

Optimal control is obtained for various combinations of parameters.

The “increasing” type of campaign is aimed at ensuring that for most individuals information is received immediately before the finish line, and that the impression of this information does not have time to weaken. In contrast, the strategy of a “decreasing” campaign implies a high role of interpersonal communication: it is based on convincing a significant number of individuals of their position at the very beginning, who will then retell it to their interlocutors.

Introduction. The work is devoted to the study of the generation and development of turbulent structures in shallow-water flows. For optimal water resource management, it is necessary to know what the consequences will be if the flow system changes as a result of human intervention. Basically, all fluid flows that relate to the practice of civil engineering are turbulent in nature. These are, for example, river and channel flows, tidal currents in the oceans and coastal seas. Shallow currents in the environment often include a wide range of vortex scales, ranging from micro-scale vortices to large-scale coherent structures with horizontal length scales that far exceed the depth of water (L >> H). The existence of such large structures is a typical characteristic of turbulence in shallow flow. This indicates the need for a systematic analysis of the problem, as well as modeling of such complex formalized systems. The purpose of this work is to model and analyze the dynamics of quasi-2D turbulence structures.

Materials and Methods. Large-scale quasi-2D coherent structures (2 DCS) are investigated depending on the source and localization in the liquid column. Turbulent flows in the channel satisfying incompressible Navier-Stokes equations are considered. The numerical experiment was carried out on the basis of the “large eddy simulation” (LES) approach.

The Results of the Study. Scenario of the dynamics of quasi-2D turbulence structures of the coastal zone is constructed, the formation of vortex structures is predicted.

Discussion and Conclusions. The development of two-dimensional turbulence in shallow flows illustrates the processes that control quasi-two-dimensional turbulence, including the merging of individual vortices. The main mechanism controlling the decay of 2DCS is the loss of energy due to friction on the bottom, while the larger the size of the vortex relative to the depth, the faster the direct dissipation of its kinetic energy occurs.

To effectively regulate traffic on highways and networks of modern megacities, it is necessary to introduce Intelligent Transport Systems, which include many innovative solutions, in particular, mathematical models for describing the dynamics of traffic flows.

The article is devoted to a brief description of the current state in this area in its development — from the simplest macroscopic and microscopic models that have become classic to modern developments.

Special attention is paid to the original multilane models developed by the authors of the article within both approaches.

The macroscopic model is based on the quasigasdynamic approach, while the microscopic one uses the ideology of cellular automata and constitutes a generalization of the Nagel-Schreckenberg model for the multilane case.

The difference in the representation method and the mathematical apparatus for the mac-roscopic and microscopic description of traffic flows is briefly described, followed by the review of the main models at different stages of their development, presented by foreign and Russian authors.

Special attention is paid to the three-phase theory of Boris Kerner and models built in the framework of this theory.

Examples of modern software for traffic modeling are given.

The original quasigasdynamic model of traffic flows, which uses the continuum approximation and is constructed by analogy with the well-known model of gas dynamics, is briefly described. Due to the introduction of the lateral speed, the model is generalized to the multilane case.

An original microscopic model based on the cellular automata theory and representing a generalization of Nagel- Schreckenberg model for the multilane case is described. The model has been further developed by taking into account various driving strategies and behavioral aspects.

The article presents a brief overview of the state of the art in the field of mathematical modeling of traffic flows, as well as original macroscopic and microscopic models developed by the authors for the case of multilane traffic.

Introduction. This work is devoted to the study of a non-stationary two-dimensional model of sediment transport in coastal marine systems. The model takes into account the complex multi-fractional composition of sediments, the gravity effect and tangential stress caused by the impact of waves, turbulent exchange, dynamically changing bottom topography, and other factors. The aim of the work was to carry out an analytical study of the conditions for the initialboundary value problem existence and uniqueness corresponding to the specified model.

Materials and Methods. Linearization of the initial-boundary value problem is performed on a temporary uniform grid. The nonlinear coefficients of a quasilinear parabolic equation are taken with a “delay” by one grid step. Thus, a chain of correlated by initial conditions is the final solutions of problems is built. The study of the existence and uniqueness of the problems included in this chain, and therefore the original problem as a whole, is carried out involving the methods of mathematical and functional analysis, as well as methods for solving differential equations.

Results. Earlier, the authors investigated the existence and uniqueness of the initial-boundary value problem of the transport of sediments of a single-component composition. In the present work, the result obtained is extended to the case of multi-fractional sediments.

Discussion and Conclusions. The non-linear spatial two-dimensional model of sediment transport was previously investigated by the team of authors in the case of bottom sediments consisting of particles having the same characteristic dimensions and density (single-component composition) based on the analysis of the existing results of mathematical modeling of hydrodynamic processes. In this paper, the previous results of the study are extended to the case of sediments of a multicomponent composition, namely, the conditions for the existence and uniqueness of the solution of the initial-boundary value problem corresponding to the considered model are determined.