We consider interpolation-characteristic schemes approximating the radiative transfer equation corresponding to the P_1 model. The model equations are modified by adjusting the rate of radiation energy transfer. This correction can reduce the influence of nonphysical effects in calculating radiative heat transfer in a medium with nonuniform opacity.

A closed system of equations for describing turbulent flows is obtained. Additional equations for the cross pulsation moments ρΔu_iΔu_k are derived using a balanced kinetic equation, which was previously used to obtain a quasi-gasdynamic system of equations. Numerical results for the problem of a two-dimensional mixing layer between two flows are presented.

Based on a simple kinetic model that is used in the derivation of a quasigasdynamic system, additional equations for turbulent moments are obtained. The properties of the additional equations are demonstrated by the example of turbulent mixing layer simulation.

На основе системы нестационарных осредненных по Рейнольдсу уравнений Навье-Стокса (URANS), дополненной моделями турбулентности Спаларта-Аллмараса (SA) и Ментера (SST) проведено численное моделирование сверхзвукового обтекания модели воздухозаборника вязким теплопроводным газом при различных числах Маха набегающего потока ($M = 4$ и $5$) и температурных факторах поверхности. Исследованы трехмерные аспекты течения и зависимость теплового потока от температуры стенки. Проведено сравнение результатов моделирования с экспериментальными данными.

Numerical simulation of a supersonic flow of viscous heat-conducting gas past an inlet model for various Mach numbers (M = 4 and 5) and temperatures of the model surface is carried out on the basis of an unsteady Reynolds averaged Navier–Stokes (URANS) equations system with the Spalart–Allmaras (SA) and Menter (SST) turbulence model. The three-dimensional features of the flow and heat flux’s dependence on the wall temperature are investigated. The simulation results are compared with the experimental data.

An original method for processing large factor models based on graph condensation using machine learning models and artificial neural networks is developed. The proposed mathematical apparatus can be used to plan and manage complex organizational and technical systems, to optimize large socioeconomic objects of national scale, and to solve problems of preserving the health of the nation (searching for compatibility of medications and optimizing health care resources).

We develop an algorithm for the numerical treatment of nonlinear heat conduction problems, which is adapted to the architecture of high-performance computing systems. The technique is based on a hyperbolic heat conduction model and explicit difference scheme. The new scheme provides the second-order temporal resolution of nonlinearity with an acceptable time step. Its application to plasma dynamic simulations is discussed.

Изучаются задачи Коши для многомерной симметричной линейной гиперболической системы уравнений 1-го порядка с переменными коэффициентами и ее сингулярных возмущений - сильно параболической и гиперболической 2-го порядка систем уравнений с малым параметром $\tau>0$ при вторых производных по $x$ и $t$. Доказываются существование и единственность слабых решений всех трех систем и равномерные по $\tau$ оценки решений систем с возмущениями. Даются оценки разности решений исходной системы и систем с возмущениями, в том числе в норме $C(0,T;L^2(\mathbb{R}^n))$ порядка $O(\tau^{\alpha/2})$ при начальной функции $\mathbf w_0$ из пространств Соболева $H^\alpha(\mathbb{R}^n)$ для $\alpha=1,2$ и пространств Никольского $H_2^{\alpha}(\mathbb{R}^n)$ для $0<\alpha<2$, $\alpha\neq 1$ и соответствующих условиях на свободный член системы 1-го порядка. При $\alpha=1/2$ охватывается широкий класс разрывных $\mathbf w_0$. Выводятся также оценки производных любого порядка по $x$ как решений, так и их разностей порядка $O(\tau^{\alpha/2})$. Указывается приложение результатов к линеаризованной на постоянном решении системе уравнений газовой динамики 1-го порядка и ее возмущениям - линеаризованным параболической и гиперболической 2-го порядка квазигазодинамическим системам уравнений. Библиография: 34 названия.

This article is focused on the modeling of multiphase hydrodynamic flows within “digital core” technology for the needs of the oil and gas industry. The essenc e of this technology is direct numerical simulation of the flows on the scale of the pore space of oil and gas reservoir rocks with direct resolution of the structure of this space and the dynamics of interphase boundaries. The importance of the development of high-performance computing facilities (supercomputers) for the successful implementation of this technology is emphasized. Work carried out by the Keldysh Institute of Applied Mathematics, RAS, in the field of mathematical models, computational algorithms, and their software implementation is described.

A method for the numerical solution of a nonlinear equation describing the diffusion transfer of radiation energy has been developed. According to the method, the second time derivative with a small parameter is introduced into the parabolic equation and an explicit difference scheme is applied. The explicit approximation of the initial equation makes it possible to implement on its basis an algorithm that is effectively adapted to the architecture of high-performance computing systems. In comparison with the original scheme, the new one allows for a larger time integration step and a sufficiently high resolution of the solution structure, providing the second order of accuracy. A heuristic algorithm for choosing the parameters of the three-level difference scheme is proposed. A promising field of application of the method can be problems in plasma physics and astrophysics.

The mathematical aspects of algorithms for assessing the influence of external factors in the cognitive simulation of complex systems are considered. The systems are represented as directed graphs, to the nodes and edges of which certain weights are assigned. In the conventional approach, the influence of external factors (input nodes) on the state (weight) of system elements (internal nodes) is computed successively on the set of all paths using the method of additive convolution of the input nodes and the corresponding edges. In the new approach proposed here, such influences are defined for each internal node as a partial derivative of a functional dependence that corresponds to the graph. Formulas for calculating the influence coefficient are derived that take into account the graph structure, partitions into cycles, and paths between corresponding nodes. The results provided by the proposed method and the conventional one are compared using as an example a simple cognitive model that assesses the impact of a certain viral disease on the population and production and measures used to counteract it. The introduced definition of the concept of influence is also valid for more complex, nonlinear formulas or fuzzy characteristics of nodes and edges.

The first ideas concerning the principle of operation of non-mechanical magneto-hydrodynamic pumping systems were originally created at the end of the 1960s during studies of efficient heat exchange systems in nuclear energy sources using liquid metals as coolant. Currently, these ideas are of increasing interest in the creation of many critical technologies that require precise flow control in distributed energy transfer systems, in particular for a new generation of environmentally friendly efficient nuclear power sources.In this regard, computational fluid dynamic methods are an effective instrument for studying complex energy transfer processes, the possibilities of which have been significantly expanded with the creation of modern high-performance computing systems and modern computing technologies.The purpose of this article is to describe an advanced method of the mathematical modeling of magneto -hydrodynamic processes based on kinetic and kinetically consistent models. The novelty of this approach lies in more advanced physical models derived from the Boltzmann kinetic equation with a complex-valued statistical distribution function that includes electromagnetic interactions. The possibility of applying this description to incompressible viscous media (liquid metals) in complex magneto-hydrodynamic systems is shown.The system of equations of the proposed method, a numerical algorithm and the results of the mathematical modeling of a test system for energy transfer using a magneto-hydrodynamic pump and an exchange cavity are presented.

A mathematical model for viscous gas flows modeling is considered taking into account the time scales limitation by the characteristic time between molecular collisions. This approach leads to a variant of the quasi-gas-dynamic equations system (QGD), based on the relationship between the kinetic and macroscopic descriptions of a continuous medium motion. Based on the presented model, a numerical algorithm is constructed for modeling viscous compressible gas flows. QGD equations are discretized by the finite volume method. Numerical investigation of the supersonic tip vortex interaction with the finite span wing has been carried out. The incoming flow has a Mach number $$M = 3$$ , and a unit Reynolds number $$\hbox {Re}=10^{7}$$ . A hexagonal unstructured grid containing 4.1 *  $$10^{7}$$ cells was used for the simulation. Simulations were performed on the multiprocessor computer system K-60 at the Keldysh Institute of Applied Mathematics RAS. The deformation of the tip vortex was revealed. Numerical data obtained from QGD system were compared with those from the averaged Navier–Stokes equations.

The COVID-19 pandemic has created a public health emergency in Russia and across the world. The wavelike spread of the new coronavirus infection, caused by newly emerging variants of the coronavirus, has led to a high incidence rate in all subjects of the Russian Federation. It is becoming extremely topical to get the opportunity to manage the development of the epidemic and assess the impact of certain regulatory measures on this process. This will help government agencies make informed decisions to control the burden on healthcare organizations. It is often impossible to obtain such assessments without using modern mathematical models.

In the paper, we study a design and stability of contrast-independent partially explicit time discretizations for Quasi-Gas-Dynamics (QGD) Equations in multiscale high-contrast media. In our previous works, we have introduced contrast-independent partially explicit time discretizations. In this paper, we extend these ideas to multiscale QGD problems. Because of high contrast, explicit methods require a very small time stepping. By designing appropriate spatial splitting and temporal splitting, partially explicit methods remove this constraint. The proposed partially explicit time discretization consists of two steps. First, we split the space into contrast dependent (fast) and contrast independent (slow) components on a coarse grid that is much larger compared to spatial heterogeneities. Secondly, we design a temporal splitting algorithm in a such way that it is stable and the time step is independent of the contrast and only depends on the coarse mesh size. Using proposed method, a few degrees of freedom are treated implicitly and the approach is mostly explicit. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space (slow). We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.

Разработан алгоритм численного решения нелинейных задач теплопроводности, хорошо адаптированный к архитектуре высокопроизводительных вычислительных систем. Метод основан на гиперболической модели теплопроводности с малым параметром при второй производной по времени и явной разностной схеме. Новая схема обеспечивает разрешение нелинейности со вторым порядком по времени с приемлемым временным шагом. Предложен эвристический алгоритм выбора параметров трехслойной разностной схемы. Обсуждается приложение к моделированию динамики плазмы.

We study the Cauchy problems for a first-order symmetric hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter τ > 0 in front of the second derivatives with respect to x and t. The properties of solutions of all three systems are formulated, and estimates of order O(τ^α /2) are given for the difference between the solutions of the original system and systems with perturbations for an initial function w0 of smoothness α in the sense of L^2(ℝ^n) , 0 < α⩽ 2 . For α = 1/2 , a broad class of discontinuous functions w0 is covered. Applications to the linearized system of gas dynamics equations and to the linearized parabolic and hyperbolic second-order quasi-gasdynamic systems of equations are given.

An implementation of the particle method on a hybrid cluster with graphics processing units (GPU) for modeling bulk generation, propagation, and scattering of electrons in a self-consistent electromagnetic field is presented. For synchronization of recording of results and data exchange in the calculation of particle parameters, solutions are proposed that reduce GPU RAM usage.

We developed a technique for the numerical solution of a nonlinear equation describing the diffusion transfer of radiation energy. The method is based on the introduction of the second time derivative with a small parameter into the parabolic equation and an explicit difference scheme. The explicit approximation of the original equation makes it possible to implement an algorithm that is effectively adapted to the architecture of high-performance computing systems. The new scheme provides a second-order resolution of the nonlinearity in time with an acceptable time step. A heuristic algorithm for choosing the parameters of a three-level difference scheme is proposed. Perspective applications of the method are problems in astrophysics, for example, the simulation of a strongly radiating shock wave breakout at the surface of a star at the stage of its evolution known as a supernova explosion.

This paper deals with the creation of parallel algorithms implementing macro-and microscopic traffic flow models on modern supercomputers. High-performance computing contributes to the development of intelligent transportation systems based on information technologies and aimed at the effective regulation of traffic in large cities. As a macroscopic approach, the quasi-gas-dynamic traffic model approximated by explicit finite-difference schemes is proposed. One- and two-dimensional variants of the system are considered, and the concept of lateral velocity and different equations for obtaining it are discussed. The microscopic approach is represented by the multilane cellular automata model. The previously developed model is extended to reproduce synchronized flow in accordance with Kerner’s three-phase theory. The new version starts from the Kerner–Klenov–Schreckenberg–Wolf model and operates with the concept of the synchronization gap. Macroscopic models are relevant for determining the common characteristics of road traffic, while microscopic models are useful for a detailed description of cars’ movement. Both approaches possess inner parallelism. The parallel algorithms are based on the geometrical parallelism principle with different boundary conditions at interfaces of the subdomains. Sufficiently high speedups were reached when up to 100 processors were involved in calculations. The proposed algorithms can serve as the core of ITS.