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# A. S. KholodovRussian Academy of Sciences | RAS · Institute for Computer Aided Design, RAS

22.39

· Corresponding Member of RAS, Head of ICAD RASAbout

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Research Items (100)

- May 2019

Physical and comprehensive numerical studies of the generation of plasma bunches with a high specific energy have been carried out with the use of a plasma gun. The parameters of the plasma bunch upon exit from the plasma accelerator and during propagation in the ionosphere (h > 200 km) to considerable distances (≈100 km) have been calculated. A special numerical algorithm is presented to determine the results of the impact of a rarefied high-velocity gas flow (\({v}\) ∼ 5 × 10⁷ cm/s) on the surface of crystalline and amorphous solid bodies.

- Aug 2018

A distinctive feature of hyperbolic equations is the finite propagation velocity of perturbations in the region of integration (wave processes) and the existence of characteristic manifolds: characteristic lines and surfaces (bounding the domains of dependence and influence of solutions). Another characteristic feature of equations and systems of hyperbolic equations is the appearance of discontinuous solutions in the nonlinear case even in the case of smooth (including analytic) boundary conditions: the so-called gradient catastrophe. In this paper, on the basis of the characteristic criterion for monotonicity, a universal algorithm is proposed for constructing high-order schemes monotone for arbitrary form of the sought-for solution, based on their analysis in the space of indefinite coefficients. The constructed high-order difference schemes are tested on the basis of the characteristic monotonicity criterion for nonlinear one-dimensional systems of hyperbolic equations.

- Mar 2017

The objective of this study is the construction of the numerical mathematical model for the propagation of fine impurities in ventilation networks using gas dynamic equations by the numerical coupling of domains with different dimensions.

An airflow in the first four generations of the tracheobronchial tree was simulated by the 1D model of incompressible fluid flow through the network of the elastic tubes coupled with 0D models of lumped alveolar components, which aggregates parts of the alveolar volume and smaller airways, extended with convective transport model throughout the lung and alveolar components which were combined with the model of oxygen and carbon dioxide transport between the alveolar volume and the averaged blood compartment during pathological respiratory conditions. The novel features of this work are 1D reconstruction of the tracheobronchial tree structure on the basis of 3D segmentation of the computed tomography (CT) data; 1D−0D coupling of the models of 1D bronchial tube and 0D alveolar components; and the alveolar gas exchange model. The results of our simulations include mechanical ventilation, breathing patterns of severely ill patients with the cluster (Biot) and periodic (Cheyne-Stokes) respirations and bronchial asthma attack. The suitability of the proposed mathematical model was validated. Carbon dioxide elimination efficiency was analyzed in all these cases. In the future, these results might be integrated into research and practical studies aimed to design cyberbiological systems for remote real-time monitoring, classification, prediction of breathing patterns and alveolar gas exchange for patients with breathing problems.
http://www.mdpi.com/2079-3197/5/1/11/pdf

- Sep 2016

Using the Duhamel–Neumann equations, we consider the stationary heat-loading problem of a bulk specimen of a two-dimensional material (like grapheme) as an approximation of small elastic deformations. We present a numerical method for solving the heat-loading problem of a specimen of a complex shape with the use of a Friedrichs-monotonic finite-difference scheme on chaotic grids in a multiply connected integration domain. Then we demonstrate the results of the computational experiments.

- Jul 2016

The 3D MHD algorithm developed by us has been adapted to modeling the interaction between two plasma bunches in the ionosphere, mainly in order to sufficiently correctly describe the physics of the interaction between two plasma regions with regard to the ionospheric inhomogeneity and the geomagnetic field action. Modeling has been performed for several versions of location of the plasma region centers.

- Jul 2016

The spatiotemporal pattern for the development of a plasma cloud formed in the ionosphere and the main cloud gas-dynamic characteristics have been obtained from 3D calculations of the explosion-type plasmodynamic flows previously performed by us. An approximate method for estimating the plasma temperature and ionization degree with the introduction of the effective adiabatic index has been proposed based on these results.

For quasilinear systems of hyperbolic equations, the nonclassical boundary value problem of controlling solutions with the help of boundary conditions is considered. Previously, this problem was extensively studied in the case of the simplest hyperbolic equations, namely, the scalar wave equation and certain linear systems. The corresponding problem formulations and numerical solution algorithms are extended to nonlinear (quasilinear and conservative) systems of hyperbolic equations. Some numerical (grid-characteristic) methods are considered that were previously used to solve the above problems. They include explicit and implicit conservative difference schemes on compact stencils that are linearizations of Godunov’s method. The numerical algorithms and methods are tested as applied to well-known linear examples.

In this paper we present a second-order hydrodynamic traffic model that generalizes the existing second-order models of Payne-Whithem, Zhang and Aw-Rascle. In the proposed model, we introduce the pressure equation describing the dependence of "traffic pressure" on traffic density. The pressure equation is constructed for each road segment from the fundamental diagram that is estimated using measurements from traffic detectors. We show that properties of any phenomenological model are fully defined by the pressure equation. We verify the proposed model through simulations of the Interstate 580 freeway segment in California, USA, with traffic measurements from the Performance Measurement System (PeMS).

In this paper, we propose a unified procedure for calibration of macroscopic second-order multilane traffic models. The focus is on calibrating the fundamental diagram using the combination stationary detector data and GPS traces. GPS traces are used in estimation of the deceleration wave speed. Thus calibrated model adequately represents the three phases of traffic: free flow, synchronized flow and the wide moving jam. The proposed approach was validated in simulation using stationary detection data and GPS traces from the Moscow Ring Road. Simulation showed that the proposed second-order model is more accurate than the first-order LWR model.

At present, scientometric assessment is an attribute of both the practice of the internal management of scientific establishments and the system of state and private financial support. Note that the attitude of scientists to scientometrics and its use for determining the efficiency of scientific activity varies, from seething resentment to (significantly less often) willingness to offer proposals on correcting scientometric instrumentation and the practice of its use. The author of the article published below is an advocate of the latter strategy. With account for the drawbacks of the existing indicators, he shows how we can make them more adequate to the targets for the sake of which they are developed.

Изложены методы построения монотонных разностных схем высокого порядка аппроксимации для решения одномерных уравнений и систем гиперболического типа. Использование таких схем является актуальным при численном решении задач, содержащих в области интегрирования разрывы искомых функций и другие особенности. Навыки практического использования рассматриваемых методов могут быть получены при выполнении лабораторной работы, задание к которой вместе с образцом выполнения работы приведено в пособии.

- Feb 2014

Several questions of scientometrics parameters organization are considered.
Two new indices for scientific works citation analysis are proposed. They
provide more detailed and reliable scientific significance assessment of
individual authors and scientific groups basing on the publication activity

- Apr 2013

Numerical simulation of the acoustic-gravity waves (AGW) generated by
long-period oscillations of the Earth's (oceanic) surface, earthquakes,
explosions, thermal heating, seiches, and tsunami is carried out.
Wavelike disturbances are quite frequent phenomena in the atmosphere and
ionosphere. These events can be caused by the impacts from space and
atmosphere, by oscillations of the Earth'as surface and other
near-surface events. These wavelike phenomena in the atmosphere and
ionosphere appear as the alternating areas of enhanced and depleted
density (in the atmosphere) or electron concentration (in the
ionosphere). In the paper, AGW with typical frequencies of a few
hertz - millihertz are analyzed. AGW are often observed after the
atmospheric perturbations, during the earthquakes, and some time (a few
days to hours) in advance of the earthquakes. Numerical simulation of
the generation of AGW by long-period oscillations of the Earth's and
oceanic surface, earthquakes, explosions, thermal heating, seiches, and
tsunami is carried out. The AGW generated by the near-surface phenomena
within a few hertz-millihertz frequency range build up at the
mid-atmospheric and ionospheric altitudes, where they assume their
typical spatial scales of the order of a few hundred kilometers.
Oscillations of the ionospheric plasma within a few hertz-millihertz
frequency range generate electromagnetic waves with corresponding
frequencies as well as travelling ionospheric irregularities (TIDs).
Such structures can be successfully monitored using satellite radio
tomography (RT) techniques. For the purposes of RT diagnostics, 150/400
MHz transmissions from low-orbiting navigational satellites flying in
polar orbits at the altitudes of about 1000 km as well as 1.2-1.5 GHz
signals form high-orbiting (orbital altitudes about 20000 km) navigation
systems like GPS/GLONASS are used. The results of experimental
studies on generation of wavelike disturbances by particle precipitation
are presented. The ionospheric footprints of atmospheric disturbances
are given. The effects of AGW evolution after launching the rockets are
studied. One of the possible applications of RT imaging of wavelike
disturbances is the study of AGW and TID as probable precursors of the
earthquakes. The key difficulty here is to distinguish between the AGW
and atmospheric and ionospheric disturbances of non-seismic nature (for
example, those caused by the enhanced solar-geomagnetic activity), which
can be done by analyzing spatial two-dimensional and three-dimensional
structures revealed by tomographic methods. The examples of AGW RT
imaging based on the real experimental satellite data measured in
regions of the Europe, North America and Asia are demonstrated. The
example of AGW and TID generation by the tsunami wave after the Tohoku
earthquake is presented. Our results prove the capability of RT methods
to detect wavelike disturbances in the ionosphere, which are caused by
the near-surface sources, and to distinguish between these disturbances
and the influence from the atmosphere and space. The work was
supported by the Russian Foundation for Basic Research (grants
11-05-01157 and 13-05-01122 ).

- Sep 2012

The results of three-dimensional calculations of a plasma flow caused by a cosmic nuclear explosion, performed in an MHD approximation, are presented. The main regularities and specific features of the development of a large-scale plasma flow have been analyzed for a later stage (up to several hundreds of seconds) depending on the altitude and plasma bulge energy.

- Jul 2012

The physicomathematical features of the plasma explosion region dynamics at altitudes higher than 100 km at the early stage of its development, which culminates in the transfer of a considerable portion of the initial plasma energy into the environment, are discussed in the first part of the work. The radiation gas dynamic, inertial, and shock stages of the disturbed region development are calculated.

- Jul 2012

In this paper, data flows from different nodes in the packet network of data transmission are simulated as immiscible continuum flows. Conceptual and mathematical models are proposed. Computer simulation has been performed and compared with the ns-2 system of packet-level simulation. The developed model demonstrated fairly good accuracy, requiring at the same time much lower consumption of computing resources.

- May 2012
- International Conference "Instabilities and Control of Excitable Networks: From Macro- to Nano-Systems"

The work is focused on extending the model of closed human circulation with mathematical model of skeletal-muscle pumping that is described in the second part. Numerical simulations are presented showing blood flow redistribution during intensive exercise. Basing on the purely mechanical approach we conclude that effective blood flow through lower extremities during intensive running can be optimized by the stride frequency that depends on the cardiovascular network morphological properties.

- Feb 2011

The generation and propagation of acoustic gravity waves (AGWs) in a two-dimensional Earth atmosphere was numerically simulated
on the basis of a magnetohydrodynamic model. Due to the divergent form of the equations and the conservativeness of the numerical
method, strong perturbations, including discontinuous solutions, were considered and computed by applying a shock capturing
algorithm. The results were compared with previous ones obtained using the Euler equations. The comparison results confirmed
the generation of atmospheric AGWs by pulsed ground-based sources and supported the possibility of using simpler gasdynamic
models for the given class of problems in a fairly wide range of governing parameters.
KeywordsMHD-based simulation of ionospheric perturbations–hyperbolic equations–conservative numerical method–numerical solution of the Euler equations

- Feb 2011

The earlier developed monotonous conservative variant of a net-characteristic method of the second- and third-order approximation
based on 3D equations of magnetic gasodynamics with consideration for the magnetic field diffusion is expanded for the solution
of dynamic problems on the flow of the upper atmosphere of the Earth by solar plasma. It is modified, which results in support
for the solenoidality of the magnetic field and the removal of the magnetic charge. Calculations are carried out and preliminary
results of using software complexes to simulate the passage of the leading edge of solar plasma through the upper atmosphere
of the Earth are given. Unlike existing studies in this direction, the dynamics of the passage of the leading edge upon a
strong burst in the intensity of solar wind over the entire altitude range from the surface of the Earth to the upper magnetosphere
is considered.
Keywordsupper atmosphere of the Earth–magnetosphere–magnetogasodynamics–computational models–methods for splitting over coordinates and physical processes

- Jan 2011

Due to the very long timescales involved (ms-s), theoretical modeling of fundamental biological processes including folding, misfolding, and mechanical unraveling of biomolecules, under physiologically relevant conditions, is challenging even for distributed computing systems. Graphics Processing Units (GPUs) are emerging as an alternative programming platform to the more traditional CPUs as they provide high raw computational power that can be utilized in a wide range of scientific applications. Using a coarse-grained Self Organized Polymer (SOP) model, we have developed and tested the GPU-based implementation of Langevin simulations for proteins (SOP-GPU program). Simultaneous calculation of forces for all particles is implemented using either the particle based or the interacting pair based parallelization, which leads to a ∼90-fold acceleration compared to an optimized CPU version of the program. We assess the computational performance of an end-to-end application of the SOP-GPU program, where all steps of the algorithm are running on the GPU, by profiling the associated simulation time and memory usage for a number of small proteins, long protein fibers, and large-size protein assemblies. The SOP-GPU package can now be used in the theoretical exploration of the mechanical properties of large-size protein systems to generate the force-extension and force-indentation profiles under the experimental conditions of force application, and to relate the results of singlemolecule experiments in vitro and in silico.

Blood system functions are very diverse and important for most processes in human organism. One of its primary functions is matter transport among different parts of the organism including tissue supplying with oxygen, carbon dioxide excretion, drug propagation etc. Forecasting of these processes under normal conditions and in the presence of different pathologies like atherosclerosis, loss of blood, anatomical abnormalities, pathological changing in chemical transformations and others is significant issue for many physiologists. In this connection should be pointed out that global processes are of special interest as they include feedbacks and interdependences among different regions of the organism. At the modern level of computer engineering the most adequate physical model for the dynamical description of cardiovascular system is the model of non-stationary flow of incompressible fluid through the system of elastic tubes. Mechanics of such flow is described by nonlinear set of hyperbolic equations including mass and momentum conservation joined with equation of state that determines elastic properties of the tube [1]. As we interested in global processes the models of the four vascular trees (arterial and venous parts of systemic and pulmonary circulation) must be closed with heart and peripheral circulation models. Heart operation is described by the model of fluid flow averaged by volume through the system of extensible chambers that results in the set of stiff ordinary differential equations [1]. When combined these models allow us to consider functional changes and responses as during one cardiac cycle and at a longer periods upon 10 minutes that
Fig. 1.
Pressure wave propagation through the large pulmonary arteries during one cardiac cycle. Grayscale designates divergence from the minimum pressure in each vessel.

Blood system functions are very diverse and important for most processes in human organism. One of its primary functions is matter transport among different parts of the organism including tissue supplying with oxygen, carbon dioxide excretion, drug propagation etc. Forecasting of these processes under normal conditions and in the presence of different pathologies like atherosclerosis, loss of blood, anatomical abnormalities, pathological changing in chemical transformations and others is significant issue for many physiologists. In this connection should be pointed out that global processes are of special interest as they include feedbacks and interdependences among different regions of the organism. Thus the main goal of this work is to develop the model allowing to describe effectively blood flow in the whole organism. As we interested in global processes the models of the four vascular trees (arterial and venous parts of systemic and pulmonary circulation) must be closed with heart and peripheral circulation models. As one of the model applications the processes of the blood loss is considered in the end of the paper.

Numerical model of the peripheral circulation and dynamical model of the large vessels and the heart are discussed in this paper. They combined together into the global model of blood circulation. Some results of numerical simulations concerning matter transport through the human organism and heart diseases are represented in the end.

Frequently during its lifetime a human organism is subjected to the acoustical and similar to them vibrating impacts. Under the certain conditions such influence may cause physiological changes in the organs functioning. Thus the study of the oscillatory mechanical impacts to the organism is very important task of the numerical physiology. It allows to investigate the endurance limits of the organism and to develop protective measures in order to extend them. The noise nuisances affects to the most parts of the organism disrupting their functions. The vibrating disturbances caused to the lung function as one of the most sensitive to the acoustical impacts is considered in this work. The model proposed to describe the air motion in trachea-bronchial tree is based on the one dimensional no-linear theory including mass and momentum conservation for the air flow in flexible tubes.

A complex dynamic model of the closed cardiovascular and respiratory systems including their interaction is considered. It
is proposed to substitute a single-component model of the alveolar volume with a multi-component one. A refined structure
of the vascular network based on experimental data is suggested. Resonance properties of the alveolar volume components are
investigated with the help of the proposed multi-component model. Computational results are presented showing an amplitude-dependent
decrease in oxygen concentration in the venous part of the pulmonary circuit when the alveolar volume is subject to periodic
disturbances at its eigenfrequencies

- Jan 2008

Injection of electronic beams, neutral and plasma clots in an ionosphere and magnetosphere is now one of fruitful active methods of research of the geophysical phenomena. One of the purposes of such experiments consists in studying interaction of beams of the charged particles with the top atmosphere, its movement in a geomagnetic field and generation of electromagnetic disturbance. Sources of the charged particles can be special generators or sources of explosive type, as in experiments "Argus", "Starfish" and others. One of the main physical features of the considered phenomena is their large-scale character: the plasma source can is at height of the order of hundred and more than kilometres and to have the average cross-section size of the order of 10-500 km. Thus the stream of MeV-electrons, extending along power lines of a magnetic field, can get deep into atmospheres up to heights of 30-40 km, forming rather narrow tube of a current in the extent into hundreds kilometres up to the nearest and thousand kilometres up to distant aurora region. On process of formation current systems defining influence renders a longitudinal electric field and the return current of conductivity arising under action of this field and ionization of an atmosphere by electronic current systems have been developed for the decision of a problem on formation the special numerical algorithm representing movement electrons in the form of final number of discrete plasma sheets. Such model allows to describing adequately a turn electrons by electric field, interoscillation of direct and return currents, and accumulation of charges into the certain area of space. The space charge of a tube of a current possesses own electric field, influencing on dynamics electrons. Therefore there is the self-coordinated electrodynamics problem defining evolution of electronic current system. The current system generates in a surrounding atmosphere and on a surface of the Earth electromagnetic fields with characteristic time of change of the order of ten seconds, corresponding characteristic recession of activity of the electrons source. Using the developed numerical algorithm, characteristics of a current of relativistic electrons, extending in distant aurora region have been calculated. Density of a current makes the order 1011 1/cm2 s, that will be coordinated with experimental data on operation "Starfish". In the field of absorption of electrons electromagnetic disturbance with amplitude of an electric field of the order 0,1 V/m is formed. Close values of a field are formed and near to a surface of the Earth.

- Jan 2008

Explosive plasma experiments remain the important means of research of geophysical effects in the top ionosphere and magnetosphere. In particular their results can be useful for development of full model of powerful geomagnetic storms. Scientific and applied value of such experiments depends on our ability to simulate them numerically and to understand the physical processes. Complexity of mathematical modelling of such experiments is caused by two circumstances - complexity and variety of physical processes, and large-scale three-dimensional current of plasma. It's important to note that not all features of the processes under consideration are well known and well modelled. And plasma parameters in the indignant area can vary up to 5-7 orders. During last several years we have developed universal enough 3D algorithm for the simulation of large-scale movement of the plasma, based on MHD approach. Diffusion of a magnetic field and the ionization structure of plasma and air is considered. The full algorithm includes the most initial the radiation-gas dynamic stage, a stage of inertial scattering when the charging structure of plasma is formed, a stage of braking of plasma a geomagnetic field and the rarefied ionosphere and later (down to 100-500 s) the stage of convective movements of plasma in a geomagnetic field and the rarefied ionosphere. The algorithm is based on special updating of a monotonous conservative variant of grid-characteristic method 2-3 orders of the approximation, including splitting on spatial variables. Calculations of explosion of energy about 1015 J are executed for some heights from a range of 100-1000 km. Character of development of current essentially varies depending on height. For 100-120 km current is close to bi-dimensional, in an initial stage the shock wave is formed, and for the period of 40-60 seconds the plasma area rises up to 300 km. At heights more than 150 km current, for a while more than 5 seconds are got with character of a powerful ascending jet. The wave comes off plasma the magneto sonic wave and quickly extends along a surface of globe. With increase in height of explosion (400-700 km) the jet gets flat character with primary distribution of weight in a plane of a magnetic meridian. It is gradually developed on a magnetic field, saving the certain inclination in relation to it. At explosions at heights more than 400 km scales current of plasma make more than 1000 km. It is shown, that the plasma area is a source of global low-frequency electromagnetic disturbance. Their parameters are estimated. At energy more than certain size, becomes possible having dug magnetosphere and global infringements in its structure, which depends on height and breadth of explosion. The developed numerical method allows to investigate a relaxation magnetosphere after such artificial indignations and at powerful magnetic storms.

- Jan 2007
- III Asian Pacific Congress on Computational Mechanics, Kyoto, Japan

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K.
Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils.
The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear
monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several
new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point)
stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives.
The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic
equations.

- Jan 2006

For each graph edge with length Xk we consider 1D nonlinear hyperbolic system of equations
n® t +F® xk ( n® ) = f® ,n® = { n1 , ¼,n1 },t \geqslant 0,0 \leqslant xk \leqslant Xk ,k = 1, ¼,K
\overrightarrow \nu _t + \overrightarrow F _{x_k } \left( {\overrightarrow \nu } \right) = \overrightarrow f ,\overrightarrow \nu = \left\{ {\nu _1 , \ldots ,\nu _1 } \right\},t \geqslant 0,0 \leqslant x_k \leqslant X_k ,k = 1, \ldots ,K
(1) with initial conditions
n® ( 0,xk ) = n® 0 ( xk ),k = 1, ¼,K
\overrightarrow \nu \left( {0,x_k } \right) = \overrightarrow \nu ^0 \left( {x_k } \right),k = 1, \ldots ,K
and the next boundary conditions: for graph enters
( l0 = 1, ¼L0 ,xk* = 0 )jli0 ( t,n® ( t,0 ) ) = 0,i = 1, ¼rk0 \leqslant I
\left( {l^0 = 1, \ldots L^0 ,x_{k_ * } = 0} \right)\varphi _{li}^0 \left( {t,\overrightarrow \nu \left( {t,0} \right)} \right) = 0,i = 1, \ldots r_k^0 \leqslant I
(2), for graph exits
( l* = 1, ¼L* ,xk = Xk )jli ( t,n® ( t,Xk ) ) = 0,i = 1, ¼,rk* \leqslant I
\left( {l^ * = 1, \ldots L^ * ,x_k = X_k } \right)\varphi _{li} \left( {t,\overrightarrow \nu \left( {t,X_k } \right)} \right) = 0,i = 1, \ldots ,r_k^ * \leqslant I
(3) and for graph branchpoints
l* = 1, ¼Lylm ( t,wl ,n® l1 , ¼n® lM1 ) = 0m = 1, ¼Ml
l^ * = 1, \ldots L\psi _{lm} \left( {t,w_l ,\overrightarrow \nu _{l1} , \ldots \overrightarrow \nu _{lM_1 } } \right) = 0m = 1, \ldots M_l
(4). Here K is the number of graph edges, LO - enters, LO - exits, L - branchpoints, Ml - incoming and outgoing graph edges for the lth branchpoint,
n® l1 , ¼n® lMl
\overrightarrow \nu _{l1} , \ldots \overrightarrow \nu _{lM_l }
- required vectors in the ends of edges adjoining to branchpoin l, Wl - required vector for the branchpoint l. The matrix
¶F® \mathord/
\vphantom ¶F® ¶n® ¶n® = A = { aij }i,j = 1, ¼,I
{{\partial \overrightarrow F } \mathord{\left/
{\vphantom {{\partial \overrightarrow F } {\partial \overrightarrow \nu }}} \right.
\kern-\nulldelimiterspace} {\partial \overrightarrow \nu }} = A = \left\{ {a_{ij} } \right\}i,j = 1, \ldots ,I
is Jacobi matrix and we can apply the identity
A = W - 1 L W
A = \Omega ^{ - 1} \Lambda \Omega
, where
L = { li }
\Lambda = \left\{ {\lambda _i } \right\}
is the diagonal matrix of the matrix A eigenvalues, Ώ is the nonsingular matrix whose rows are linearly independent left-hand
eigenvectors of the matrix A
( DetW ¹ 0 )
\left( {Det\Omega \ne 0} \right)
and Ώ −1 is the matrix inverse to Ώ.
To enclose boundary conditions (2)-(4) we can use compatibility conditions
wi dn® \mathord/
\vphantom dn® dti dti = 0,( dn® \mathord/
\vphantom dn® dti dti = ¶n® \mathord/
\vphantom ¶n® ¶t ¶t + li ¶n® \mathord/
\vphantom li ¶n® ¶xk ¶xk ),i = 1, ¼,I
\omega _i {{d\overrightarrow \nu } \mathord{\left/
{\vphantom {{d\overrightarrow \nu } {dt_i }}} \right.
\kern-\nulldelimiterspace} {dt_i }} = 0,\left( {{{d\overrightarrow \nu } \mathord{\left/
{\vphantom {{d\overrightarrow \nu } {dt_i }}} \right.
\kern-\nulldelimiterspace} {dt_i }} = {{\partial \overrightarrow \nu } \mathord{\left/
{\vphantom {{\partial \overrightarrow \nu } {\partial t}}} \right.
\kern-\nulldelimiterspace} {\partial t}} + {{\lambda _i \partial \overrightarrow \nu } \mathord{\left/
{\vphantom {{\lambda _i \partial \overrightarrow \nu } {\partial x_k }}} \right.
\kern-\nulldelimiterspace} {\partial x_k }}} \right),i = 1, \ldots ,I
along the characteristics of the system (1)
dx = li dt
dx = \lambda _i dt
directed inside integration domain. These compatibility conditions can be used to analyze correctness of the problem definition
for system (1) through the all graph segments.
The main idea of this approach is that the solution of the global 3D problem for the whole graph can be split out on the set
of the 1D problem for the single graph elements (edges). Then on the each step of numerical integration we join these 1D problems
by the additional equation systems in the graph nodes to get the common solution for the original problem. By this way we
guarantee the global coupling for the all original problem variables. The same computational model can be applied for nonlinear
parabolic equations.
The numerical results are presented for the problem solution on the different graph systems: The global numerical models of
blood circulation in the human body. The model of global regional electrical power systems. The model of bar structures and
frames behavior under the different impacts. The model of the intensive information flows in the computer networks. The model
of heavy traffic in the big cities. The model of flood water and pollution propagation in the large river systems.

- Jan 2006

As a result of three-dimensional calculations it is shown that at explosions in the top ionosphere the stage of intensive scattering of plasma continuously passes in ascending plasma current with the certain inclination to a gradient of density and a geomagnetic field Movement of ionized area across power lines of a geomagnetic field results in its polarization and occurrence of an electric field Under this field in the indignant area and surrounding ionosphere the current system of complex topology which is the reason of generation of longtime electromagnetic disturbance is formed The uniform 3D procedure of dynamics of plasma and electromagnetic indignations calculation is developed parameters of these disturbance are determined It is shown that the topology and a role of some components of current system in generation of electromagnetic disturbance varies depending on latitude and height of a point of explosion and also in time It is explained by the change of a direction of movement of plasma and change of various components of unisotropic conductivity of ionosphere It is shown that the researched mechanism is capable to generate disturbance only in the near connected area

- Jan 2006

For an explanation of results and forecasting of experiments in the top
ionosphere and magnetosphere with use of plasma jets and powerful
explosions physical models and three-dimensional numerical algorithm on
basis of MHD approximation are developed In the algorithm a new updating
of a grid-characteristic method of 2-3 order of approximation with
splitting on spatial variable and physical processes is developed
One-speed approximation with plasma and air components and with the
account of diffusion of a geomagnetic field in the ionized indignant
environment is used Calculations of the plasma currents formed by
powerful explosions in an ionosphere and magnetosphere E sim 10 19 -10
23 erg carried out on various latitudes varphi in a range of heights of
100--1000 km are executed It is shown that character of development of
plasma current for t ge 0 3-0 5 s depends much on all three parameters
for smaller time -- is determined mainly by initial specific energy E M
At explosions in a range of heights of 100--120 km there is a basic
change in character of current of plasma initial rigid radiation starts
to leave plasma area up to the big distance the scale of the thermal
area starts to exceed the height of a homogeneous atmosphere and
vertical movement is not formed inside it because of quite high density
of air the current is two-dimensional long time up to time about 60 s
and magnetoacoustic wave extending in the top hemisphere is weak For
heights more than 150 km and average energies the current becomes
three-dimensional for

- Jan 2006

On the basis of the detailed theoretical analysis the physical model is
developed and the opportunity of formation of jet current of plasma in
magnetosphere is appreciated as a result of development of flutter
instability at the front of a plasma cloud with high initial specific
energy It is shown that for plasma with E approx 10 21 erg and M 10 6 gr
at its scattering in equatorial area varphi le 15 0 the fastest growth
of disturbance with wave number k 6 is possible Restriction of wave
number k from above is caused by viscosity The range of explosions of
400--700 km where conditions for development flutter instability are
optimum is determined As the description of nonuniform at the front
plasma demands a detailed settlement grid numerical researches were
carried out on 2D to algorithm Initial disturbance were not set they
were generated during calculation and not focused on borders of
settlement area The quantity disturbance stayed as a result of evolution
of indignations approximately met to theoretical estimations k approx 6
It is shown that during the further evolution one ascending plasma jet
is formed Special laboratory modelling formation of jet current of
plasma in a magnetic field with use of laser plasma is executed
Correctness of the developed theoretical representations about
development of flutter instability is confirmed and the explanation of
an inclination of a plasma jet to the west from a plane of a magnetic
meridian is given

We proposed a mathematical model and estimated the parameters of adsorption of albumin-bilirubin complex to the surface of carbon pyropolymer. Design data corresponded to the results of experimental studies. Our findings indicate that modeling of this process should take into account fractal properties of the surface of carbon pyropolymer.

- Dec 2004

In this paper the numerical investigation of parameters of the plasmoid expanding into rarefied ionosphere and the geomagnetic field are performed. The range of the initial energy change is ɛ∼1012–1015 [J]ɛ∼1012–1015 [J] and of the mass change is M=(0.3–1)⋅103 [kg]M=(0.3–1)⋅103 [kg]. These ranges conform to the well-known large-scale geophysical experiments “Argus” and “Starfish”. The first stages of the development of the plasma flow are considered: the radiation-gas-dynamic, the inertial and the braking stage by the geomagnetic field. For this goal a special numerical algorithm is created.

- Dec 2004

It was developed the 3D numerical MHD model meant for the simulation of the plasma cloud expansion into nonuniform partly ionized ionosphere with magnetic field. The plasma cloud has high energy density and was formed by the strong spherical explosion. The MHD equations were written down in the divergence form of one-speed and one-temperature approach by using Eulerian coordinates. The diffusion of magnetic field and the angle between the vectors of vertical atmospheric density gradient and magnetic field was also taken into account. As the initial condition for 3D numerical simulation was used the data from the initial stage of the plasma deceleration found by the solution of 1D problem in Lagrange statement. The numerical simulations of the plasma cloud expanding into rarefied ionosphere with geomagnetic field are made up to 6 [s] in time by means of the present 3D numerical algorithm. It was found the geomagnetic field evident effect along the whole length of the plasma cloud.

- Jan 2004
- ASME/JSME 2004 Pressure Vessels and Piping Conference

The problems in the form of nonlinear partial derivative equations on graphs (nets, trees) arise in different applications. As the examples of such models we can name the circulatory and respiratory systems of the human body, the model of heavy traffic in the big cities, the model of flood water and pollution propagation in the large river systems, the model of bar structures and frames behavior under the different impacts, the model of the intensive information flows in the computer networks and others.

- Nov 2000

New unconditionally stable (in the limit of a linear boundary problem) majorizing explicit finite difference schemes for solving systems of hyperbolic and parabolic equations in many spatial dimension with complex unconnected boundaries have been developed. The construction procedure for these schemes is based on (i) analysis of their properties in the space of undefined coefficients, and (ii) using of unstructured (chaotic) grids, that require no node connectivity information at all. Similar schemes have been developed for elliptic equations as well. A numerical method for hyperbolic equations in the hierarchically branching systems is discussed. A few benchmarking examples will be presented along with the scheme applications to several problems of human physiology (such as dynamics of incompressible liquid in a network of elastic blood vessels, respiratory system, etc.) [1] Magomedov K.M., Kholodov A.S., Grid-Characteristic Numerical Methods, Nauka, Moscow, 1988 [2] Kholodov A.S., Mat. Mod. 3, 104, 1991 [3] O.M.Belotserkovskii, A.S. Kholodov, Comp. Math. Mat. Phys. 39, 1730, 1999.

- Jun 1999

Fluid modelling of an edge plasma is usually performed using a finite-difference scheme on a fixed structured grid. However, both experimental measurement and numerical simulation show the presence of front-like regions characterized by sharp variations of the main plasma parameters such as temperature and radiation power. This is caused in part (i) by strong nonlinearities in the fluid equation coefficients due to abrupt changes of various plasma reaction rates as a function of temperature and (ii) by high anisotropy of the plasma transport along and across magnetic field lines. Manual mesh adoption is usually applied to allow better resolution of the regions with sharp gradients. However, such an approach is very time-consuming and limited. To overcome this problem, we propose to use adaptive unstructured meshes constructed with a new quasi-one-dimensional adaption algorithm. This approach is fast and conservative because we use a new finite-volume scheme. The price of adaptation is high, because numerical algorithms became much more complicated. To avoid unwanted complexity, we suggest an alternative use of a grid-free method, which requires no connectivity of arbitrarily placed vertices. To benchmark the methods and codes in two dimensions, we find analytical and semi-analytical solutions of the nonlinear diffusion–radiation equation, which may have sharp fronts, unconnected boundaries and bifurcated solutions. We use these solutions to study the efficiency of the proposed numerical algorithms.

- Jan 1999

Based on an analysis of finite-difference schemes in the space of indeterminate coefficients for hyperbolic and parabolic equations, new explicit difference schemes characterized by absolute stability and majorizing properties (in the linear approximation for the Cauchy problem) are developed. The schemes can be used in computations on unstructured (chaotic) grids in complex domains with multiple boundaries and do not require a priori information on the valence of a vertex. Analogous schemes are constructed for elliptic equations. Test results are presented for the schemes.

- Nov 1997

Scrape-off layer (SOL) fluid simulations typically discretise partial differential equations by a finite difference method and apply the result on a fixed, commonly structured, grid. Neighboring mesh vertices are fixed and thus can be illustrated by fixed stencils in a local discretization scheme. Because SOL plasma flow in the promising detached regimes demonstrate sharp variation in plasma parameters (as seen in both experiment and modeling), there is a clear demand for local grid refinement and hence the transition from conventional to adaptive algorithms. The price for adaptivity is high, however, and the grids become unstructured, time-varying, and a challenge to manage. Fortunately there is a group of grid-free methods which effectively deal with a ``cloud of vertices''. Local stencils are determined on-the-fly from stability and accuracy conditions. We study applicability of one such method to SOL plasmas by solving two model problems, one with convection (mass transport) dominance and another with diffusion (conduction-radiation) dominance.

- Dec 1990

The three-dimensional dynamical problem of the oblique impact of a rigid pellet on a deformable elastoplastic barrier is solved using a hybrid grid-characteristic scheme for the numerical solution of non-stationary systems of hyperbolic equations. Thegrid-characteristic hybrid scheme is adapted for the numerical solution of multidimensional non-stationary problems in the mechanics of deformable bodies.

- Dec 1989

The numerical solution of the problem of three-dimensional supersonic inviscid thermally non-conducting gas flow past bodies is considered when mutual influence occurs. The characteristic-mesh method is used to solve the equations of gas dynamics. Computational results are given for the example of flow past a sphere.

- Dec 1987

The simultaneous problem of external supersonic flow and the elastic deformation of a solid in the flow is examined. An explicit grid-characteristic scheme, related to schemes with positive approximation, is used to solve the problem.

- Jan 1986

A method for constructing difference schemes for multidimensional quasi-linear equations of a hyperbolic type is proposed. The method is based on a successive transition from the simplest one-dimensional transport equation to a system of linear, and then quasi-linear, one-dimensional hyperbolic equations followed by multidimensional equations. The development of difference schemes for one-dimensional hyperbolic equations is described. The application of the difference schemes to one-dimensional quasi-linear hyperbolic systems, and multidimensional systems of hyperbolic equations using grid-characteristic methods is discussed. Numerical data on the aerodynamics of blunt bodies of complex geometric shapes, numerical simulation of one-dimensional and two-dimensional nonstationary problems in plasma physics, and the mechanics of a deformable body are presented.

- Nov 1985

This paper presents some results of two-dimensional calculations (using Euler variables) of conical targets, performed on the basis of the coservative variant of the grid characteristic method. The mathematical model of the physical processes occurring in the high temperature plasma includes absorption of external laser radiation, the hydrodynamic motion, electron thermal conductivity, and electronion collision relaxation. The basic system of two-dimensional non-steady-state equations for the two-temperature plasma is presented and supplemented by an equation for the mass concentration of one of the components of the two-component mixture of ideal gases.

- Dec 1984

Explicit and implicit hybrid difference schemes are considered for one-dimensional quasilinear equations of hyperbolic type and typical mesh patterns. The different ways of regularising the discontinuous numerical solutions are analysed in the context of the unusual approach, involving the introduction of linear spaces of coefficients of the difference schemes.

- Dec 1984

The mesh-characteristic method is used to study some two-dimensional dynamic problems of the mechanics of a deformable rigid body, notably, the problem of small elastic and elastic-plastic deformations of a plate or shell of finite thickness (including the case when thermal effects are present), and the problem of the high-speed impact of absolutely rigid strikers with a plate of finite thickness. The influence of the choice of coordinate system (Lagrangian, moving Euler, or a combination of these) in problems with large deformations is studied.

- Mar 1978

Two algorithms for numerical solution of the problem of the interaction
of powerful laser radiation with matter are investigated. One algorithm
is based on a Lagrangian description of the hydrodynamic processes
developing in the plasma as a result of the interaction, while the other
algorithm uses an Euler representation of the equations on which the
mathematical model is based. Results of calculations of the interaction
of radiation with a spherical target in one-dimensional and
two-dimensional approximations are presented and discussed. The effect
of anomalous heat transfer is examined.

- Feb 1976

THE RESULTS ARE GIVEN OF COMPUTER NUMERICAL CALCULATIONS OF THE FLOW AROUND SPHERICALLY BLUNTED CONES WITH LARGE SEMIANGLES AT HYPERSONIC SPEEDS WITHIN THE FRAMEWORK OF A MODEL OF THE COMPLETED GAS WITH THE ATTACK ANGLE ALPHA = 0. THE PECULIARITIES OF THE FLOW AROUND SUCH BODIES WITH A CHANGED SEMIANGLE AND ADIABATIC VALUE GAMMA ARE DESCRIBED.

- Mar 1975

This paper presents results of numerical calculations of hypersonic flow over spherically blunted cones with large vertex angles ?=40‡, using a perfect gas model at angle of attack a=0. There are notable peculiarities in the flow over such bodies, for variations in the semivertex angle and in the adiabatic index ?. Specific numerical results on investigation of such bodies have been given in [1, 2].

- Jan 1975

The method of aerodynamic derivatives [1–3] can be used for the investigation of the flow around a body executing oscillations with a small amplitude. The characteristics of the flow are expressed in the form of functions which are determined from the solution of the linearized equations of gasdynamics and describe the flow pattern with adequate accuracy. The present article is devoted to the discussion of the results of solution of the general nonstationary problem in nonlinear formulation. Supersonic flows around a hemisphere and a cylindrical front end executing arbitrary harmonic oscillations along the axis of symmetry or experiencing the corresponding oscillations of the flow (turbulent atmosphere) are discussed as examples. The effect of the nonlinearity on the flow pattern is demonstrated for different Strouhal numbers. The results are compared with those of the linear theory and with the results obtained from the solution of the corresponding stationary problems. The solution is obtained by using the characteristic method in form [4].

- Jan 1970
- Sixth International Conference on Numerical Methods in Fluid Dynamics

In a large-scale geophysical experiment "Starfish" (weight of plasma 10 6 [g], energy 6*10 22 [erg]) the formation of an ascending plasma jet was observed where the most of the mass was distributed in the plane of a magnetic meridian. This kind of stream can be formed as the result of flutter mode growth on the front of the plasma stream. We created a physical model, which shown that this growth is determined by dissipation processes, and the main of them is viscosity, it's affect is significant after the time 2 / 1 ~ k η τ , where λ πR k 2 = is a wave number. Viscosity destroys disturbances of high frequency, which grow (in ideal plasma) for the time k / 1 ~ τ . As the result, -radius, pressure, density and viscosity of plasma. Our estimations shown, that in the moment of plasma braking, m k =5÷6. Numerical modeling of such a large-scale three-dimensional stream of plasma with sated physical contents is an extremely difficult problem. In the same time, due to extremely low pressure of surrounding ionized atmosphere, which exists on the heights over 400 km (we have no reason to look at jet stream formation on the lower heights), speed of Alfven waves is about 600÷1000 [km/s] which exceeds initial plasma speed ~ 500 [km/s]. MHD-perturbation reaches the boarder of the area very fast, so we must choose in this moment – either we need to follow MHD-wave and increase the computational area very fast, or we want to work with main plasma streams, and grow the area bit slower. In this work we chose second approach, which led us to quite a small mistakes taking place due to absence of information about stream structure behind the front of MHD-perturbation, which has already run away from the area.

- Dec 1969

THERE have recently been intensive developments in various numerical methods for solving multi-dimensional problems for partial differential equations [1–9]. In particular, schemes using the characteristic equations of gas dynamics have found wide application [4–8]. Numerical methods, using the characteristics of hyperbolic equations, have definite advantages over ordinary finite-difference methods, in particular, in taking into account the physical nature of the problem, weak variation along the characteristics of some complexes of the required functions, and the possibility of predicting the instant when discontinuities will occur, for instance, suspended jumps. These advantages stand out very sharply in the solution of problems with two independent variables. But when the number of independent variables and the amount of information to be processed is increased a fixed choice of nodes becomes desirable, as in the network method. In this respect the numerical schemes of [6–8] for analyzing supersonic space flows of gas, for which a fixed network and the use of some form of characteristic relations are usual, are most effective.

- Jul 1967

We consider the problem of steady flow of an inviscid, non-heat-conducting gas about a delta wing which is spherically blunted at the nose and cylindrlcally blunded on the leading edges, at an angle of attack.
Several experimental and theoretical studies have been devoted to the investigation of this problem, of which we note [1–4], In the following the three-dimensional method of characteristics using the scheme proposed in [5] is used to calculate the flow fields about such bodies for freestream Mach numbers M=6, 7, 8, and ∞, sweep angle χ=70°, and angles of attack α from 0 to 15°.