| Russian Mathematical Surveys |
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Mathematical Life Boris Nikolaevich Chetverushkin (on his eightieth birthday) A. I. Aptekarev, S. I. Bezrodnykh, M. A. Guzev, S. I. Kabanikhin, B. S. Kashin, S. V. Kislyakov, V. V. Kozlov, N. Yu. Lukoyanov, M. B. Markov, D. O. Orlov, Yu. S. Osipov, I. B. Petrov, V. P. Platonov, I. A. Taimanov, V. F. Tishkin, D. V. Treschev, E. E. Tyrtyshnikov, M. V. Yakobovskiy  On 26 January 2024 Boris Nikolaevich Chetverushkin, the scientific supervisor of the Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, a prominent expert in applied mathematics, mathematical modelling, and parallel computations, a teacher and organizer of science, a member of the Russian Academy of Sciences, a member of the Praesidium of the Russian Academy of Sciences, a deputy secretary academician of the Division of Mathematical Sciences of the Academy of Sciences, observed his 80th birthday. He was born in Moscow. In 1960 he graduated from Moscow school no. 170 with a silver medal of distinction, and in Jume 1960 he enrolled at the Faculty of Astrophysics and Applied mathematics of the Moscow Institute of Physics and Technology. His scientific background was strongly influenced by lecture courses in mathematical analysis, analytic mechanics, the theory of boundary layer, and numerical methods read by L. D. Kudryavtsev, F. R. Gantmaсher, A. A. Dorodnitsyn, A. A. Samarskii, and other prominent researchers and professors. At the end of his third years he selected specialization at the core department of the Institute of Physics and Technology at the Institute of Applied Mathematics of the USSR Academy of Sciences. He went to Department no. 3 of the latter, headed by A. A. Samarskii, where he then became a junior researcher in 1966. His scientific advisor for the diploma thesis and then the Ph.D. thesis (1971) was V. Ya. Gol’din, one of the pioneers of Soviet numerical mathematics and an active participant of the Soviet atomic project. In 1981 Chetverushkin became a Doctor of Sciences (physical and mathematical sciences). His research style mainly shaped at about that time by accumulating the best features of the scientific school of Tychonoff and Samarskii. These include a deep multifaceted understanding of the nature of the physical, mechanical, chemical, and other processes and phenomena under study, an art of designing adequate mathematical models that can be investigated comprehensively using the methods of contemporary science, an ability to propose relevant computational models and algorithms taking account of the available hardware, a combination of the high theoretical level of research with its practical focus, and a systematic holistic view of problems to be solved. Starting with his first publications in the late 1960s, Chetverushkin research interests were connected with the numerical solution of kinetic equations and with modelling problems in continuum mechanics in which the influence of kinetic transport is significant. In the 1970s-early 1980s, in joint works with Gol’din, Chetverushkin created algorithms and software for the solution of problems in the dynamics of a radiating gas and, in particular, for the description of processes of high-temperature gas dynamics. Even when realized on hardware of the early 1980s, these methods allowed one to solve spatially two-dimensional multigroup problems (taking the dependence of the coefficients of photons on frequency into account) in radiation gas dynamics. The results obtained were generalized in Chetverushkin’s monograph Mathematical modelling of dynamical problems of radiating gas (Nauka, Moscow, 1985; in Russian). Among these methods we can mention an algorithm for averaging over the energies of photons in the case when the absorbtion coefficient has the form where $f_1>0$ and $f_2>0$, $\nu$ is the photon frequency, and $T$ and $\rho$ are the temperature and density of the gas, respectively.Even in the simplest case of a plane layer the intensity of radiation energy is described by the transfer equations where is the equilibrium radiation intensity, $h$ is the Planck constant, $k$ is the Boltzmann constant, and $c$ is the speed of light. Apart from the space variable $x$ and time $t$, the energy intensity $I(t,x,\nu,\mu)$ depends on the frequency $\nu$ of the photons and the angular variable $\mu=\cos\theta$ (where $\theta$ is the angle between the direction of motion of a photon and the $x$-axis).Note that the dependence on $\nu$ is the most resource-consuming one in terms of computational costs because the dependence of $\kappa$ on $\nu$ is complicated. The flow of radiation energy $W$ can be determined from (2): It was proposed to calculate $W$ using integrals of lower multiplicity, by introducing the variable $z=\mu f_1^{-1}(\nu)$:
$$
\begin{equation}
W=\int_{-\max f_1^{-1}(\nu)}^{\max f_1^{-1}(\nu)}zI\,dz.
\end{equation}
\tag{4}
$$
In its turn, the intensity $I(t,x,z)$ can be found from the equation of lower dimension where the source of radiation $F$ is determined in advance by
$$
\begin{equation}
F(T,|z|)=\int_{\omega_{|z|}} f_1^2(\nu)I\nu \rho\,d\nu,
\end{equation}
\tag{6}
$$
and the domain of integration is $\omega_{|z|}:=\{f_1^{-1}(\nu)\geqslant |z|\}$ (see Fig. 1).
In this way we can reduce the dimension of the transfer equation by eliminating the most complicated dependence on the frequency $\nu$. This approach can also be generalized to the case of two space dimensions. Note that defining $F(T,\rho)$ by (6) corresponds to the definition of a Lebesgue integral. The solution of systems of linear equations arising in difference approximation of parabolic and elliptic equations is an important area of applied mathematics. Chetverushkin put forward an original iterative ‘$\alpha$-$\beta$-method’ for the solution of such equation, which we present using the example of a five-point difference scheme:
$$
\begin{equation}
\begin{gathered} \, BU_{i,n-1}+KU_{i-1,n}-CU_{in}+EU_{i+1,n}+VU_{i,n+1}+F_{in}=0, \\ 1\leqslant i\leqslant N_i,\qquad 1<n\leqslant N_n. \end{gathered}
\end{equation}
\tag{7}
$$
Assume that the solution of (7) satisfies the relations
$$
\begin{equation}
\begin{alignedat}{2} U_{in}&=\alpha_{i+1,n}U_{i+1,n}+\beta_{i+1,n},&\qquad i&=1,\ldots,N_i-1, \\ U_{in}&=\gamma_{i-1,n}U_{i-1,n}+d_{i-1,n},&\qquad i&=N_i,\ldots,2; \end{alignedat}
\end{equation}
\tag{8}
$$
$$
\begin{equation}
\begin{alignedat}{2} U_{in}&=\widetilde{\alpha}_{i,n+1}U_{i,n+1}+\widetilde{\beta}_{i,n+1},&\qquad n&=1,\ldots,N_n-1, \\ U_{in}&=\widetilde{\gamma}_{i,n-1}U_{i,n-1}+\widetilde{d}_{i,n-1},&\qquad n&=N_n,\ldots,2. \end{alignedat}
\end{equation}
\tag{9}
$$
If we know the pair of coefficients $\alpha$, $\beta$, then we can find $U$ by back substitution along rows or columns. To find the eight substitution coefficients we proceed as follows. Using (9) we obtain a three-point difference scheme for $U_{i-1,n}$, $U_{in}$, and $U_{i+1,n}$, and using (8) we obtain a scheme for $U_{i,n-1}$, $U_{in}$, and $U_{i,n+1}$. These schemes can be used to derive the substitution relations along rows and columns. As a result, we obtain a system of eight equations with respect to $\alpha$, $\gamma$, $\widetilde{\alpha}$, $\widetilde{\gamma}$, $\beta$, $d$, $\widetilde{\beta}$, and $\widetilde{d}$. The nonlinear system with respect to $\alpha$, $\gamma$, $\widetilde{\alpha}$, $\widetilde{\gamma}$ can be solved separately. After that, given $\alpha$, $\gamma$, $\widetilde{\alpha}$, and $\widetilde{\gamma}$, we can solve the linear system with respect to $\beta$, $d$, $\widetilde{\beta}$, and $\widetilde{d}$. One advantage of this method is that it is robust, so that one can successfully solve problems with strong dependence of the absorbtion coefficients on the temperature and density, which is characteristic of high-temperature gas dynamics. In addition, this method suits for multiprocessor realization. Among important results of Chetverushkin’s are kinetically consistent difference schemes and a quasi-gasdynamic system of equations, which he has been developing in conjunction with his students and colleagues, among which we can name D.Sc. T. G. Elizarova. For 40 years the algorithm has successfully been used for the solution of problems in fluid and gas dynamics (including modelling of processes in a viscous heat conducting gas, and of unsteady or turbulent flows), problems in aeroacoustics, and for modelling combustion processes. The main difference of this approach from the others is that it takes explicitly into account the relationship between the kinetic and hydrodynamic descriptions of a continuous medium, which is well known in theoretical mechanics but was never used before in actual calculations. The quasi-gasdynamic system, for all its explicit differences, is actually different from the Navier–Stokes equations only by terms of the second order of smallness (in terms of the Knudsen number). It is not a coincidence that the results of computations obtained on the basis of the quasi-gasdynamic system and the Navier–Stokes equations are virtually the same. On the other hand, modelling on the basis of the quasi-gasdynamic system has a number of advantages. First of all, the quasi-gasdynamic system is well posed, which is explained on the physical level by the fact that in its construction we take account of the smoothing of the solution at lengths of the order of the mean free path $l$ of molecules or the time $\tau$ between collisions. In fact, the dissipative terms in the quasi-gasdynamic system are regularizers constructed on the basis of natural physical constraints. By analogy with quasi-gasdynamic systems, a guiding principle was stated that in problems of continuum mechanics there exist natural spatial and temporal limits such that going below these limits is meaningless. The Navier–Stokes equations hold for Knudsen numbers $ \operatorname{Kn}$ less than $10^{-3}$. For greater $\operatorname{Kn}$ more complex kinetic models must be used. Using the quasi-gasdynamic system we can give a macroscopic description for $\operatorname{Kn} \leqslant 0.1$. This opens additional opportunities for the description of moderately rarefied gases, capillary flows, flows in porous media, for instance, in the numerical simulation of core samples. In contrast to the Navier–Stokes equations the quasi-gasdynamic system is hyperbolic: it contains terms with small coefficients multiplying the second time derivatives of gas-dynamic parameters. Thus more stable explicit schemes can be designed, and additional opportunities arise in modelling on high-performance systems with extramassive parallelism. As a generalization of this trick, Chetverushkin stated the hyperbolization principle, when to a parabolic equation one adds a term with a small coefficient multiplying a second time derivative. As a result, one can design fairly stable explicit schemes which are perfectly adapted to the architecture of systems with extramassive parallelism. The results of these investigations were generalized in Chenverushkin’s monograph Kinetic schemes and quasi-gasdynamic system of equations (MAKS Press, Moscow, 2004; in Russian). It looks interesting to extend this approach to the description of magnetohydrodynamic processes. If for the description of the distribution of particles in a magnetic field $\overline{B}$ we use the complex-values distribution function
$$
\begin{equation}
f=\frac{\rho(f,\vec{x}\,)}{(2\pi RT(t,\vec{x}\,))^{3/2}} \exp\biggl\{\frac{-\bigl(\vec{\xi}-\vec{u}-{\rm i}\vec{B}/{\sqrt{4\pi\rho}}\,\bigr)^2}{2RT}\biggr\}
\end{equation}
\tag{10}
$$
(where $\rho$ is the density, $T$ the temperature, $\vec{u}(t,\vec{x}\,)$ the macroscopic velocity of the system $R$ is the gas constant, $\vec{\xi}$ the velocity of a molecule, and ${\rm i}$ is the imaginary unit), then we can derive the equations of magnetohydrodynamics without using Maxwell’s equations.
The magnetohydrodynamic analogue of the quasi-gasdynamic system was used to model the spatially three-dimensional problems of the absorbtion of matter by a black hole and the formation of an astrophysical jet, and also the problem of the melt flow in the coolant system of a nuclear reactor. In the astrophysical problem computations were performed on a spatial grid of $8\cdot 10^9$ knots. It should be noted that using coarser grids with fewer than $5\cdot 10^7$ nodes, we cannot describe the formation of a jet. An important line of Chetverushkin’s research is the development of methods of parallel calculations and their applications to problems in science and technology. Starting with the early 1990s, together with Elizarova, in the framework of a project led by the Nobel Prize winner I. R. Prigogine, they managed to realize an explicit scheme based on quasi-gasdynamic systems for the solution of current problems in fluid and gas dynamics, which is nicely scalable to many processors. A brilliant example of Chetverushkin’s activity as an organizer of science was putting into operation the first Russian heterogeneous computing cluster K-100, carried out at the Keldysh Institute of Applied Mathematics of RAS in cooperation with the Research Institute “Kvant”. In this cluster graphics cards are used as accelerators. Such systems are currently dominating among computing systems of high and superhigh performance. On the other hand, for the successful use of heterogeneous systems algorithms must combine the (difficult to correlate) properties of logical simplicity and efficiency. Chenverushkin and his scientific school concentrate currently on the development of such methods of computation. In recent years Chetverushkin was actively involved in the problems of artificial intelligence and big data processing, by developing the trend for hybrid artificial intelligence. A characteristic feature of this development is combining methods of neural data processing and traditional mathematical models. These and some subsequent results were strongly based on ideas due to Chetverushkin and his students, who proposed new approaches to the creation of discrete models and algorithms adapted to the capabilities of the modern computer technologies. By implementing these approaches Chenverushkin’s scientific school rose to leading positions in Russian applied mathematics. This school includes currently a number of well-known researchers: two corresponding members of the Russian Academy of Sciences, a dozen of doctors of science, more than 30 Ph.D.’s, and a large group of talented youths. Their results have been reflected in hundreds of publications, including a few monographs. In 2000 Chetverushkin became a corresponding member of the Russian Academy of Sciences, and in 2011 he was elected a full member of the academy. In 1986, by decision of the Central Committee of the Communist Party and the Council of Ministers of the Soviet Union, on the basis of Department no. 3 of the Keldysh Institute of Applied Mathematics, the All-Union Center for Mathematical Modelling was organized under the auspices of the Keldysh Institute (which was transformed in 1990 into the independent Institute for Mathematical Modelling of the Academy of Sciences). Samarskii became the organizing director of this Center and Chetverushkin was a deputy director. From 1998 to 2008 Chetverushkin was a director of the Institute for Mathematical Modelling. In 2008, in the course of consolidation of research institutes, the staff of the Institute for Mathematical Modelling was transferred to the Keldysh Institute, and Chetverushkin became the director of the latter. Since 2016 he is its scientific supervisor. In 2013 he was elected to the Praesiduim of the Russian Academy of Sciences. Chetverushkin’s management style cannot be separated from his research style and features a thorough analysis of the tasks facing the institute, the selection of a strategic goal, the development of ways towards this goal, an active and, usually, successful search for sources of funding, taking rapid and balanced decisions concerning the interaction between departments of the institute and external, including foreign, organizations, and a careful selection of personnel. Concentrated on the practical realization of the results of research as he is, Chetverushkin is also concerned about their relationship with fundamental science and about promotion of the institute’s achievements. He is the editor-in-chief of the journal Matematicheskoe Modelirovanie1, the organiser and lead contributor to many respected academic forums, the head of departments of the Moscow Institute of Physics and Technology and the Faculty of Computational Mathematics and Cybernetic at Lomonosov Moscow State University. He sees has main task in maintaining the high level of Russian applied mathematics, and first of all in the area of massive computations and large-scale numerical experiments with the use of state-of-the arts supercomputers, one of which works in the Keldysh Institute. Academician Chetverushkin is a Honoured Scientist of the Russian Federation, he was awarded the A. N. Krylov Prize of the Russian Academy of Siences (2001), the Order of Friendship (2016), and the M. V. Keldysh Gold Medal of the Academy of Sciences (2021). In 2023 he was awarded the Demidov Prize and in 2024 the Order for Services to the Fatherland of the 4th degree. Chetverushkin’s colleagues are well aware of his love of the Russian culture and history, his respect and courtesy to people, frankness, caring nature, and readiness to help. We congratulate Boris Nikolaevich on his birthday and wish him good health, long creative life, and new successes for the good of Russian science. 
Citation in format AMSBIB
\Bibitem{AptBezGuz24} |
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