## Participation of Academician A.A. Samarskii in the Atomic Project

Alexander Andreyevich Samarskii considered his contribution to the creation of atomic and thermonuclear bomb as one of his main scientific results. Below, there is a report of Academician A.A. Samarskii presented at the international conference held in the town of Dubna of the Moscow region on May, 14-18, 1996, and devoted to the history of Russian atomic project. The report is followed by a chapter on the same topic from the book about Academician A.N. Tikhonov.

### Direct calculation of explosive power

**A.A. Samarskii**

**Science and society: history of Soviet atomic project (40-50s). Proceedings of international conference ISAP-96.**

The report describes the history of the development of numerical methods and their application to the calculation of complete models of atomic and thermonuclear explosions as well as the history of the scientific group created for this purpose by A.N. Tikhonov in 1948, who for the first time in the world made such calculations.

There is one side of the problem under discussion, about which they talked and still talk little – the mathematical support of the nuclear program.

The explosion of a nuclear bomb is the simultaneous occurrence of many interrelated processes: fission of nuclear fuel by neutrons, spread of resulting neutrons, release of energy and its transfer through the material, gas-dynamic expansion of highly heated matter. All of these processes are described by a system of nonlinear partial differential equations. Neither physicists nor mathematicians could solve such problems in 1947-48.

In 1947, the work of creation of the Soviet atomic bomb came to an end. There was a question about the theoretical prediction of the power of the explosion. This problem was discussed at the seminar of I.V. Kurchatov in the beginning of 1948. By this time, simplified models of the atomic bomb had been proposed, described by a system of ordinary differential equations for the average values.

A.N. Tikhonov, who was attending the seminar, proposed to use the method of finite differences for direct numerical calculation of the explosion on the basis of complete models of physical processes (propagation of neutrons and heat, nuclear burning and gas dynamics), described by a system of nonlinear differential equations using their representation in the Lagrangian coordinates.

At that time, neither theory nor experience of practical application of the difference schemes for complex problems in mathematical physics actually existed. Therefore, this statement was a surprise to physicists and caused a remark of L.D. Landau, that such calculation would be a scientific feat.

On the initiative of I.V. Kurchatov to carry out the computational work in order to study the process of the nuclear explosion, a resolution of the Council of Ministers was issued about the creation of a special laboratory under the direction of A.N. Tikhonov at the Geophysical Institute of the Academy of Sciences. At this time, A.N. Tikhonov was the head of the Chair of Mathematics at the Physics Faculty of Moscow State University and the head of the mathematics department of the Geophysical Institute of the Academy of Sciences of the USSR.

Selection of personnel for the laboratory was a difficult problem. The work that had to be made required joint efforts of physicists, mathematicians, specialists in differential equations and calculations. There were actually no qualified specialists in numerical methods. Therefore, these specialists were supposed to be prepared in the process of solving of the real problems.

It was about 6 years before the advent of computers. Only the calculators "Felix" and later keyboard calculators "Mercedes" could be used for the calculations.

In the spring of 1948, I graduated from the post-graduate school at the Physics Department of Moscow State University and defended candidate thesis. We, together with A.N. Tikhonov, in 1945-1948 performed a series of computational and theoretical work on the theory of a gas mask (problems of the dynamics of adsorption and desorption of gas mixtures) and the theory of excitation of radio waveguides. It gave me some experience of calculations. V.Ya. Goldin graduated in the same year from the same Faculty of Physics of MSU, who, under the guidance of A.N. Tikhonov, studied the neutron transport equation.

In addition, a graduate student of Mechanical and Mathematical Faculty of MSU N.N. Yanenko, who was specialized in geometry, was enrolled into the laboratory, and two years later - another student of Andrey Nikolaevich, a graduate of the Faculty of Physics of MSU, B.L. Rozhdestvenskii (since 1951). In the very early period, Professor E.S. Kuznetsov took part in the discussion stage of the analytical methods for the solution of the neutron transport.

A group of calculation specialists was created to perform calculations. Most of them were the graduates of Geodesic Institute and also Moscow State University.

An important role in the training of the calculation specialists was played by candidate of physical-mathematical sciences O.P. Kremer, who had experience in the numerical solution of ordinary differential equations in the group of Academician Fesenkov during the processing of the materials of astrophysical observations. In the future, O.P. Kremer excelled as one of the first programmers in our country working on the first computers.

Our laboratory was organized at the Integrated Geophysical Expedition of the Geophysical Institute of the USSR and was placed first on Pjatnitskaja Street, then at the Kirov Street, and finally, from 1952 at the Miuss square, in a former building of Physical Institute of Academy of Sciences.

The development of numerical methods for the complete system of partial differential equations describing the nuclear explosion was entrusted to me; V.Ya. Goldin and O.P. Kremer were entrusted to make the calculations based on the tasks from L.D. Landau, E.M. Lifshitz and I.M Khalatnikov, who built a model of the atomic explosion in the form of a system of ordinary differential equations for the characteristic values averaged over the space.

V.Ya. Goldin was also entrusted to the verification of the ODE system. To do this, he built a complete system of equations of the explosion in partial derivatives and the integro-differential equation of neutron transport and used it to write the ODE system using the approximations specified by I.M. Khalatnikov.

Theoretical and physical aspects of the calculations, including the formulation of the Cauchy problem for a system of ordinary differential equations and analysis of singular points, were made by V.Ya. Goldin together with N.N. Yanenko. All these calculations were extremely urgent and their results were used immediately. A.A. Samarskii was engaged in the development and conduction of a direct calculation of the nuclear explosion.

The calculation of the nuclear explosion required joint solution of the kinetic equation for the neutron transport, the gas dynamics equations with thermal conductivity. In 1948, for the non-stationary transport equation in spherical coordinates, I proposed and tested a monotone difference scheme, and early in 1949 I made the first calculation of the complete system of equations of the explosion: first, of a ball of plutonium, and then of a shell of uranium. Furthermore, an averaging of the partial differential equations in space and angles was made using the profiles from the complete calculation of these equations, leading to a system of ordinary differential equations.

Thus, in less than a year, a group of 3 scientific researchers and a group of calculators with the adding machines managed "from scratch" to build the methods, to establish calculations and to get the first production results.

More complex calculations were made in 1949-1950. We used the differential-finite-difference approximations (finite-difference in space and differential in time), which were solved using iteratively-difference methods.

To speed up the computations, a method for parallel computing was developed, involving 30-40 calculators. This method allowed us to perform calculations on electric adding machines "Mercedes" in a short time, that it was extremely important at the time.

The processing of the calculations obtained in 1949-1950 allowed me in 1950 to formulate the general principle of conservatism, i.e. the conservation laws at the discrete level for the difference schemes. The experience of two years of computational work has shown that it is necessary to pay attention to the development of theoretical studies. The idea of conservatism of homogeneous difference schemes was further studied in detail in the works of A.N. Tikhonov and A.A. Samarskii, who found the necessary conditions of conservatism of the difference schemes for the studied classes of differential equations.

In 1950 we were approached by I.E. Tamm with a proposal to calculate a more complex design. Initially, we did not know the physical and technical ideas in this design. Then A.D. Sakharov and Yu.A. Romanov contacted us. It was about the creation of the hydrogen bomb. It required complication and further development of the methods. The first calculation was conducted in 1951. An important role was played by A.A. Samarskii who made the transition to conservative difference schemes. For the equations of nonlinear heat conduction and diffusion of neutrons, we used implicit schemes using iterative methods for the determination of solution at the new layer, and we used explicit schemes for gas dynamics.

This work required concentration of our forces to obtain the finite-difference equations, preparation of detailed tasks, organization of calculations, refinement of the physical characteristics. This work was made with joint participation of A.N. Tikhonov, A.A. Samarskii, V.Ya. Goldin, N.N. Yanenko and B.L. Rozhdestvenskii. Groups of calculators were created to make the calculations. We can recall a number of outstanding calculators, who felt the solution without knowing its meaning (A.A. Trofimova, V.N. Ravinskaya, M.I. Volchinskaya, V.A. Lokhina, V.K. Kameron and many others). The calculations were led by A.A. Samarskii, V.Ya. Goldin, B.L. Rozhdestvenskii.

To refine the physical aspects of the problem, an important role was played by the seminar of I.E. Tamm. That’s where V.Ya. Goldin learned about the work of Feynman, Teller and Metropolis, which led him to formulate and carry out calculations of the equation of state and the creation of interpolation formulas. As a result, the ideal gas equation of state has been replaced by a more accurate model. When analyzing these equations, N.N. Yanenko constructed the asymptotic behavior, which helped to clarify the interpolation formulae. An important role was played by the evaluation of the effect of mixing (Rayleigh-Taylor instability), the model for which was proposed by S.Z. Belenkii, an employee of I.E. Tamm.

I.E. Tamm has promoted the research of the compression process at the boundary of the active and passive areas. I found the appropriate self-similar solutions and used it as a test in the calculation of the contact discontinuities.

As a result, in 1950-51 we have developed numerical methods, in 1951 we made the first calculation of the "layer cake" of A.D. Sakharov and made a report. Then in 1951-53 we calculated a range of options of the "layer cake". These calculations helped physicists to see clearly all the processes in the explosion and choose the final design. The results of successful tests in 1953 confirmed the ideas of the physicists laid into the design, and showed that our models and calculations carried out before the advent of computers, corresponded to physics with a good accuracy.

This work has been praised. A.N. Tikhonov was awarded the title of Hero of Socialist Labor, he was awarded the State Prize of the 1st degree. State prizes and medals were also awarded to A. Samarskii, V.Ya. Goldin, N.N. Yanenko and B.L. Rozhdestvenskii. A group of our employees has been awarded orders and medals.

In 1953, for the mathematical support of the nuclear and space programs, a special Institute was created - Department of Applied Mathematics - OPM (subsequently renamed the Institute of Applied Mathematics) by combining the groups of M.V. Keldysh from the Mathematical Institute of Academy of Sciences and the laboratory of A.N. Tikhonov of GEOFIAN. M.V. Keldysh was appointed the director of the institute, and A.N. Tikhonov - the deputy director. The laboratory of A.N. Tikhonov was transformed into the 3rd division of the institute. I was entrusted to head the department. In 1949-1953, a significant replenishment came in our department. Among them: from MSU I.M. Sobol’, S.P. Kurdjumov, V.B. Uvarov. Particularly large replenishment came in 1953 from a number of universities in the country (from Gorky, Saratov, Tomsk, and others.) Among them, P.P. Volosevych, N.N. Anuchina, N.N. Kuchumov, D.A. Sidorova, G.V. Danilova et al.

In 1954, the first production computer "Strela" began working in OPM, headed by A.N. Myamlin. Transition to a computer made us to upgrade the procedure, to make it homogeneous without explicit separation of the boundaries and singularities.

The transition of the calculations to the "Strela" computer allowed us to obtain the results significantly faster. This was particularly important in connection with the development of new products. In this work, we had to work closely with A.D. Sakharov, Yu.A. Romanov, Ya.B. Zel'dovich, K.I. Shchelkin, Y.N. Babaev, G.A. Goncharov, Yu.A. Trutnev, V.M. Zagrafov, L.P. Feoktistov, E.I. Zababakhin, E.N. Avrorin and other theorists.

The transition to the calculation of the "layer cakes" demanded the development of numerical methods for solving systems of differential equations in heterogeneous media with coefficients that vary by hundreds of times during the transition from one area to another.

Because of nonlinearity of the processes and heterogeneity of the environment it was necessary to build such difference approximations, which would allow to consider and pass with sufficient accuracy through breaks (contact, weak and strong shock waves) of the solutions. Particular difficulties had to be overcome in the calculation of temperature waves, which largely affected the behavior of the process.

Moreover, various processes in the kinetics during combustion of the light layer of the "layer cake", gas dynamics, heat transfer and diffusion of neutrons had different timescales.

The ideas of conservatism and homogeneity of difference schemes (A.N. Tikhonov, A.A. Samarskii) allowed obtaining the pass-through schemes without explicit separation of discontinuities, providing not only the end result of the energy release with sufficient accuracy, but also a correct description of the dynamics of the combustion process.

In the discussion that took place in those years, I created an example showing that the refusal to comply with the laws of conservation of difference schemes can lead to complete loss of accuracy in the case of a heterogeneous medium with discontinuous coefficients (due to the appearance of the fictitious approximative sources of uncontrolled power). Thus, the necessity of conservatism for the difference approximations was verified.

Dropping of explicit consideration of discontinuities using homogeneous difference schemes and compliance with the laws of conservation at the discrete level provided sufficient accuracy of calculations of the products.

The need to develop theoretical studies at a level sufficient for the class of the solved tasks was understood in a timely manner. The practical needs stimulated the development of theoretical work, especially in our team.

As a result, were built the foundations of modern theory of difference schemes for a broad class of stationary and non-stationary equations of mathematical physics. I point out the following sections of this theory: the theory of the stability of difference schemes, including the theory of iterative methods for solving finite-difference equations, the general theory of regularization of difference schemes in order to obtain a given quality schemes and its application to the solution of the inverse (or incorrect) tasks, the new principles of the approximation of multidimensional problems (such as the method of full or weak approximation).

The main theoretical results of our work were reflected in numerous publications and production of a number of books, many of which have been translated into foreign languages. It is important to note that the methodological basis of the book of A.N. Tikhonov and A.A. Samarskii "Partial Differential Equations" (1951) were used and developed further in the books (over 20 books) on numerical methods by A.A. Samarskii, his students and colleagues.

Along with the development of a common methodology, much attention was paid to the refinement of the environment model. The model of the absorption coefficient was refined and the absorption coefficient were calculated. Initially, these calculations were carried out by N.N. Yanenko using the tasks from E.S. Fradkin, and then V.B. Uvarov and A.F. Nikiforov closely with Y.N. Babaev had developed the new techniques with an accurate account of the quantum theory. The technique of neutron calculations was refined and developed, as it will be discussed in the report of V.Ya. Goldin.

All aspects of the mathematical calculations of the thermonuclear products were collected and developed in OPM. An important role in the development of techniques for solving problems was played by regular discussion at the seminars of M.V. Keldysh.

This work extremely stimulated the theoretical comprehension, which we used as the basis for the development of our efficient numerical methods for the solution of complex nonlinear problems.

An important role in the urgent calculations for the new products was played by the programming of "Strela" computer. I should note a large contribution of staff from the Department of Programming of I.B. Zadykhailo, E.Z. Lyubimskii et al., and the members of our department, mentioned above. As a result of these works, the process of the explosion of new product was designed in detail and all the main characteristics were identified. The results of the tests carried out in the autumn of 1955 were in good agreement with our calculations.

In 1956 a new nuclear facility was created - VNIIP (now VNIITF). N.N. Yanenko, our staff member, was recommended to create a mathematical sector in the new institute. He created a large group of graduates from several universities. These new staff members have almost a one-year internship at the Institute of Applied Mathematics, including our department. The result was a team that mastered the techniques and programs created in the departments of IAM. This allowed to create an effective math center in a short period of time. It played an important role in the success of VNIIP. This "matrix" method of creating of research teams in the new directions is extremely effective.

At the same time, all our methods, tasks and programs have been transferred to VNIIEF as well. Thus, over the years the basic calculations of explosions at VNIIEF and VNIIP were conducted using to our methods and programs, and in the future using their modifications.

Some members of the mathematical sector of VNIIEF also went through our school. On the other hand, we had a very useful collaboration with the physicists from VNIIEF and VNIITF.

The main part of the staff of our department received a physical education (Faculty of Physics, MIPT, Moscow Engineering Physics Institute), which had a beneficial impact on our work. Knowledge of physics allowed to independently formulate and solve many problems and significantly affected the approach to the numerical methods, which led ultimately to the creation of concepts and methods of mathematical modeling.

When our first computer was put into operation in 1954, the calculations started to be transferred into it. In 1954, we debugged the production programs. Despite the modest computer settings (memory of 1024 numbers and speed of 2000 operations per second), the grids could be taken already considerably larger than in the manual calculations.

Our finite-difference methods were so good that the mathematical error in the calculation had become less than the uncertainty in the data on the equations of state, paths of photon and neutron constants.

So we had to do a refinement of the properties of matter. The calculations of the paths of photons based on the quantum theory of radiation, taking into account all relevant processes, including absorption in lines, were assigned to V.B. Uvarov and A.F. Nikiforov (1956). The original physical formulation of the problem was given by Yu.N. Babaev and E.S. Fradkin. But pretty soon the problem began to refine significantly (A.F. Nikiforov, V.V. Uvarov, V.V. Novikov, N.Yu. Orlov). Gradually, this work has grown into an independent scientific discipline, which with some shift in time began to develop at VNIIEF and VNIITF. Despite this, for many years the work of our group did not stop. Recently, N.Yu. Orlov has achieved high accuracy not only for the Rosseland average, but for the spectral curves as well.

The calculation of the equation of state has been entrusted to a graduate of the Physics Faculty of Moscow State University, N.N. Kalitkin (1958). It started with the refinement of the data of Feynman, Metropolis and Teller on the basis of the adjustments of D.A. Kirzhnits to the Thomas-Fermi model. This work has also grown into a separate research area that connects the theory of gases and plasma with condensed matter. It was possible to establish a link between the optical and thermodynamic properties of hot substances because they both have been caused by the fluctuating microscopic electric field in the plasma. These ideas are allowed to refine the calculations of photon paths.

The need to calculate the generators of superstrong magnetic fields and currents forced N.N. Kalitkin to construct a theory of the conductivity of strongly non-ideal plasma (1966). This theory was ahead of the experiments for a few years and the experimental results were well predicted by it.

All these works were brought to the complex systems of programs, which were used to calculate detailed tables of various properties of substances: the photon path, equations of state, transport coefficients. The first tables yet in the late 50s were included in the programs for calculating the power of the explosion on the basis of the equations of gas dynamics with heat conduction and neutron transport of nuclear and thermonuclear reactions. In subsequent years, the data were refined by taking into account an increasing number of physical effects, which led to a substantial increase in the authenticity of the results.

For many years there has been an intense work on both the mathematical and physical aspects of the calculation of the explosion. The results were tested hard in the experiments, and the responsibility for the quality was high. This led us to form the foundations of modern mathematical modeling and computing experiment, which became the basis of our work in all subsequent years.

Theoretical and algorithmic studies were later applied to many other applications of science and technology. It is enough to mention many years of work (in collaboration with a team of N.G. Basov) in laser fusion, which allowed us at the first stage (early 70s) to be ahead of our colleagues from the United States.

The works associated with the creation of atomic and thermonuclear weapons led to an enormous acceleration of the development not only of many sections of engineering, physics, chemistry, but also to the restructuring of mathematical sciences in connection with the advent of computers and computational methods. The leading role in the knowledge is now played by mathematical modeling with the technology of computational experiment. Its kernel is the triad of "model-algorithm-program".

### Participation in the works on atomic project

**Andrey Nikolayevich Tikhonov. Series "Remarkable scientists of the Physical Faculty of MSU".****Issue VIII. Moscow, Physical Faculty of MSU, 2004.**

The solution of the atomic problem in the Soviet Union took place in a very difficult war and post-war period, with partially destroyed industry, generally about 20 years after science became organized in the new post-revolutionary society. For targeted execution of works, which were carried out with tremendous stress, I.V. Kurchatov created a well-organized structure uniting the scientists from different disciplines (physicists, chemists, mathematicians, geologists), engineers and technicians, and a huge number of other professionals.

One of the aspects of the problem, which is still not enough known, is mathematical support of the nuclear program. All work on the atomic project was carried out in an atmosphere of highest secrecy and were classified as "top secret" or "special folder". The participating people did not know neither the general scope of the work, nor the number of people working on related topics. Moreover, mathematicians then were not allowed to know what the problem solved by them relates to. Therefore, scientific reports directly about the results of the work are absent, and only some publications appeared in recent years.

The proposed text relating to the participation of the team of mathematicians led by A.N. Tikhonov in the work on the atomic problem, is based on the materials from a Senior Researcher at Institute of Mathematical Modeling of the Russian Academy of Sciences, Prof. V.Ya. Goldin. Vladimir Yakovlevich graduated from the nuclear branch of the Physics Department of Moscow State University. At the same time he worked at the Department of Mathematics, in particular, he conducted calculations, in which he used the numerical techniques. His diploma work was devoted to the methods of solution of the neutron transport equation, and the leaders were: in physics – E.L. Feinberg, and in math – A.N. Tikhonov. After the graduation in 1948, Andrei Nikolayevich invited him to work in the new team. Vladimir Yakovlevich is a witness and an active participant in the history of the development of the national nuclear project.

The explosion of a nuclear bomb is a simultaneous occurrence of many interrelated processes: fission of nuclear fuel by neutrons, propagation of neutrons produced at the same time, the release of energy and its transfer through the material, gas-dynamic expansion of highly heated matter. All these processes are more or less accurately can be described by a system of nonlinear differential equations in partial derivatives. Neither physics nor mathematics could solve such problems in 1947-1948.

In 1947, the work to create the Soviet atomic bomb came to an end. There was a question about the theoretical prediction of the explosion power. In the beginning of 1948, this problem was discussed at the seminar of I.V. Kurchatov. There was a discussion of the results of the work carried out at the Theoretical Department of the Institute of Physical Problems of Academy of Sciences of USSR under the direction of L.D. Landau by E.M.Livshits and I.M. Khalatnikov. Initially, a simple model was proposed describing the atomic explosion of a "naked ball", which was reduced to a system of ordinary differential equations for the average values over the space. It was a system of nonlinear equations, which solution was carried out from a singular point - from minus infinity, because the initial data could not be set.

A.N. Tikhonov, who attended the seminar, expressed the idea that such a problem can be solved straight by direct methods, it is possible to carry out numerical calculations of the system of partial differential equations by finite difference method in the Lagrangian variables. It should be noted that now the use of finite difference methods to solve the most difficult problems is not surprising, it is natural. But at that time, neither theory nor practical experience in the application of finite difference schemes for complex problems in mathematical physics did actually exist. Therefore, the proposal of Andrey Nikolayevich caused a remark of Lev Davidovich Landau that if this is done, it will be a scientific feat. In response to Igor Vasilyevich Kurchatov, Andrey agreed to perform computing operations with the aim of studying the process of a nuclear explosion.

On the initiative of the I.V. Kurchatov, on June 10, 1948 a Resolution of the Council of Ministers of the USSR №1990-774 SS/OP was issued to create a special laboratory No. 8 at the Integrated Geophysical Expedition of the Geophysical Institute of the USSR under the leadership of Corresponding Member of Academy of Sciences A.N. Tikhonov. Andrey Nikolaevich had serious tasks of both scientific and organizational nature.

The problem of recruitment of employees was very urgent. During the war, many of the younger generation of scientists were killed, many scientific schools were destroyed. The group was established in a short time, with the basis from the students and post-graduates of Andrey Nikolayevich. The senior staff in the new team were: Alexander Andreyevich Samarskii, who graduated from the post-graduate school at under the leadership of Andrey Nikolayevich and finished his Ph.D. thesis in 1948, Vladimir Yakovlevich Goldin, who had just defended his diploma at the Chair of Mathematics, Nicholay Nikolaevich Yanenko, who received his Ph.D. in 1948 at the Faculty of Mechanics on differential geometry at Prof. P.K. Rashevskii, and later, in 1951, Boris Leonidovich Rozhdestvenckii, a graduate of the Chair of Mathematics of the Faculty of Physics. In addition, Andrey Nikolayevich invited an experienced candidate of physical-mathematical sciences Olga Pavlovna Kramer, who had the experience in the numerical solution of ordinary differential equations in the processing of the materials of astrophysical observations in the group of Academician Fesenkov. Andrei Nikolayevich understood that the mathematical calculations on paper are not enough for this work, it will require a lot of calculations. He recruited several graduates of the Mechanical Faculty especially for the calculations, but mostly he recruited the graduates of the Moscow Institute of Geodesy, Aerial Photography and Mapping, which were actually prepared for the calculations.

V.Ya. Goldin subsequently emotionally recalled: "Andrei Nikolayevich, of course, knew what a big deed he starts to do, but neither Alexander Andreyevich, neither I, nor, especially, others had any idea about what we undertook. And if we knew, possibly we did not dare to do it. Fortunately, we had no idea about the complexity of the problem."

In the beginning, it was necessary to deal with the system of equations describing the model of the atomic explosion. Andrei Nikolayevich connected V.Ya.Goldin with the colleagues of L.D. Landau. After the discussion with I.M. Khalatnikov and E.M. Lifshitz, V.Ya. Goldin built a complete system of equations of the explosion - partial differential equations together with the equations of neutron transport - and brought out from it a system of ordinary differential equations, using the approximations used by I.M. Khalatnikov. The obtained result has been accounted for in the task for calculations sent from the Institute of Physical Problems.

Based on this system of equations, Andrey Nikolayevich and Alexander Andreyevich started the main work to create a partial difference method to solve this system of equations. It was quite unusual, as at the time no numerical methods for the solution of such complex systems of equations existed. In the 1920s, the famous paper by Courant, Friedrichs and Lewy was published, which have proved the convergence of difference schemes for solving differential equations. But there was no any practical development of the ideas that it contained. It was necessary not only to develop numerical methods for calculating the total system of equations in Lagrange variables, to build an efficient algorithm for the calculation of the difference problem, but also to take care about its realization using the available computing resources. We note that it was 6 years prior to the usage of the first computers.

In the autumn of 1948, the laboratory No. 8 settled on Kirov street, in the courtyard of the building designed by architect Bazhenov (the building of the former Senior art technical workshops, and then the Mechanical Institute), in an unnoticeable house with the label "Small wholesale vegetable base." The laboratory was given a separate premise opposite to the entrance to the base. It contained 5 or 6 rooms, there was a large hall where 30-40 people worked on the trophy electromechanical adding machines "Mercedes". Externally, these machines looked like typewriters, performing arithmetic operations accompanied by the clanging trolleys. All work was carried out in an atmosphere of top secrecy, female janitors carefully checked passes at the door. Once Andrei Nikolayevich saw as a cat went into the room and asked; "And do you have a pass?" The cat immediately jumped out through the window leaf.

As already mentioned, the main task was the solution of ordinary differential equations. This work was related to the fact that it was required to obtain an interpolation formula for the energy release as a function of parameters. It was necessary to calculate a lot of options to get this interpolation formula. The calculations that were made on the tasks of the Institute for Physical Problems were very urgent.

A.N. Tikhonov and A.A.Samarskii were working to develop the methods for the calculation of the complete system of partial differential equations in the Lagrange variables. V.Ya. Goldin, O.P. Kramer and N.N.Yanenko were primarily engaged in the support of these calculations. Alexander Andreyevich came up with the method of parallelization, which significantly increased the speed of calculations. The problem was solved by 10 or 15 calculators at once, who calculated some individual piece each, and the data was exchanged via the audio signals – the person who calculated his part, shouted the result to the neighbor. Thus it was what is now called multiprocessing. This allowed to create the methods of calculation and calculate the tasks in a very short time. The work started in the late summer of 1948, and within less than a year, a group of three researchers and calculators by starting the work "from scratch" managed to build the methods, to establish calculations and to get the first production results. In 1949, we made the first calculation of the complete system of equations of the explosion, first of a ball of plutonium, and then of a shell of uranium.

At the next stage of the work, we received from Landau a more difficult task: the system described the case of a sphere with a shell. This system was much more complex, it was necessary to bring it to a form suitable for solution with then-existing methods, and it took a month of work. In addition, without proper physical data about the values of the coefficients in the system of equations, these calculations would be meaningless. The first calculations and the expansion of this work on partial differential equations just used in a fully ionized ideal gas as the equation of state, and the results of astrophysical research were taken for the absorption coefficient. According to V.Ya.Goldin, these calculations helped in the preparation of the interpolation formula and were used to estimate then forthcoming atomic explosion.

The explosion of an atomic bomb for the first time in the USSR occurred on August 29, 1949, on a specially constructed and equipped range, 170 km west of Semipalatinsk. As reported then, the calculation of the energy very well coincided with the experimentally observed value. For this work, Andrey Nikolayevich was awarded the Order of Red Banner of Labor, and the employees got big prizes.

A new phase of work began in 1950: on February 26, 1950, the Resolution of the Council of Ministers was issued about the connection of the team with the work on the hydrogen bomb. The center of this work was located in the KB-11 near the town of Arzamas (Sarov). The work was preceded by a fairly long conversation of Igor Evgeniyevich Tamm with Andrey Nikolayevich. Then, we established a working contact with the staff of I.E. Tamm and A.D. Sakharov and Yu.A. Romanov who began immediate work. Initially, there was just the task for to the calculations, but from its contents it could be assumed that we were talking about a thermonuclear explosion. It was directly explained after a while.

The equations of gas dynamics with radiative thermal conductivity, the birth of neutrons and their transport due to fission and fusion reactions were the basis of the considered mathematical model. An important role in the mathematical models was played by the values of the physical characteristics of the process: the equations of state, the absorption coefficient of light. For the calculation of a new design, we had to transform greatly the previously developed methods.

The department did not have the computer equipment, which was at that time established in the United States under the leadership of Neumann. That made the problem of development of efficient and stable algorithms very urgent. At this time, many ideas on the theory of difference schemes appeared, which were later presented in the works of A.N. Tikhonov and A.A. Samarskii. A closed seminar was held regularly in the department, where we discussed the new ideas, work progress, encountered obstacles and the obtained results.

So, in 1950, A.A. Samarskii formulated the general principle of conservatism, i.e. realization of the same conservation laws at a discrete level for the difference schemes as for the original differential problem. As a result, he suggested the use of what is now called conservative difference schemes. Then Andrei Nikolayevich and Alexander Andreyevich justified the use of this principle in the calculation of homogeneous difference schemes. With the joint efforts, the conservative difference schemes were written rather quickly for the complete system, which gave us much easier life.

In another case, during the work discussion A.N. Tikhonov suggested to partially use analytical solutions during the construction of the difference scheme. This led to the construction of conservative difference scheme with quasianalytic interpolation by B.L. Rozhdestvenskii, which allowed increasing the grid. This was particularly important at a time when there were no computers. An important role was played by the offer of Yu.A. Romanov (KB-11), who by that time has developed a new simple method for solving the neutron transport equation. This method was introduced in the developed scheme.

"The question of the of the reliability of calculations was important. The task was issued to two persons who had no right to talk during the work, and at the end the results were compared. The tasks themselves were controlled 3 times. A.A.Samarskii recalls that if he wrote the task, then N.N. Yanenko and V.Ya. Goldin checked it, the next time the roles were reversed. The duty to control the results was on the head of the department". [7]

A story from Andrei Nikolayevich told by A.H. Pergament indicates the responsibility for the accuracy of the results. Once it happened that the field test results significantly did not coincided with the results of the calculations. It threatened a big troubles up to the repression of the participants of the work. A commission was set up, which confirmed that both teams of the calculators made the same mistake, and it saved the project managers. Andrei Nikolayevich said that they were saved by a miracle.

"Work of the department was well organized, it was carried out quickly, but without excessive nervousness. At that time, Andrei Nikolayevich had time to both lead the department, and to give lectures and conduct seminars at Moscow State University, and to continue basic research in the field of geophysics and the theory of differential equations with small parameters, and to work on the textbook "Equations of mathematical physics." The way of his life at this time looked no different from the usual "[7].

As a result, in 1950-1951 the numerical methods were developed, and in 1951, the first numerical calculation of the "layer cake" of A.D. Sakharov was produced and the report was released. Then in 1951-1953, there were calculations of a range of options of the "layer cake". The calculations helped the physicists to see clearly in the explosion processes and select the final design.

In 1952, to assess the progress of the work, a commission was established under the leadership of D.I. Blokhintsev. The commission reviewed and compared the results obtained in the group fo A.N. Tikhonov and the group of L.D. Landau, which was engaged in parallel solution of the same problem. Some improvements of the methods were introduced as a result of the work of the commission.

On November 1, 1952, at Eniwetok Atoll, the Americans managed to carry out a thermonuclear reaction. The detonated device had a huge weight and size (in fact it was a small plant) and was not transportable.

On August 12, 1953, on the range in Central Asia, there have been successfully tested a Soviet hydrogen bomb. It was dropped from an airplane. B.L.Rozhdestvenskii was present at the tests. The results of successful tests confirmed the ideas of the physicists laid into the design, and showed that the mathematical models and calculations (conducted before the advent of computers) with good accuracy corresponded to the real processes. The discrepancy between the results of calculations and the explosive yield recorded by the instruments was less than 30%.

American specialists have managed to build a bomb, suitable for military purposes, only by March of 1954.

In 1954-1955, in connection with the development of thermonuclear designs based on a new principle, the development of mathematical aspects of the calculations for these products and the modernization of the numerical techniques actively continued. When the first national computer "Strela" appeared in 1954, the calculations started to be transferred to it, and there was an extensive series of calculations. They allowed to calculate the explosion process of a new design with sufficient details and to determine its basic characteristics. The results of the field tests carried out in the autumn of 1955 were in very good agreement with the calculated results. A.N. Tikhonov, A.A.Samarskii and V.Ya. Goldin attended the test, which, according to V.Ya. Goldin "was looking absolutely stunning."

The results were highly praised, its executors received the government awards. A.N. Tikhonov became a Hero of Socialist Labor, he was awarded the Order of Lenin and the Stalin Prize of the 1st degree, A.A.Samarskii and V.Ya.Goldin received the Order of Lenin and the Stalin Prize of the 2nd degree, and B.L. Rozhdestvenskii and N.N.Yanenko – the Order of the Red Banner of Labor and the Stalin Prize of the 3rd degree. A number of employees-solvers were awarded the Order of Honor, medals and big prizes.

V.Ya. Goldin recalls, "In those days, they tried to create a pretty decent working conditions for us. We were given trophy "Mercedes", which could be considered pretty decent to calculate on. The salaries were higher than in the Academy of Sciences of the USSR - research staff have received 200 rubles, while in the Academy - 120 rubles. We were helped, and we were excited to work, we had a young team. Note that at the time, when we started, Andrey Nikolayevich was only 42 years old. Everyone else was much younger. Our customers were about the same age as we were."

The work on the creation of mathematical models and methods of the developed of the numerical calculations of atomic and thermonuclear designs were a convincing example of the practical application of finite difference schemes for solving complex problems in mathematical physics. They had led to a significant development of computational mathematics and mathematical modeling, and further to the creation of the new faculty of Moscow State University - Faculty of Computational Mathematics and Cybernetics. In the course of this work under the head of A.N. Tikhonov, a large collective of specialists in applied mathematics had grown.

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