## Main scientific results of Academician A.A. Samarskii in the filed of mathematical modeling.

The first works of Academician A.A. Samarskii in the field of mathematical modeling were classified and were associated with the calculation of the explosion of the atomic bomb. This work started with the creation of the laboratory headed by A.N. Tikhonov in 1948.

First calculations were carried out under the simplified averaged model, and then using the full model with the solution of a system of more than 100 partial differential equations by difference methods. The calculations were done in just a few months and coincided with the full-scale tests of the nuclear charge with the deviation less than 30%, which was a very good result. Next A.A. Samarskii was switched to the more difficult task of calculating the thermonuclear explosion. Prior to these calculations, such a task seemed extremely difficult and was named by L.D. Landau "a scientific feat".

It was A.A. Samarskii who coined the term "computer experiment". In 1965, as a result of a computational experiment, A.A. Samarskii and a group of authors discovered T-layer effect - formation of self-sustaining high-temperature conductive layer of gas in the non-stationary processes of magnetohydrodynamics. This effect was the first recorded discovery made in the computer first, and only later in the field experiment.

This effect is produced as a result of the pre-emptive release of Joule heat in a certain mass of the gas. By its nature, it is close to the well-known skin effect and the phenomena of dissipative instability in magnetohydrodynamics. However, it is not limited to any of them and substantially differs from them. The mathematical model on which the calculations were carried out, accounted for a significant non-linearity of nonstationary processes of magnetohydrodynamics.

Many scientists doubted the physical sense of the discovery, being guided by the principle that there is only what that is observable.

I.M. Sobol recalls about the preparation of an article, together with A.A. Samarskii, "Examples on numerical calculation of temperature waves" (J. Comp. Math. and Math. Phys., V.3, No. 4, 1963. pp.702-719): "During the work, we found an unusual solution: the end of the rod is heated, the temperature increases without limit, but the length of the heated region does not change, and the front of the heat does not move." So it was even questioned whether or not to publish such a solution, as there was no known real-world problem described by such a solution. Such relevant infinite solutions were subsequently called blow-up regimes. But even earlier, in a paper from 1950, the unbounded solution was also obtained by Ya.B. Zeldovich and A.S. Kompaneets.

The decisive role in the recognition of the effect of the T-layer was played by Academicians Ya.B. Zeldovich, N.G. Basov and A.E. Sheidlin. The discovery of the T-layer, although not made at the tip of the pen but on the computer screen, was then almost simultaneously discovered by several groups of experimenters. The usage of the effect of the T-layer allowed creating pilot installations of magnetic hydrodynamic generators of a new type, directly converting the energy of the plasma flow into the electric current. The observed effect has a general physical character. One of the co-discoverers of the T-layer, V.S. Sokolov and his colleagues, later published the papers about new non-traditional explanations of the nature of chromospheric flares on the Sun and their accompanying phenomena.