Modern Problems of Numerical Analysis. On the Centenary of the Birth of Alexander Andreevich Samarskii

Alexander Andreevich Samarskii (1919–2008) is one of the greats from the pleiad of Russian scientists who brought prominence, international recognition, and fame to the Soviet science. He is the founder of the mathematical modeling in Russia, the leadingworld expert in the field of computational mathematics, mathematical physics, theory of difference schemes, and numerical simulation of complex nonlinear systems. In 1999, A. A. Samarskii was awarded the State Prize for his work on the theory of difference schemes, [7]. Since receiving his Ph.D. in 1947 has held a series of appointments both at various universities and at the Russian Academy of Science. He has been Professor of M. V. Lomonosov Moscow State University and Keldysh Institute of Applied Mathematics of the Soviet Academy of Sciences for more than 50 years. In 1966, A. A. Samarskiiwas elected a correspondingmember, and in1976–a fullmember of theAcademyof Sciences of the USSR. He is a Hero of Socialist Labor, Laureate of the Lenin and State Prizes andM.V. Lomonosov Prize, and awardee of numerous medals and orders as a veteran of the World War II. His early works were in the defense projects on the explosive power of nuclear, and later, of the hydrogen devices. A. A. Samarskii is the co-author of the scientific discovery “The effect of the T-layer”, which is listed in the State Register of Discoveries of the USSR – the first recorded event detected first trough mathematical modeling and computer simulations, and only then in the real experiment. His cutting edge research in laser fusion, magnetic and radiation gas dynamics, high-power lasers, aerodynamics, nuclear energy, and plasma physics was the base for developing understanding of the principles and the organization of the concept of mathematical modeling and computational experiment. A. A. Samarskii created the general theory of operator-difference schemes, [8], including stability of difference schemes for elliptic, [9] and time-dependent problems, [10]. The main trust of Samarskii’s scientific heritage is the theory of operator-difference schemes, [11]. After a proper discretization in space by finite differences or finite elements any initial value problem for time dependent linear partial differential equations (or systems) is transformed into Cauchy problem for a system of ordinary differential equations. This problem is set as andifferential-operator equation in a proper finite dimensionalHilbert space. Timediscretization leads to a operator-difference equation, which is studied by the developed by A. A. Samarskii theory of stability of such schemes. Fundamental result of Samarskii is the formulation of general and abstract criteria, namely, sufficient and necessary conditions, for the stability of twoand three-level schemes. These criteria are based on quite simple and easy to check operator inequalities. Using this approach, he developed the general theory of operator-difference schemes – the stability with respect to the initial data and the right-hand side and also stability with respect to problem coefficients (strong stability).