In these notes, I will tell how in 1953-1954 the first version of the "Godunov's scheme" was invented and how it was modified in the subsequent works by myself (until 1969) and by others at the Institute of Applied Mathematics in Moscow (now named after its founder, academician M. V. Keldish).
Parallel to the modifications which I will describe below (Sections 1, 2), other algorithms were being developed as well at that time, in particular second-order accurate methods for gas-dynamical flows with a small number of strong and weak discontinuities [1,27,26]. Our research activity was related to performing a large number of calculations. The first codes for this purpose were written by V. V. Lucikovich. As the problems became more complex, more artful techniques were required for splitting the computational domain into sub-domains. A. V. Zabrodin was a master in this field. Usage of such splitting techniques lead to the necessity to make the grid generation automatic.
Between 1961 and 1968, together with G. P. Prokopov, we worked long and hard on the methodology for the generation of moving grids, which in 1968-1969 were included into the commercial codes by A. V. Zabrodin and programmers G. M. Novozhilova and G. B. Alalikin [39,13,31]. The problems that arose in the grid generation focused our attention on the solution of elliptic systems [20,38]. Later these methods were used in elliptic spectral problems [30,21], which attracted my attention to the problems of numerical linear algebra. A whole host of surprising observations made at that time provided a source of seminar discussions at Moscow University and, after 1969, at Novosibirsk University for years to come. These discussions resulted in the development of spectral dichotomy methods [18,16] 1 Originally published as "Vospominaniya o raznostnyh shemah" by Nauchnaya Kniga, Novosibirsk, 1997.