Reminiscences about numerical schemes

de recherche Thème NUM INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Reminiscences about numerical schemes S. K. Godunov N° 6666 October 2008 Unité de recherche INRIA Sophia Antipolis Reminiscences about numerical schemes S. K. Godunov∗†‡ Thème NUM — Systèmes numériques Projet Smash Rapport de recherche n° 6666 — October 2008 — 24 pages Abstract: This preprint appeared firstly in Russian in 1997. Some truncated versions of this preprint were published in English and French, here a fully translated version is presented. The translation in English was done by O. V. Feodoritova and V. Deledicque to whom I express my gratitude. Key-words: Godunov’s Scheme, hyperbolic systems of conservation laws. ∗Sobolev Institute of Mathematics, Novosibirsk, Russia. † godunov@math.nsc.ru ‡ This preprint is a talk that was given at the International Symposium entitled “Godunov’s Method in Gas Dynamics”, Michigan University (USA), May 1997. Reminiscences about numerical schemes Résumé : Une version, écrite en russe, est apparu la première fois en 1997. D’autres versions (anglaise et française) on été publiées mais sous forme abrégée. On présente ici une traduction complète du document original en anglais, traduction effectuée par O. V. Feodoritova et V. Deledicque, envers lesquels j’exprime ici toute ma gratitude. Mots-clés : Schéma de Godunov, systèmes hyperboliques de lois de conservation. Reminiscences about numerical schemes 3 Introduction In the present paper I will describe how the first variant of the “Godunov’s scheme” has been elaborated in 1953-1954 and tell about all modifications realized by myself (until 1969) and the group of scientists from the Institute of Applied Mathematics in Moscow (which has become the M.V.Keldysh Institute of Applied Mathematics). At the time these modifications (see Sections 2,3) were carried out, other algorithms were developed, in particular second order schemes for gas dynamics problems with a small number of strong and weak discontinuities [1-3]. We performed many calculations based on the first codes written by V. V. Lucikovich. More complicated problems resulted in an elaboration of very artful approaches to divide the whole computational domain into sub-domains which have been developed by A.V.Zabrodin. This procedure resulted in the necessity to develop algorithms of grid construction. In 1961-1968 G. P. Prokopov and I carried out approaches to the construction of moving grids which were used in serial calculations by A. V. Zabrodin, G. N. Novozhilova and G. B. Alalikin (see [4-6]). The problems appearing in the grid construction forced us to solve elliptical systems (see [7-8]). The methods elaborated here, have later been employed in elliptical spectral problems and have been presented in my papers on numerical linear algebra (see [9,10]). The number of interesting observations made during the analysis of my calculations gave many discussions at the Moscow University and, after 1969 – at the Novosibirsk University. As a result of such discussions the criterion of spectral dichotomy [11,12] has been developed and high-precision algorithms to calculate singular vectors have been constructed (see [11,13]). It is difficult to imagine that the reason of such investigations has been the elaboration of approaches to the gas dynamics calculus and numerical grid constructions. During my studies at Moscow University I learned the differential equations theory in seminars of I. M. Gelfand and I. G. Petrovskii. The latter focused my attention on gas dynamics problems and proposed to me to use stationarization methods to study transi- tional flows (with sub- and supersonic regions). My qualification work was devoted to the stationarization of a flow inside a nozzle (however, only in subsonic regime and with artifi- cially added time derivatives introduced in Chaplygin’s equation). Petrovskii’s idea about stationarization in practical form was published in 1961 (see [6]). The technical statement was presented in the qualification work of G. P. Prokopov performed under my supervision. The coding was made by G. N. Novozhilova. The elaboration of numerical schemes was carried out at the same time with attempts to have a better understanding of the notion of generalized solutions to quasi-linear systems of equations. As a rule, the hypothesis about possible definitions and properties of the generalized solutions were preceded to the construction of numerical schemes, which used these properties. At the same time, I tried to prove the formulated hypothesis. To my deep disappointment, these attempts had no success. On the contrary, they often led to contradictory examples. But, at the same time, a numerical scheme, more precisely its modification, based on using the Euler coordinates, moving grids and tracking methods for strong and weak discontinuous, was used in customary calculations. 1 How the scheme has been elabor

…(Full text truncated)…