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Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics

Abstract : The method of characteristics used for numerical computation of solutions of fluid dynamical equations is characterized by a large degree of non standardness and therefore is not suitable for automatic computation on electronic computing machines, especially for problems with a large number of shock waves and contact discontinuities. In 1950 v. Neumann and Richtmyer proposed to use, for the solution of fluid dynamics equations, difference equations into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over several mesh points. Then, it was proposed to proceed with the computations across the shock waves in the ordinary manner. In 1954, Lax published the "triangle'' scheme suitable for computation across the shock" waves. A deficiency of this scheme is that it does not allow computation with arbitrarily fine time steps (as compared with the space steps divided by the sound speed) because it then transforms any initial data into linear functions. In addition, this scheme smears out contact discontinuities. The purpose of this paper is to choose a scheme which is in some sense best and which still allows computation across the shock waves. This choice is made for linear equations and then by analogy the scheme is applied to the general equations of fluid dynamics. Following this scheme we carried out a large number of computations on Soviet electronic computers. For a check, some of these computations were compared with the computations carried out by the method of characteristics. The agreement of results was fully satisfactory.
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Sergei Godunov, I. Bohachevsky. Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Matematičeskij sbornik, Steklov Mathematical Institute of Russian Academy of Sciences, 1959, 47(89) (3), pp.271-306. ⟨hal-01620642⟩

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