Abstract
An implicit finite difference scheme approximating the equations of barotropic gas flow is proposed. This scheme ensures the positivity of density and the validity of an energy inequality and the mass conservation law. The continuity equation is approximated implicitly. It is proved that the resulting system of nonlinear equations has a solution for any time and space stepsizes. An iterative method for solving the system of nonlinear equations at each time step is proposed.
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O. M. Belotserkovskii, M. O. Vasil’ev, A. B. Vedernikov, V. P. Dymnikov, B. V. Zamyshlyaev, B. Yu. Krysanov, N. V. Kovshov, V. E. Kunitsyn, E. A. Molokov, A. Yu. Repin, N. A. Sidorenkova, E. L. Stupitskii, Ya. A. Kholodov, and A. S. Kholodov, On the numerical simulation of certain problems concerning the interaction of the lithosphere, hydrosphere, and atmosphere of the Earth,” in Fragments of the History and Achievements of the Institute for Analytical Instrumentation of the Russian Academy of Sciences 1986–2011 (Inst. Analit. Pribor. Ross. Akad. Nauk, Moscow, 2011), pp. 14–71 [in Russian].
I. V. Popov and I. V. Fryazinov, Method of Adaptive Numerical Viscosity for the Numerical Solution of Gas Dynamics Equations (URSS, Moscow, 2015) [in Russian].
S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids (Nauka, Novosibirsk, 1983; North Holland, Amsterdam, 1990).
A. A. Amosov and A. A. Zlotnik, USSR Comput. Math. Math. Phys. 27 (4), 46–57 (1987).
A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).
P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2: Compressible Models (Clarendon, Oxford, 1998).
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow (Gordon and Breach, New York, 1969; Nauka, Moscow, 1970).
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Original Russian Text © F.B. Imranov, G.M. Kobel’kov, A.G. Sokolov, 2018, published in Doklady Akademii Nauk, 2018, Vol. 478, No. 4, pp. 388–391.
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Imranov, F.B., Kobel’kov, G.M. & Sokolov, A.G. Finite Difference Scheme for Barotropic Gas Equations. Dokl. Math. 97, 58–61 (2018). https://doi.org/10.1134/S1064562418010179
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DOI: https://doi.org/10.1134/S1064562418010179