Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory
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Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory, 2018, Volume 148, Pages 13–19 (Mi into298)  
Solution of Equations of a One-Dimensional Two-Phase Filtration Problem in a Porous Medium with Account of Thermodynamical Effects by Using Geometric Methods

I. A. Boronin, A. A. Shevlyakov

V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow
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Abstract: One-dimensional problems of two-phase filtration of liquids (water and oil) in porous media are described by the Buckley–Leverett equations, the Darcy law, and the law of conservation of energy under certain initial and boundary conditions. In this paper, we propose an asymptotic method of constructing a solution of the problem and methods for resolution of singularities associated with shock waves that arise in the process. The method proposed is implemented numerically in the Maple software.
Keywords: shock waves, partial differential equations, geometric methods.
Funding agency Grant number
Russian Science Foundation 15-19-00275
This work was supported by the Russian Science Foundation (project No. 15-19-00275).
English version:
Journal of Mathematical Sciences, 2020, Volume 248, Issue 4, Pages 385–391
DOI: https://doi.org/10.1007/s10958-020-04878-y
Bibliographic databases:
Document Type: Article
UDC: 517.922
MSC: 65N06, 34K06, 34K10
Language: Russian
Citation: I. A. Boronin, A. A. Shevlyakov, “Solution of Equations of a One-Dimensional Two-Phase Filtration Problem in a Porous Medium with Account of Thermodynamical Effects by Using Geometric Methods”, Proceedings of the International Conference “Geometric Methods in Control Theory and Mathematical Physics: Differential Equations, Integrability, and Qualitative Theory,” Ryazan, September 15–18, 2016, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 148, VINITI, M., 2018, 13–19; Journal of Mathematical Sciences, 248:4 (2020), 385–391
Citation in format AMSBIB
\Bibitem{BorShe18}
\by I.~A.~Boronin, A.~A.~Shevlyakov
\paper Solution of Equations of a One-Dimensional Two-Phase Filtration Problem in a Porous Medium with Account of Thermodynamical Effects by Using Geometric Methods
\inbook Proceedings of the International Conference ``Geometric Methods in Control Theory and Mathematical Physics: Differential Equations, Integrability, and Qualitative Theory,'' Ryazan, September 15--18, 2016
\serial Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz.
\yr 2018
\vol 148
\pages 13--19
\publ VINITI
\publaddr M.
\mathnet{http://mi.mathnet.ru/into298}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3847703}
\transl
\jour Journal of Mathematical Sciences
\yr 2020
\vol 248
\issue 4
\pages 385--391
\crossref{https://doi.org/10.1007/s10958-020-04878-y}
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