Geodesic
Solution of Equations of a One-Dimensional Two-Phase Filtration Problem in a Porous Medium with Account of Thermodynamical Effects by Using Geometric Methods
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Geometric Methods in Control Theory and Mathematical Physics: Differential Equations, Integrability, and Qualitative Theory,” Ryazan, September 15–18, 2016, Tome 148 (2018), pp. 13-19.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
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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. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Geometric Methods in Control Theory and Mathematical Physics: Differential Equations, Integrability, and Qualitative Theory,” Ryazan, September 15–18, 2016, Tome 148 (2018), pp. 13-19. http://geodesic.mathdoc.fr/item/INTO_2018_148_a2/

[1] Arnold V. I., Obyknovennye differentsialnye uravneniya, Nauka, M., 1984

[2] Akhmetzyanov A. V., Kushner A. G., Lychagin V. V., “Mass and heat transport in the two-phase Buckley—Leverett model”, J. Geom. Phys., 113 (2017), 2–9 | DOI | MR | Zbl