Convergence of an implicit difference scheme for the problem of saturated filtration consolidation with a limiting gradient

V.L. Gnedenkova ∗ , M.F. Pavlova ∗∗ , E.V. Rung ∗∗∗

Kazan Federal University, Kazan, 420008 Russia

E-mail: ∗Valentina.Gnedenkova@kpfu.ru, ∗∗M.F.Pavlova@mail.ru, ∗∗∗HelenRung@mail.ru

Received April 19, 2021

 

ORIGINAL ARTICLE

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DOI: 10.26907/2541-7746.2021.3-4.250-260

For citation: Gnedenkova V.L., Pavlova M.F., Rung E.V. Convergence of an implicit difference scheme for the problem of saturated filtration consolidation with a limiting gradient. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2021, vol. 163, no. 3–4, pp. 250–260. doi: 10.26907/2541-7746.2021.3-4.250-260. (In Russian)

Abstract

This work is devoted to the study of the convergence of an implicit difference scheme for a one-dimensional initial-boundary problem that simulates the process of filtration consolidation with a limiting gradient. From a mathematical point of view, this model is a system of partial differential equations for the displacements of an elastic medium and fluid pressure. In addition, the equation for pressure is degenerate, with nonlinearity in the spatial operator, which generates a non-smooth solution. In this regard, the study of the convergence was carried out under minimal conditions on the smoothness of the initial data. It was based on obtaining a number of a priori estimates that allow, using the monotonicity method, to establish the convergence of piecewise constant completions of the difference solution to a generalized solution of the problem. The spatial operator was approximated using the method of summation identities.

Keywords: filtration, filtration consolidation, difference schemes, difference scheme convergence

References

  1. Kadyrov F.M., Kosterin A.V., Skvortsov E.V. Plane problem of filtration consolidation for an elastic half-space with discontinuous initial conditions. J. Appl. Mech. Tech. Phys., 2016, vol. 57, no. 6, pp. 1076–1082. doi: 10.1134/S0021894416060158.
  2. Zaretsky Yu.K. Teoriya konsolidatsii gruntov [The Theory of Soil Consolidation]. Moscow, Nauka, 1967. 270 p. (In Russian)
  3. Nikolaevskii V.N. Mekhanika poristykh i treshchinovatykh sred [Mechanics of Porous and Cracked Media]. Moscow, Nedra, 1984. 232 p. (In Russian)
  4. Biot M.A. Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys., 1962, vol. 33, no. 4, pp. 1482–1498. doi: 10.1063/1.1728759.
  5. Egorov A.G., Kosterin A.V., Skvortsov E.V. Konsolidatsiya i akusticheskie volny v nasyshchennykh poristykh sredakh [Consolidation and Acoustic Waves in Saturated Porous Media]. Kazan, Izd. Kazan. Univ., 1990. 102 p. (In Russian)
  6. Pavlova M.F., Rung E.V. On the solvability of the problem of saturated-unsaturated filtration consolidation. Differ. Equations, 2012, vol. 48, no. 7, pp. 990–1004. doi: 10.1134/S0012266112070105.
  7. Pavlova M.F., Rung E.V. On the existence of a generalized solution of the saturated-unsaturated filtration problem. Differ. Equations, 2018, vol. 54, no. 3, pp. 352–362. doi: 10.1134/S0012266118030072.
  8. Dautov R.Z., Drobotenko M.I., Lyashko A.D. Investigation of the well-posedness of the generalized solution of the filtration consolidation problem. Differ. Equations, 1997, vol. 33, no. 4, pp. 518–525.
  9. Diersch H.-J.G., Perrochet P. On the primary variable switching technique for simulating unsaturated-saturated flows. Adv. Water Resour., 1999, vol. 23, no. 3, pp. 271–301. doi: 10.1016/S0309-1708(98)00057-8.
  10. Akhtareev A.A., Dautov R.Z. Mixed variable technique for simulating unsaturated-saturated flows. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2007, vol. 149, no. 4, pp. 58–72. (In Russian)
  11. Williams G.A., Miller C.T., Kelley C.T. Transformation approaches for simulating flow in variably saturated porous media. Water Resour. Res., 2000, vol. 36, no. 4, pp. 923–934. doi: 10.1029/1999WR900349.
  12. Lions J.-L. Nekotorye metody resheniya nelineinykh kraevykh zadach [Some Methods of Solving Nonlinear Boundary Value Problems]. Moscow, Mir, 1972. 587 p. (In Russian)

 

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