Computational Mathematics and Mathematical Physics

, Volume 46, Issue 12, pp 2176–2186 | Cite as

Three-dimensional numerical simulation of the inverse problem of thermal convection using the quasi-reversibility method

  • A. T. Ismail-Zadeh
  • A. I. Korotkii
  • I. A. Tsepelev
Article

Abstract

Inverse (time-reverse) simulation of three-dimensional thermoconvective flows is considered for a highly viscous incompressible fluid with temperature-dependent density and viscosity. The model of the fluid dynamics is described by the Stokes equations, the incompressibility and heat balance equations subject to the appropriate initial and boundary conditions. To solve the problem backward in time, the quasi-reversibility method is applied to the heat balance equation. The numerical solution is based on the introduction of a two-component vector potential for the velocity of the medium, on the application of the finite element method with a special tricubic spline basis for computing this potential, and on the application of the splitting method and the method of characteristics for computing the temperature. The numerical algorithm is designed to be executed on parallel computers. The proposed numerical algorithm is used to reconstruct the evolution of diapiric structures in the Earth’s upper mantle. The computational efficiency of the algorithm is analyzed on the basis of the appropriate functionals of residuals.

Keywords

inverse problem ill-posed problem thermal convection incompressible fluid singularly perturbed problem quasi-reversibility method parallel algorithm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. P. McKenzie, J. M. Roberts, and N. O. Weiss, “Convection in the Earth’s Mantle: Towards a Numerical Simulation,” J. Fluid Mech. 62, 465–538 (1974).CrossRefMATHGoogle Scholar
  2. 2.
    H. Frick, F. H. Busse, and R. M. Clever, “Steady Three-Dimensional Convection at High Prandtl Number,” J. Fluid Mech. 127, 141–153 (1983).CrossRefMATHGoogle Scholar
  3. 3.
    G. Houseman, “The Dependence of Convection Planform on Mode of Heating,” Nature 332, 346–349 (1988).CrossRefGoogle Scholar
  4. 4.
    L. Cserepes, M. Rabinowicz, and C. Rosemberg-Borot, “Three-Dimensional Convection in a Two-Layer Mantle,” J. Geophys. Res. 93, 12009–12025 (1988).CrossRefGoogle Scholar
  5. 5.
    B. Travis, P. Olson, and G. Schubert, “The Transition from Two-Dimensional to Three-Dimensional Planforms in Infinite-Prandtl-Number Thermal Convection,” J. Fluid Mech. 216, 71–91 (1990).CrossRefGoogle Scholar
  6. 6.
    U. Christensen and H. Harder, “3-D Convection with Variable Viscosity,” Geophys. J. Int. 104, 213–226 (1991).Google Scholar
  7. 7.
    M. Ogawa, G. Schubert, and A. Zebib, “Numerical Simulation of Three-Dimensional Thermal Convection in a Fluid with Strongly Temperature-Dependent Viscosity,” J. Fluid Mech. 233, 299–328 (1991).CrossRefMATHGoogle Scholar
  8. 8.
    P. J. Tackley, “Effects of Strongly Temperature-Dependent Viscosity on Time-Dependent, Three-Dimensional Models of Mantle Convection,” Geophys. Res. Lett. 20, 2187–2190 (1993).Google Scholar
  9. 9.
    A. T. Ismail-Zadeh, A. I. Lobkovskii, and B. M. Naimark, “A Hydrodynamic Model of the Formation of Sedimentary Basins as a Result of the Formation and Subsequent Phase Transition of a Magma Lens in the Upper Mantle,” in Computational Seismology (Nauka, Moscow, 1994), No. 26, pp. 139–155 [in Russian].Google Scholar
  10. 10.
    V. V. Rykov and V. P. Trubitsyn, “Numerical Simulation of the Three-Dimensional Mantle Convection and the Tectonics of Continental Plates,” in Computational Seismology (Nauka, Moscow, 1994), No. 26, pp. 94–102 [in Russian].Google Scholar
  11. 11.
    A. T. Ismail-Zadeh, A. I. Korotkii, B. M. Naimark, et al., “Implementation of a Three-Dimensional Hydrodynamic Model for Evolution of Sedimentary Basins,” Zh. Vychisl. Mat. Mat. Fiz. 38, 1190–1203 (1998) [Comput. Math. Math. Phys. 38, 1138–1151 (1998)].Google Scholar
  12. 12.
    A. T. Ismail-Zadeh, A. I. Korotkii, B. M. Naimark, and I. A. Tsepelev, “Numerical Simulation of Three-Dimensional Viscous Flows with Gravitational and Thermal Effects,” Zh. Vychisl. Mat. Mat. Fiz. 41, 1399–1415 (2001) [Comput. Math. Math. Phys. 41, 1331–1345 (2001)].Google Scholar
  13. 13.
    G. Schubert, D. L. Turcotte, and P. Olson, Mantle Convection in the Earth and Planets (Cambridge Univ. Press, Cambridge, 2001).Google Scholar
  14. 14.
    A. T. Ismail-Zadeh, I. A. Tsepelev, C. J. Talbot, and A. I. Korotkii, “Three-Dimensional Forward and Backward Modelling of Diapirism: Numerical Approach and Its Applicability to the Evolution of Salt Structures in the Pricaspian Basin,” Tectonophysics 387, 81–103 (2004).CrossRefGoogle Scholar
  15. 15.
    A. T. Ismail-Zadeh, I. A. Tsepelev, C. Talbot, and P. Oster, “Three-Dimensional Modeling of Salt Diapirism: A Numerical Approach and Algorithm of Parallel Calculations,” Comput. Seis. Geodyn. 6, 33–41 (2004).Google Scholar
  16. 16.
    A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1979; Winston, Washington, 1977).Google Scholar
  17. 17.
    R. Lattes and J.-L. Lions, The Method of Quasi-Reversibility; Applications to Partial Differential Equations (Elsevier, New York, 1969; Mir, Moscow, 1970).Google Scholar
  18. 18.
    A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer (Editorial URSS, Moscow, 2002) [in Russian].Google Scholar
  19. 19.
    A. T. Ismail-Zadeh, A. I. Korotkii, B. M. Naimark, and I. A. Tsepelev, “Three-Dimensional Numerical Simulation of the Inverse Problem of Thermal Convection,” Zh. Vychisl. Mat. Mat. Fiz. 43, 614–626 (2003) [Comput. Math. Math. Phys. 43, 587–599 (2003)].Google Scholar
  20. 20.
    A. Ismail-Zadeh, G. Schubert, I. Tsepelev, and A. Korotkii, “Inverse Problem of Thermal Convection: Numerical Approach and Application to Mantle Plume Restoration,” Phys. Earth Planet. Inter. 145, 99–114 (2004).CrossRefGoogle Scholar
  21. 21.
    A. Ismail-Zadeh, G. Schubert, I. Tsepelev, and A. Korotkii, “Data Assimilation in Mantle Dynamics: Theory and Applications,” in Proc. Int. Workshop on Stagnant Slab: A New Keyword for Mantle Dynamics (Kyushu Univ., Fukuoka (Japan), 2005), pp. 134–137.Google Scholar
  22. 22.
    A. Ismail-Zadeh, G. Schubert, I. Tsepelev, and A. Korotkii, “Three-Dimensional Forward and Backward Numerical Modeling of Mantle Plume Evolution: Effects of Thermal Diffusion,” J. Geophys. Res. 111, B06401, doi:10.1029/2005JB003782 (2006).Google Scholar
  23. 23.
    A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Inverse Problems of Mathematical Physics (Editorial URSS, Moscow, 2004) [in Russian].Google Scholar
  24. 24.
    L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Nauka, Moscow, 1986; Pergamon Press, Oxford, 1987).Google Scholar
  25. 25.
    O. A. Ladyzhenskaya, Mathematical Issues of the Dynamics of Viscous Incompressible Fluid (Nauka, Moscow, 1970) [in Russian].Google Scholar
  26. 26.
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon, Oxford, 1961).MATHGoogle Scholar
  27. 27.
    B. M. Naimark, “Local Existence and Uniqueness of a Solution to the Rayleigh-Benard Problem,” in Computational Seismology (Nauka, Moscow, 1988), No. 21, pp. 94–114 [in Russian].Google Scholar
  28. 28.
    R. E. Ewing and H. Wang, “A Summary of Numerical Methods for Time-Dependent Advection-Dominated Partial Differential Equations,” J. Comput. and Appl. Math. 128, 423–445 (2001).CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    A. Staniforth and J. Cote, “Semi-Lagrangian Integration Schemes for Atmospheric Models—a Review,” Monthly Weather Rev. 119, 2206–2223 (1991).CrossRefGoogle Scholar
  30. 30.
    M. Falcone and R. Ferretti, “Convergence Analysis for a Class of High-Order Semi-Lagrangian Advection Schemes,” SIAM J. Numer. Anal. 35, 909–940 (1998).CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    A. McDonald and J. R. Bates, “Improving the Estimate of the Departure Point Position in a Two-Time Level Semi-Lagrangian and Semi-Implicit Scheme,” Monthly Weather Rev. 115, 737–739 (1987).CrossRefGoogle Scholar
  32. 32.
    N. S. Bakhvalov, Numerical Methods (Nauka, Moscow, 1975) [in Russian].Google Scholar
  33. 33.
    F. P. Vasil’ev, Optimization Methods (Faktorial, Moscow, 2002) [in Russian].Google Scholar

Copyright information

© MAIK “Nauka/Interperiodica” 2006

Authors and Affiliations

  • A. T. Ismail-Zadeh
    • 1
    • 2
  • A. I. Korotkii
    • 3
  • I. A. Tsepelev
    • 3
  1. 1.International Institute of Earthquake Prediction Theory and Mathematical GeophysicsRussian Academy of SciencesMoscowRussia
  2. 2.Geophysical InstituteKarlsruhe UniversityKarlsruheGermany
  3. 3.Institute of Mathematics and Mechanics, Ural DivisionRussian Academy of SciencesYekaterinburgRussia

Personalised recommendations