Abstract
A retrospective problem consisting in the reconstruction of an apriori unknown initial state of a highly viscous, incompressible fluid from its known final state is studied. The fluid dynamics model in the Boussinesq approximation is described by the equations of Stokes, incompressibility, and thermal balance with the corresponding initial and boundary conditions. The problem is solved in the inverse direction of time using a new, specially developed iterative method which makes it possible to reduce the original unstable problem to a set of stable problems. The algorithm based on this method is realized on a parallel computer using the OpenFOAM package for engineering calculations. The computational efficiency of the algorithm is analyzed.
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References
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon, Oxford (1987).
O.A. Ladyzhenskaya, Mathematical Questions of Viscous Incompressible Fluid Dynamics [in Russian], Nauka, Moscow (1970).
A.T. Ismail-Zadeh, A.I. Korotkii, B.M. Neimark, and I.A. Tsepelev, “Numerical Modeling of Three-Dimensional Viscous Flows under Gravitational and Thermal Effects,” Zh. Vychisl. Mat. Mat. Fiz. 41, 1399 (2001).
A.N. Tikhonov and V.Ya. Arsenin, Methods for Solving Ill-Posed Problems [in Russian], Nauka, Moscow (1979).
V.K. Ivanov, V.V. Vasin, and V.P. Tanana, Theory of Linear Ill-Posed Problems and its Applications [in Russian], Nauka, Moscow (1978).
M.M. Lavrent’ev, Certain Ill-Posed Problems of Mathematical Physics [in Russian], Siberian Division of the USSR Academy of Sciences, Novosibirsk (1962).
A. Ismail-Zadeh, A. Korotkii, A. Schubert, and I. Tsepelev, “Numerical Techniques for Solving the Inverse Retrospective Problems of Thermal Evolution of the Earth Interior,” Computers Structures 87, 802 (2009).
A.I. Korotkii and I.A. Tsepelev, “Solution of a Retrospective Inverse Problem for One Nonlinear Evolutionary Model,” Proc. Steklov Inst. Math. Suppl. 2, 80 (2003).
A.T. Ismail-Zadeh, A.I. Korotkii, and I.A. Tsepelev, “Three-Dimensional Numerical Modeling of the Inverse Retrospective Problem of Thermal Convection on the Basis of the Quasiconversion Method,” Zh. Vychisl. Mat. Mat. Fiz. 46, 2279 (2006).
A.A. Samarskii, P.N. Vabishchevich, and P.P. Matus, Difference Schemes with Operator Multipliers [in Russian], Minsk (1998).
Y. Wang and K. Hutter, “Comparisons of Numerical Methods with Respect to Convectively Dominated Problem,” Int. J. Numer. Meth. Fluid 37, 721 (2001).
A.A. Samarskii and A.V. Gulin, Numerical Methods [in Russian], Nauka, Moscow (1989).
A.A. Samarskii, P.N. Vabishchevich, and V.I. Vasil’ev, “Iterative Solution of the Retrospective Inverse Heat-Conduction Problem,” Mat. Model. 9(5), 119 (1997).
http://foam.sourceforge.net/doc/Guides-a4/UserGuide.pdf (last accession date August 10, 2010).
http://foam.sourceforge.net/doc/Guides-a4/ProgrammersGuide.pdf (last accession date August 10, 2010).
S. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York (1980).
O. Axelsson, Iterative Solution Methods, Cambridge Univ. Press, Cambridge (1994).
A. Ismail-Zadeh, G. Schubert I. Tsepelev, and A. Korotkii, “Thermal Evolution and Geometry of the Descending Lithosphere beneath the SE-Carpathians: An Insight from the Past,” Earth Planet. Sci. Lett. 273(1–2), 68 (2008).
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Original Russian Text © I.A. Tsepelev, 2011, published in Vychislitel’naya Mekhanika Sploshnykh Sred, 2011, Vol. 4, No. 2, pp. 119–127.
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Tsepelev, I.A. Iterative algorithm for solving the retrospective problem of thermal convection in a viscous fluid. Fluid Dyn 46, 835–842 (2011). https://doi.org/10.1134/S0015462811050164
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DOI: https://doi.org/10.1134/S0015462811050164