Abstract
We propose and implement new, more general versions of the method of collocations and least squares (the CLS method) and, for a system of linear algebraic equations, an orthogonal method for accelerating the convergence of an iterative solution. The use of the latter method and the proper choice of values of control parameters, based on the results of investigating the dependence of the properties of the CLS method on these parameters, as well as some other improvements of the CLS method suggested in this paper, allow one to solve numerically problems for Navier-Stokes equations in a reasonable time using a single-processor computer even for grids as large as 1280 × 1280. In this case, the total number of unknown variables is ∼ 25 · 106. The numerical results for the problem of the lid-driven cavity flow of a viscous fluid are in good agreement with known results of other authors, including those obtained by means of schemes of higher approximation order with a small artificial viscosity. This and some other facts prove that the new versions of the CLS method make it possible to obtain an approximate solution with high accuracy.
Similar content being viewed by others
References
G. F. Carey, Y. K. Cheung, and S. L. Lau, Comput. Methods Appl. Mech. EngRG. 22, 121 (1980).
T. Mizusawa and T. Kajita, Int. J. Numer. Methods Engrg. 18(6), 897 (1982).
G. Burgess and E. Mahajerin, Comput. & Fluids, 12, 311 (1984).
S. B. Dong and A. E. Lopez, Int. J. Solids and Structures 21, 515 (1985).
B.-N. Jiang and G. F. Carey, Int. J. Numer. Methods Engrg. 24, 569 (1987).
L. R. Bentley, G. F. Pinder, and I. Herrera, Numer. Methods Partial Differ. Equations 5(3), 227 (1989).
L. G. Semin, A. G. Sleptsov, and V. P. Shapeev, Vychisl. Tekhnol. 1(2), 90 (1996).
V. Karamyshev, V. Kovenya, and A. Sleptsov, in Computational Fluid Dynamics (Proc. 3rd ECCOMAS Conf.), 1996, Vol. 1, pp. 301–307.
L. G. Semin and V. P. Shapeev, in Proc. of the Siberian School-Workshop “Mathematical Problems of Mechanics of Continua”, Novosibirsk, 1997, pp. 125–126.
L. G. Semin and V. P. Shapeev, Vychisl. Tekhnol. 3(3), 72 (1998).
P. Bochev, Z. Cai, T. A. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal. 35(3), 990 (1998).
L. G. Semin and V. P. Shapeev, in Proc. Int. Conf. on the Methods of Aerophysical Research, Novosibirsk, 1998, pp. 186–191.
V. V. Belyaev and V. P. Shapeev, Vychisl. Tekhnol. 5(4), 12 (2000).
V. P. Shapeev, L. G. Semin, and V. V. Belyaev, Proc. Steklov Inst. Math., Suppl. 2, 115 (2003).
L. G. Semin, Vychisl. Tekhnol. 11(1), 18 (2006).
V. I. Isaev, V. P. Shapeev, and S. A. Eremin, Vychisl. Tekhnol. 12(3), 53 (2007).
V. I. Isaev and V. P. Shapeev, in Proc. Int. Conf. on the Methods of Aerophysical Research (Parallel’, Novosibirsk, 2007), pp. 147–152.
R. D. Russell and L. F. Shampine, Numer. Math. 19, 1 (1972).
C. de Boor and B. Swartz, SIAM J. Numer. Anal. 10(4), 582 (1973).
U. Ascher, J. Christiansen, and R. D. Russel, Math. Comp. 33, 659 (1979).
A. G. Sleptsov, E. Yu. Letova, K. G. Salomatov, and I. V. Shmykov, Vychisl. Tekhnol. 2(5), 192 (1993).
A. V. Shapeev, Proc. of the 38th Int. Sci. Student Conf. “Student and the Scientific and Technical Progress”, Novosibirsk, 2000, p. 16.
H. K. Moffat, J. Fluid Mech. 18, 1 (1964).
E. Barragy and G. F. Carey, Comput. & Fluids 26(5), 453 (1997).
V. A. Garanzha and V. N. Kon’shin, Zh. Vychisl. Mat. Mat. Fiz. 39(8), 1378 (1999).
D. V. Beklemishev, Additional Chapters in Linear Algebra (Nauka, Moscow, 1983).
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1979; Mir, Moscow, 1981).
O. A. Ladyzhenskaya, Mathematical Problems of Dynamics of a Viscous Incompressible Fluid (Nauka, Moscow, 1970) [in Russian].
Y. Saad, Math. Comp. 37(155), 105 (1981).
A. G. Sleptsov, Model. Mekh. 3(3), 132 (1989).
A. G. Sleptsov, Model. Mekh. 3(5), 118 (1989).
R. P. Fedorenko, Introduction to Computational Physics (MFTI, Moscow, 1994) [in Russian].
V. B. Karamyshev, V. M. Kovenya, A. G. Sleptsov, and S. G. Cherny, Comput. & Fluids 25(5), 467 (1996).
M. A. Ol’shanskii, Doctor’s Dissertation in Physics and Mathematics (Moscow, 2006).
A. V. Shapeev, Dinamika Sploshn. Sredy 116, 119 (2000).
U. Ghia, K. N. Ghia, and C. T. Shin, J. Comput. Phys. 48, 387 (1982).
C. J. Chen and H. J. Chen, J. Comput. Phys. 53(2), 209 (1984).
C. H. Bruneau and C. Jouron, J. Comput. Phys. 89(2), 389 (1990).
G. B. Deng, J. Piquet, P. Queutey, and M. Visonneau, Int. J. Numer. Methods Fluids 19(7), 605 (1994).
O. Botella and R. Peyret, Comput. & Fluids 27(4), 421 (1998).
P. N. Shankar and M. D. Deshpande, Annu. Rev. Fluid Mech. 32, 93 (2000).
M. Sahin and R. G. Owens, Int. J. Numer. Methods Fluids 42(1), 57 (2003).
Y. Wu and S. Liao, Int. J. Numer. Methods Fluids 47, 185 (2005).
E. Erturk, T. C. Corke, and C. Gokcol, Int. J. Numer. Methods Fluids 48, 747 (2005).
M. Cheng and K. C. Hung, Comput. & Fluids 35, 1046 (2006).
B. N. Chetverushkin, Introductory talk and plenary talk at the 5th Int. Congress on Math. Modeling (Dubna, 2002).
A. I. Tolstykh, Compact Difference Schemes and Their Application in Aerohydrodynamics Problems (Nauka, Moscow, 1990) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.I. Isaev, V.P. Shapeev, 2008, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2008, Vol. 14, No. 1.
Rights and permissions
About this article
Cite this article
Isaev, V.I., Shapeev, V.P. Development of the collocations and least squares method. Proc. Steklov Inst. Math. 261 (Suppl 1), 87–106 (2008). https://doi.org/10.1134/S0081543808050088
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543808050088