Abstract
Adaptive methods for solving convection-diffusion problems are developed. A variant of the adaptive modified alternating triangular method of minimal corrections is constructed and its convergence rate is estimated given that the grid Peclet number is bounded, which holds for monotone difference approximations of diffusion-convection problems. Results of the numerical calculation of spatial three-dimensional currents in the Sea of Azov are given; a parallel implementation of this method performed on a supercomputer at the Taganrog Institute of Technology is considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).
A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems (Editorial URSS, Moscow, 1999) [in Russian].
A. N. Konovalov, “To the Theory of the Alternating Triangle Iteration Method,” Sib. Math. J. 43, 552–572 (2002).
A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems (Editorial URSS, Moscow, 1999) [in Russian].
A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Bikhauser Verlag, Basel, 1989).
A. I. Sukhinov, “Modified Alternating Triangle Method for Heat Conduction and Filtration Problems,” in Computational Systems and Algorithms, (Izd-vo RGU, Rostov-on-Don, 1984), pp. 52–59.
A. E. Chistyakov, “Three-Dimensional Model of Motion of Water in the Sea of Azov Taking into Account Salt and Heat Transport,” Izvestiya YuFU. Tekhnicheskie nauki. Tematicheskii vypusk “ktual’-nye problemy matematicheskogo modelirovaniya” 8(97), 75–82 (Izd-Vo TTI YuFU, Taganrog, 2009).
A. E. Chistyakov, “Theoretical Estimates of Acceleration and Efficiency of Parallel Implementation of ATM of Steepest Decent,” Izvestiya YuFU. Tekhnicheskie nauki. Tematicheskii vypusk “Aktual’nye problemy matematicheskogo modelirovaniya”, 6(107), 237–249 (Izd-Vo TTI YuFU, Taganrog, 2010).
A. I. Sukhinov and A. V. Shishenya, “Improving Estimate of the parameter γ1 of the Alternating Triangle Iteration Method with A Priori Information,” Izvestiya YuFU. Tekhnicheskie nauki. Tematicheskii vypusk “Aktual’nye problemy matematicheskogo modelirovaniya”, 6(107), 7–15 (Izd-Vo TTI YuFU, Taganrog, 2010).
A. I. Sukhinov, A. E. Chistyakov, and E. V. Alekseenko, “Numerical Realization of the Three-Dimensional Model of Hydrodynamics for Shallow Water Basins on a High-Performance System,” Math. Models Comput. Simul. 3(5), 562–574 (2011).
E. A. Al’shina, A. A. Boltnev, and O. A. Kacher, “Gradient Methods with Improved Convergence Rate,” Comp. Math. Math. Phys. 45, 356–365 (2005).
O. Yu. Milyukova, “Parallel Iterative Methods with Factorized Preconditioning Matrix for Discrete Elliptic Equations on Non-Uniform Grid,” Mat. Mod. 15(4), 3–15 (2003).
O. Yu. Milyukova, “Parallel Iterative Methods Using Factorized Preconditioning Matrices for Solving Elliptic Equations on Triangular Grids,” Comput. Math. Math. Phys. 46, 1044–1060 (2006).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.I. Sukhinov, A.E. Chistyakov, 2012, published in Matematicheskoe Modelirovanie, 2012, Vol. 24, No. 1, pp. 3–20.
Rights and permissions
About this article
Cite this article
Sukhinov, A.I., Chistyakov, A.E. Adaptive modified alternating triangular iterative method for solving grid equations with a non-self-adjoint operator. Math Models Comput Simul 4, 398–409 (2012). https://doi.org/10.1134/S2070048212040084
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070048212040084