Abstract
A new numerical algorithm based on multigrid methods is proposed for solving equations of the parabolic type. Theoretical error estimates are obtained for the algorithm as applied to a two-dimensional initial-boundary value model problem for the heat equation. The good accuracy of the algorithm is demonstrated using model problems including ones with discontinuous coefficients. As applied to initial-boundary value problems for diffusion equations, the algorithm yields considerable savings in computational work compared to implicit schemes on fine grids or explicit schemes with a small time step on fine grids. A parallelization scheme is given for the algorithm.
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).
N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods, httr://rhus.srb.ru/Stud/Vooks/index.rhr.
R. P. Fedorenko, “Relaxation Method for Solving Difference Elliptic Equations,” Zh. Vychisl. Mat. Mat. Fiz. 1, 922–927 (1961).
N. S. Bakhvalov, “On the Convergence of a Relaxation Method under Natural Constraints on the Elliptic Operator,” Zh. Vychisl. Mat. Mat. Fiz. 6, 861–883 (1966).
M. E. Ladonkina, O. Yu. Milyukova, and V. F. Tishkin, “A Numerical Algorithm for Diffusion-Type Equations Based on Multigrid Methods,” Mat. Model. 19(4), 71–89 (2007).
M. E. Ladonkina, O. Yu. Milyukova, and V. F. Tishkin, “Multigrid Methods as Applied to Diffusion-Type Equations,” VANT Mat. Model. Fiz. Protsessov, No. 1, 4–19 (2008).
M. E. Ladonkina, O. Yu. Milyukova, and V. F. Tishkin, “Numerical Algorithm for Solving Diffusion Equations on the Basis of Multigrid Methods,” Zh. Vychisl. Mat. Mat. Fiz. 49, 518–541 (2009) [Comput. Math. Math. Phys. 49, 502–524 (2009)].
M. E. Ladonkina, O. Yu. Milyukova, and V. F. Tishkin, “Conservative Schemes for Solving Diffusion-Type Equations on the Basis of Multigrid Methods,” Tr. Srednevolzhsk. Mat. O-va Saransk 10(2), 21–44 (2008).
A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Birkhäuser, Basel, 1989).
Y. Saad, Iterative Methods for Sparse Linear Systems. PWS Publishing Co., Int. Tompson Publ. Co. (1995).
S. K. Godunov and V. S. Ryaben’kii, Difference Schemes: An Introduction to the Underlying Theory (Nauka, Moscow, 1977; North-Holland, Amsterdam, 1987).
R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, 2nd ed. (Wiley, New York, 1967; Mir, Moscow, 1972).
O. A. McBryan, P. O. Frederikson, J. Linden, et al., “Multigrid Methods on Parallel Computers: A Survey of Recent Developments,” Impact Comput. Sci. Eng. 3, 1–75 (1991).
H. Ritzdorf, A. Schuller, B. Steckler, and K. Stüben, “LiSS—An Environment for the Parallel Multigrid Solution of Partial Differential Equation on General Domains,” Parallel Comput. 20, 1559–1570 (1994).
O. Yu. Milyukova, “Parallel Approximate Factorization Method for Solving Discrete Elliptic Equations,” Parallel Comput. 27, 1365–1379 (2001).
I. A. Gustafsson, “Class of First Order Factorization Methods,” BIT 18, 142–156 (1978).
Author information
Authors and Affiliations
Additional information
Original Russian Text © M.E. Ladonkina, O.Yu. Milyukova, V.F. Tishkin, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 8, pp. 1438–1461.
Rights and permissions
About this article
Cite this article
Ladonkina, M.E., Milyukova, O.Y. & Tishkin, V.F. Numerical algorithm for solving diffusion-type equations on the basis of multigrid methods. Comput. Math. and Math. Phys. 50, 1367–1390 (2010). https://doi.org/10.1134/S0965542510080087
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542510080087