Skip to main content
SpringerLink
Log in

Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

A stationary convection-diffusion problem with a small parameter multiplying the highest derivative is considered. The problem is discretized on a uniform rectangular grid by the central-difference scheme. A new class of two-step iterative methods for solving this problem is proposed and investigated. The convergence of the methods is proved, optimal iterative methods are chosen, and the rate of convergence is estimated. Numerical results are presented that show the high efficiency of the methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems (Editorial URSS, Moscow, 1999) [in Russian].

    Google Scholar 

  2. K. W. Morton, Numerical Solution of Convection-Diffusion Problems (Chapmen-Hall, London, 1996).

    Google Scholar 

  3. J. Zhang, “Preconditioned Iterative Methods and Finite Difference Schemes for Convection-Diffusion,” Appl. Math. Comput. 109, 11–30 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  4. G. I. Shishkin, “Grid Approximation of a Singularly Perturbed Elliptic Equation with Convective Terms in the Presence of Various Boundary Layers,” Zh. Vychisl. Mat. Mat. Fiz. 45, 110–125 (2005) [Comput. Math. Math. Phys. 45, 104–119 (2005)].

    MATH  MathSciNet  Google Scholar 

  5. N. S. Bakhvalov, “Optimization Methods for Boundary Value Problems in the Presence of a Boundary Layer,” Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969).

    MATH  Google Scholar 

  6. V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Computations (Nauka, Moscow, 1984) [in Russian].

    Google Scholar 

  7. L. A. Krukier, “Implicit Difference Schemes and Iterative Method for Their Solution as Applied to a Class of Systems of Quasilinear Equations,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 7, 41–52 (1979).

  8. L. A. Krukier and T. S. Martynova, “Influence of the Form of Convection-Diffusion Equation on Convergence of the Successive Over-Relaxation Method,” Zh. Vychisl. Mat. Mat. Fiz. 39, 1821–1827 (1999) [Comput. Math. Math. Phys. 39, 1748–1754 (1999)].

    MathSciNet  Google Scholar 

  9. O. Taussky, “Positive-Definite Matrices and Their Role in the Study of the Characteristic Roots of General Matrices,” Adv. Math. 2, 175–186 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  10. L. A. Krukier, L. G. Chikina, and T. V. Belokon, “Triangular Skew-Symmetric Iterative Solvers for Strongly Nonsymmetric Positive Real Linear System of Equations,” Appl. Numer. Math. 41, 89–105 (2002).

    Article  MathSciNet  Google Scholar 

  11. L. Wang and Z.-Z. Bai, “Skew-Hermitian Triangular Splitting Iteration Methods for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts,” BIT Numer. Math. 44, 363–386 (2004).

    Article  MathSciNet  Google Scholar 

  12. A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Birkhäuser, Basel, 1989).

    Google Scholar 

  13. M. A. Bochev and L. A. Krukier, “Iterative Solution of Strongly Nonsymmetric Systems of Linear Algebraic Equations,” Zh. Vychisl. Mat. Mat. Fiz. 37, 1283–1293 (1997) [Comput. Math. Math. Phys. 37, 1241–1251 (1997)].

    MathSciNet  Google Scholar 

  14. Z.-Z. Bai, J.-C. Sun, and D.-R. Wang, “A Unified Framework for the Construction of Various Matrix Multisplitting Iterative Methods for Large Sparse System of Linear Equations,” Comput. Math. Appl. 32(12), 51–76 (1996).

    Article  MathSciNet  Google Scholar 

  15. T. V. Belokon, “Alternating-Triangular Skew-Symmetric Preconditioners Applied to Solving Strongly Nonsymmetric Systems of Linear Algebraic Equations by Variational Methods,” Proceedings of X All-Russia School-Seminar on Modern Problems in Mathematical Modeling (Rostov. Gos. Univ., Rostov-on-Don, 2004), issue 2, pp. 60–71.

    Google Scholar 

  16. L. A. Krukier, “Mathematical Modeling of Transport in Incompressible Media with Predominating Convection,” Mat. Model. 9(2), 4–12 (1997).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Science, 100080, P.O. Box 2719, Beijing, P.R. China

    Z. -Z. Bai

  2. Computer Center of Rostov State University, pr. Stachki 200/1, bld. 2, Rostov-on-Don, 344090, Russia

    L. A. Krukier & T. S. Martynova

Authors
  1. Z. -Z. Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. A. Krukier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. S. Martynova
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © Z.-Z. Bai, L.A. Krukier, T.S. Martynova, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 2, pp. 295–306.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bai, Z.Z., Krukier, L.A. & Martynova, T.S. Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid. Comput. Math. and Math. Phys. 46, 282–293 (2006). https://doi.org/10.1134/S0965542506020102

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542506020102

Keywords

Search

Navigation