Multigrid methods with skew-Hermitian based smoothers for the convection–diffusion problem with dominant convection




convection-diffusion equation, multigrid methods, smoothing procedure, product-type skew-Hermitian triangular splitting, local Fourier analysis, convergence


The convection–diffusion equation with dominant convection is considered on a uniform grid of central difference scheme. The multigrid method is used for solving the strongly nonsymmetric systems of linear algebraic equations with positive definite coefficient matrices. Two-step skew-Hermitian iterative methods are utilized for the first time as a smoothing procedure. It is demonstrated that using the proper smoothers enables to improve the convergence of the multigrid method. The robustness of the smoothers with respect to variation of the Peclet number is shown by local Fourier analysis and numerical experiments.

Author Biographies

Tatiana S. Martynova

Southern Federal University,
Vorovich Institute for Mathematics, Mechanics and Computer Science,
• Senior Researcher

Galina V. Muratova

Southern Federal University,
Vorovich Institute for Mathematics, Mechanics and Computer Science,
• Professor

Irina N. Shabas

Southern Federal University,
Vorovich Institute for Mathematics, Mechanics and Computer Science,
• Senior Researcher

Vadim V. Bavin

Southern Federal University,
Vorovich Institute for Mathematics, Mechanics and Computer Science,
• Junior Researcher


  1. G. Birkhoff, E. C. Gartland Jr., and R. E. Lynch, “Difference Methods for Solving Convection-Diffusion Equations,” Comput. Math. Appl. 19 (11), 147-160 (1990).
    doi 10.1016/0898-1221(90)90158-G.
  2. P. N. Vabishchevich and A. A. Samarskii, “Finite Difference Schemes for Convection-Diffusion Problems on Irregular Meshes,” Comput. Math. Math. Phys. 40 (5), 692-704 (2000).
  3. J. Zhang, “Preconditioned Iterative Methods and Finite Difference Schemes for Convection-Diffusion,” Appl. Math. Comput. 109 (1), 11-30 (2000).
    doi 10.1016/S0096-3003(99)00013-2.
  4. T. Ma, L. Zhang, F. Cao, and Y. Ge, “A Special Multigrid Strategy on Non-Uniform Grids for Solving 3D Convection-Diffusion Problems with Boundary/Interior Layers,” Symmetry, 13 (7), (2021).
    doi 10.3390/sym13071123.
  5. V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Computations (Nauka, Moscow, 1984) [in Russian].
  6. L. A. Krukier, “Implicit Difference Schemes and an Iteration Method for Their Solution for a Class of Systems of Quasilinear Equations,” Sov. Math. 23 (7), 43-51 (1979).
  7. L. A. Krukier, “Convergence Acceleration of Triangular Iterative Methods Based on the Skew-Symmetric Part of the Matrix,” Appl. Numer. Math., 30 (2-3), 281-290 (1999).
    doi 10.1016/S0168-9274(98)00116-0.
  8. Z.-Z. Bai, L. A. Krukier, and T. S. Martynova, “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. Math. Phys. 46 (2), 282-293 (2006).
    doi 10.1134/S0965542506020102.
  9. L. A. Krukier, T. S. Martynova, and Z.-Z. Bai, “Product-Type Skew-Hermitian Triangular Splitting Iteration Methods for Strongly Non-Hermitian Positive Definite Linear Systems,” J. Comput. Appl. Math. 232 (1), 3-16, (2009).
    doi 10.1016/
  10. G. Muratova and E. Andreeva, “Multigrid Method for Fluid Dynamics Problems,” J. Comput. Math. 32 (3), 233-247 (2014).
    doi 10.4208/JCM.1403-CR11.
  11. R. P. Fedorenko, “A Relaxation Method for Solving Elliptic Difference Equations,” USSR Comput. Math. Math. Phys. 1 (4), 1092-1096 (1962).
    doi 10.1016/0041-5553(62)90031-9.
  12. A. Brandt, “Multi-Level Adaptive Solutions to Boundary-Value Problems,” Math. Comput. 31, 333-390 (1977).
    doi 10.1090/S0025-5718-1977-0431719-X
  13. U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid (Academic Press, London, 2001).
  14. P. Wesseling and C. W. Oosterlee, “Geometric Multigrid with Applications to Computational Fluid Dynamics,” J. Comput. Appl. Math. 128 (1-2), 311-334 (2001).
    doi 10.1016/S0377-0427(00)00517-3.
  15. V. T. Zhukov, N. D. Novikova, and O. B. Feodoritova, “Multigrid Method for Elliptic Equations with Anisotropic Discontinuous Coefficients,” Comput. Math. Math. Phys. 55 (7), 1150-1163 (2015).
    doi 10.1134/S0965542515070131.
  16. Y. Pan and P.-O. Persson, “Agglomeration-Based Geometric Multigrid Solvers for Compact Discontinuous Galerkin Discretizations on Unstructured Meshes,” J. Comput. Phys. 449 (2021).
    doi 10.1016/
  17. S. Dargaville, A. G. Buchan, R. P. Smedley-Stevenson, et al., “A Comparison of Element Agglomeration Algorithms for Unstructured Geometric Multigrid,” J. Comput. Appl. Math. 390 (2021).
    doi 10.1016/
  18. W. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial (SIAM Press, Philadelphia, 2000).
  19. K. Stüben, “Algebraic Multigrid (AMG): Experiences and Comparisons,” Appl. Math. Comput. 13 (3-4), 419-451 (1983).
    doi 10.1016/0096-3003(83)90023-1.
  20. R. D. Falgout, “An Introduction to Algebraic Multigrid,” Comput. Sci. Eng. 8 (6), 24-33 (2006).
    doi 10.1109/MCSE.2006.105.
  21. S. I. Martynenko, “Numerical Methods for Black Box Software,” Vychisl. Metody Program. 20, 147-169 (2019).
    doi 10.26089/NumMet.v20r215.
  22. U. M. Yang, “Parallel Algebraic Multigrid Methods -- High Performance Preconditioners,” in Lecture Notes in Computational Science and Engineering (Springer, Heidelberg, 2006), Vol. 51, pp. 209-236.
    doi 10.1007/3-540-31619-1_6.
  23. H. Sterck, U. M. Yang, and J. J. Heys, “Reducing Complexity in Parallel Algebraic Multigrid Preconditioners,” SIAM J. Matrix Anal. Appl. 27 (4), 1019-1039 (2006).
    doi 10.1137/040615729.
  24. G. Muratova, T. Martynova, E. Andreeva, et al., “Numerical Solution of the Navier-Stokes Equations Using Multigrid Methods with HSS-Based and STS-Based Smoothers,” Symmetry 12 (2020).
    doi 10.3390/sym12020233.
  25. Sh. Li and Zh. Huang, “Convergence Analysis of HSS-Multigrid Methods for Second-Order Nonselfadjoint Elliptic Problems,” BIT Numer. Math. 53 (4), 987-1012 (2013).
    doi 10.1007/S10543-013-0433-5.
  26. S. Hamilton, M. Benzi, and E. Haber, “New Multigrid Smoothers for the Oseen Problems,” Numer. Linear Algebra Appl. 17 (2-3), 557-576 (2010).
    doi 10.1002/nla.707.
  27. Y. He and S. P. MacLachlan, “Local Fourier Analysis of Block-Structured Multigrid Relaxation Schemes for the Stokes Equations,” Numer. Linear Algebra Appl. 25 (3), 1-28, (2018).
    doi 10.1002/nla.2147.
  28. M. S. Darwish, T. Saad, and Z. Hamdan, “A High Scalability Parallel Algebraic Multigrid Solver,” in Proc. European Conf. on Computational Fluid Dynamics (ECCOMAS CFD 2006), Egmond aan Zee, Netherlands, September 5-8, 2006
    doi 10.1007/978-3-540-92779-2_34.
  29. 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).
    doi 10.1023/B: BITN.0000039428.54019.15.
  30. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM Press, Philadelphia, 2003).
  31. M. A. Botchev and L. A. Krukier, “Iteration Solution of Strongly Nonsymmetric Systems of Linear Algebraic Equations,” Comput. Math. Math. Phys. 37 (11), 1241-1251 (1997).
  32. L. A. Krukier, “Iterative Solution of Nonsymmetric Linear Equation Systems with Dominant Skew-Symmetric Part,” in Proc. Int. Summer School on Iterative Methods and Matrix Computation, Rostov-on-Don, Russia, June 2-9, 2002 (Rostov State Univ., Rostov-on-Don, 2002), pp. 205-259.
  33. J. H. Bramble, J. E. Pasciak, and J. C. Xu, “The Analysis of Multigrid Algorithms for Nonsymmetric and Indefinite Elliptic Problems,” Math. Comput. 51, 389-414 (1988).
    doi 10.1090/S0025-5718-1988-0930228-6.
  34. Z.-H. Cao, “Convergence of Multigrid Methods for Nonsymmetric, Indefinite Problems,” Appl. Math. Comput. 28 (4), 269-288 (1988).
    doi 10.1016/0096-3003(88)90076-8.
  35. J. Mandel, “Multigrid Convergence for Nonsymmetric, Indefinite Variational Problems and One Smoothing Step,” Appl. Math. Comput. 19 (1-4), 201-216 (1986).
    doi 10.1016/0096-3003(86)90104-9.




How to Cite

Martynova T.S., Muratova G.V., Shabas I.N., Bavin V.V. Multigrid Methods With Skew-Hermitian Based Smoothers for the convection–diffusion Problem With Dominant Convection // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2022. 23. 46-59. doi 10.26089/NumMet.v23r104



Methods and algorithms of computational mathematics and their applications