Skip to main content
SpringerLink
Log in

Matrix-equation-based strategies for convection–diffusion equations

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection–diffusion partial differential equations with separable coefficients, dominant convection and rectangular or parallelepipedal domain. Preconditioners based on the matrix equation formulation of the problem are proposed, which naturally approximate the original discretized problem. For the considered setting, we show that the explicit solution of the matrix equation can effectively replace the linear system solution. Numerical experiments with data stemming from two and three dimensional problems are reported, illustrating the potential of the proposed methodology.

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

Notes

  1. Roughly speaking, two elliptic operators are compact – equivalent if their principal parts coincide up to a constant factor, and they have homogeneous Dirichlet conditions on the same part of the boundary.

  2. A Matlab implementation of the algorithm is available at www.dm.unibo.it/~simoncin/software.html.

  3. Other variable aggregations are possible. The one we chose allowed us to explicitly treat the first derivative in the z direction in the preconditioner.

  4. In the application of \({\mathcal {P}}^{-1}\), the leading computational cost of KPIK is given by sparse direct solves with matrices of size \(n_xn_y\times n_xn_y\), together with the orthogonalization procedure with vectors having \(n_xn_y\) components. The same holds for the problem in (19).

References

  1. Axelsson, O., Karátson, J.: Symmetric part preconditioning for the conjugate gradient method in Hilbert space. Numer. Funct. Anal. Optim. 24(5–6), 455–474 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Axelsson, O., Karátson, J.: Mesh independent superlinear PCG rates via compact-equivalent operators. SIAM J. Numer. Anal. 45(4), 1945–1516 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartels, R.H., Stewart, G.W.: Algorithm 432: solution of the matrix equation \(AX+XB=C\). Commun. ACM 15(9), 820–826 (1972)

    Article  Google Scholar 

  4. Benner, Peter, Damm, Tobias: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control Optim. 49(2), 686–711 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bickley, W.G., McNamee, J.: Matrix and other direct methods for the solution of linear difference equation. Philos. Trans. Roy. Soc. Lond. Ser. A 252, 69–131 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boyle, J., Mihajlović, M.D., Scott, J.A.: HSL\_MI20: an efficient AMG preconditioner for finite element problems in 3D. Int. J. Numer. Meth. Eng. 82(1), 64–98 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Breiten, T., Simoncini, V., Stoll, M.: Fast iterative solvers for fractional differential equations. Technical report, Alma Mater Studiorum - Università di Bologna (2014)

  8. Chin, R.C.Y., Manteuffel, T.A., De Pillis, J.: ADI as a preconditioning for solving the convection–diffusion equation. SIAM J. Sci. Stat. Comput. 5(2), 281–299 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dolgov, S.V., Savostyanov, D.V.: Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems (2013). arXiv:1304.1222v2

  10. Dolgov, S.V., Savostyanov, D.V.: Alternating minimal energy methods for linear systems in higher dimensions. SIAM J. Sci. Comput. 36(5), A2248–A2271 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Clarendon Press, Oxford (1989)

  12. Ellner, N.S., Wachspress, E.L.: New ADI model problem applications. In: Proceedings of 1986 ACM Fall Joint Computer Conference, Dallas, Texas, United States, pp. 528–534. IEEE Computer Society Press, Los Alamitos (1986)

  13. Elman, H.C., Golub, G.H.: Iterative methods for cyclically reduced non-self-adjoint linear systems. Math. Comput. 54(190), 671–700 (1990)

    MathSciNet  MATH  Google Scholar 

  14. Elman, H.C., Ramage, A.: A characterisation of oscillations in the discrete two-dimensional convection–diffusion equation. Math. Comput. 72(241), 263–288 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Elman, H.C., Ramage, A.: An analysis of smoothing effects of upwinding strategies for the convection–diffusion equation. SIAM J. Numer. Anal. 40(1), 254–281 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Elman, H.C., Ramage, A., Silvester, D.J.: IFISS: a matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33(2) Article 14, (2007)

  17. Elman, H.C., Schultz, M.H.: Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations. SIAM J. Numer. Anal. 23(1), 44–57 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers, with applications in incompressible fluid dynamics. In: Numerical Mathematics and Scientific Computation, 2 edn, vol. 21. Oxford University Press, NY (2014)

  19. George, A., Liu, J.: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall Inc., Englewood Cliffs (1981)

    MATH  Google Scholar 

  20. Grasedyck, L.: Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing 72, 247–265 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grasedyck, L., Kressner, D., Tobler, Ch.: A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen 36(1), 53–78 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gunn, J.E.: The numerical solution of \(\nabla \cdot a \nabla u = f\) by a semi-explicit alternating-direction iterative technique. Numer. Math. 6, 181–184 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hackbusch, W., Khoromskij, B.N., Tyrtyshnikov, E.E.: Hierarchical Kronecker tensor-product approximations. J. Numer. Math. 13(2), 119–156 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Khoromskij, B.N.: Tensors-structured numerical methods in scientific computing: survey on recent advances. Chemom. Intell. Lab. Syst. 110, 1–19 (2012)

    Article  Google Scholar 

  25. Kressner, D., Tobler, C.: Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl. 31(4), 1688–1714 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kressner, D., Uschmajew, A.: On low-rank approximability of solutions to high-dimensional operator equations and eigenvalue problem (2014). arXiv:1406.7026v1

  27. Manteuffel, T., Otto, J.: Optimal equivalent preconditioners. SIAM J. Numer. Anal. 30(3), 790–812 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Manteuffel, T.A., Parter, S.V.: Preconditioning and boundary conditions. SIAM J. Numer. Anal. 27(3), 656–694 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  29. Matthies, H.G., Zander, E.: Solving stochastic systems with low-rank tensor compression. Linear Algebra Appl. 436, 3819–3838 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Notay, Y.: Users Guide to AGMG, 3rd edn. In: Service de Métrologie Nucléaire Universitè Libre de Bruxelles (C.P. 165/84), 50, Av. F.D. Roosevelt, B-1050 Brussels, Belgium (2010)

  31. Notay, Y.: Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput. 34(4), A2288–A2316 (2012)

  32. Palitta, D.: Preconditioning strategies for the numerical solution of convection-diffusion partial differential equations. Master’s thesis, Alma Mater Studiorum - Università di Bologna (2014)

  33. Saad, Y.: A flexible inner–outer preconditioned GMRES. SIAM J. Sci. Comput. 14, 461–469 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. Saad, Y.: Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia (2003)

  35. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shank, S.D., Simoncini, V., Szyld, D.B.: Efficient low-rank solutions of generalized Lyapunov equations. Tech.Rep. 14-11-10, Department of Mathematics, Temple University (2014)

  37. Simoncini, V.: On the numerical solution of \({AX-XB=C}\). BIT 36(4), 814–830 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29(3), 1268–1288 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Simoncini, V.: Computational methods for linear matrix equations. Technical report, Alma Mater Studiorum - Università di Bologna (2013)

  40. Starke, G.: Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 28(5), 1431–1445 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  41. Stynes, M.: Numerical methods for convection-diffusion problems or the 30 years war. Department of Mathematics, National University of Ireland (2013). arXiv:1306.5172v1

  42. Wachspress, E.L.: Iterative Solution of Elliptic Systems. Prentice-Hall Inc., Englewood Cliffs (1966)

    MATH  Google Scholar 

  43. Wachspress, E.L.: Extended application of alternating direction implicit iteration model problem theory. J. Soc. Ind. Appl. Math. 11(4), 994–1016 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wachspress, E.L.: Generalized ADI preconditioning. Comput. Math. Appl. 10(6), 405–477 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank Michele Benzi and Howard Elman for helpful discussions, and the two reviewers for their constructing criticism.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, 40127, Bologna, Italy

    Davide Palitta & Valeria Simoncini

  2. IMATI-CNR, Pavia, Italy

    Valeria Simoncini

Authors
  1. Davide Palitta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valeria Simoncini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valeria Simoncini.

Additional information

Communicated by Hans Petter Langtangen.

Version of June 15, 2015. This research is supported in part by the FARB12SIMO grant of the Università di Bologna, and in part by INdAM-GNCS under the 2015 Project Metodi di regolarizzazione per problemi di ottimizzazione e applicazioni.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Palitta, D., Simoncini, V. Matrix-equation-based strategies for convection–diffusion equations. Bit Numer Math 56, 751–776 (2016). https://doi.org/10.1007/s10543-015-0575-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-015-0575-8

Keywords

Mathematics Subject Classification

Search

Navigation