Skip to main content
SpringerLink
Log in

Higher-order schemes for the Laplace transformation method for parabolic problems

  • Published:
Computing and Visualization in Science

Abstract

In this paper we solve linear parabolic problems using the three stage noble algorithms. First, the time discretization is approximated using the Laplace transformation method, which is both parallel in time (and can be in space, too) and extremely high order convergent. Second, higher-order compact schemes of order four and six are used for the the spatial discretization. Finally, the discretized linear algebraic systems are solved using multigrid to show the actual convergence rate for numerical examples, which are compared to other numerical solution 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. Anantha Krishnaiah U., Manohar R., Stephenson J.W.: Fourth-order finite difference methods for three-dimensional general linear elliptic problems with variable coefficients. Numer. Methods Partial Differ. Equ. 3(3), 229–240 (1987)

    Article  MathSciNet  Google Scholar 

  2. Anantha Krishnaiah U., Manohar R., Stephenson J.W.: High-order methods for elliptic equations with variable coefficients. Numer Methods Partial Differ. Equ. 3(3), 219–227 (1987)

    Article  MathSciNet  Google Scholar 

  3. Bromwich T.J.I.: Normal coordinates in dynamical systems. Proc. Lond Math. Soc. 15(Ser. 2), 401–448 (1916)

    MATH  Google Scholar 

  4. Cohen A.M.: Numerical Methods for Laplace Transform Inversion. Springer, New York (2007)

    MATH  Google Scholar 

  5. Crump K.S.: Numerical inversion of Laplace transforms using a Fourier series approximation. J. ACM 23(1), 89–96 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  6. Douglas, C.C.: Multi–grid algorithms for elliptic boundary–value problems. PhD thesis, Yale University (1982)

  7. Douglas C.C.: Multi–grid algorithms with applications to elliptic boundary–value problems. SIAM J. Numer. Anal. 21, 236–254 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Douglas C.C.: Madpack: a family of abstract multigrid or multilevel solvers. Comput. Appl. Math. 14, 3–20 (1995)

    MATH  Google Scholar 

  9. Douglas C.C., Douglas J. Jr: A unified convergence theory for abstract multigrid or multilevel algorithms, serial and parallel. SIAM J. Numer. Anal. 30, 136–158 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Douglas C.C., Douglas J. Jr, Fyfe D.E.: A multigrid unified theory for non-nested grids and/or quadrature. E W J Numer. Math. 2, 285–294 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Faber V., Manteuffel T.: Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal. 21, 352–362 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Freund, R.W.: Solution of shifted linear systems by quasi-minimal residual iterations. In: Numerical Linear Algebra (Kent, OH, 1992), de Gruyter, Berlin, pp. 101–121 (1993)

  13. Freund R.W., Nachtigal N.M.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60(3), 315–339 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gander M.J., Vandewalle S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29, 556–578 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gavrilyuk I.P., Hackbusch W., Khoromskij B.N.: \({\fancyscript{H}}\) -matrix approximation for the operator exponential with applications. Numer. Math. 92, 83–111 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gupta M.M., Manohar R.P., Stephenson J.W.: A single cell high order scheme for the convection-diffusion equation with variable coefficients. Intern. J. Numer. Methods Fluids 4(7), 641–651 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gupta M.M., Manohar R.P., Stephenson J.W.: High-order difference schemes for two-dimensional elliptic equations. Numer. Methods Partial Differ. Equ. 1(1), 71–80 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Karaa S., Zhang J.: High order adi method for solving unsteady convection-diffusion problems. J. Comput. Phys. 198, 1–9 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kwon, Y., Manohar, R., Stephenson, J.W.: Single cell fourth order methods for the biharmonic equation. In: Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), vol. 34, pp. 475–482 (1982)

  20. Lee, H.: The laplace transformation method and its applications in financial mathematics. PhD thesis, Interdisciplinary Program in Computational Science and Technology, Seoul National University, Seoul (2009)

  21. Lee J., Sheen D.: An accurate numerical inversion of Laplace transforms based on the location of their poles. Comput. Math. Appl. 48(10–11), 1415–1423 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lee J., Sheen D.: A parallel method for backward parabolic problems based on the Laplace transformation. SIAM J. Numer. Anal. 44, 1466–1486 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lions J.L., Maday Y., Turinici G.: A parareal in time discretization of PDE’s. C R Acad. Sci. Paris Ser. I Math. 332, 661–668 (2001)

    MathSciNet  MATH  Google Scholar 

  24. López-Fernández M., Palencia C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289–303 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. López-Fernández M., Palencia C., Schädle A.: A spectral order method for inverting sectorial laplace transforms. SIAM J. Numer. Anal. 44(3), 1332–1350 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Murli A., Rizzardi M.: Algorithm 682: Talbot’s method for the Laplace inversion problem. ACM Trans. Math. Softw. 16, 158–168 (1990)

    Article  MATH  Google Scholar 

  27. Sheen D., Sloan I.H., Thomée V.: A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature. Math. Comp. 69(229), 177–195 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sheen D., Sloan I.H., Thomée V.: A parallel method for time-discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal. 23(2), 269–299 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Singer I., Turkel E.: High order finite difference methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 163, 343–358 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Singer I., Turkel E.: Sixth order accurate finite difference schemes for the Helmholtz equation. J. Comput. Accoustics 14, 339–351 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Talbot A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Applics 23, 97–120 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. Thomée V.: A high order parallel method for time discretization of parabolic type equations based on Laplace transformation and quadrature. Int. J. Numer. Anal. Model 2, 121–139 (2005)

    Google Scholar 

  33. Weeks W.T.: Numerical inversion of Laplace transforms using Laguerre functions. J. ACM 13(3), 419–429 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  34. Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comp. 76(259), 1341–1356 (electronic) (2007)

    Google Scholar 

  35. Widder D.V.: The Laplace transform. Princeton University Press, Princeton (1941)

    Google Scholar 

  36. Young, D.M., Dauwalder, J.H.: Discrete representation of partial differential operators. In: Error in Digital Computation, Wiley New York, vol. 2, pp. 181–207 (1965)

  37. Zhang J.: An explicit fourth-order compact finite difference scheme for three-dimensional convection-diffusion equation. Comm. Numer. Methods Eng. 14(3), 209–218 (1998)

    Article  MATH  Google Scholar 

  38. Zhang J., Sun H., Zhao J.J.: High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems. Comput. Methods Appl. Mech. Eng. 191(41-42), 4661–4674 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Author notes

  1. H. Lee

    Present address: Bell Labs Seoul, Alcatel-Lucent, 7fl DMC R&D Center, Digital Media City, E3-2, Sangam-Dong, Mapo-Gu, Seoul, 120-270, Korea

Authors and Affiliations

  1. School of Energy Resources and Department of Mathematics, University of Wyoming, Laramie, WY, 82072-3036, USA

    C. Douglas & H. Lee

  2. Department of Mathematics and Interdisciplinary Program in Computational Science & Technology, Seoul National University, Seoul, 151-747, Korea

    I. Kim & D. Sheen

Authors
  1. C. Douglas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. Sheen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Douglas.

Additional information

Communicated by: C. W. Oosterlee and A. Borzi.

The research by Prof. Douglas is based on work supported in part by NSF grants CNS-1018072 and CNS-1018079 and Award No. KUS-C1-016-04, made by the King Abdullah University of Science and Technology (KAUST). The research by Prof. Sheen was partially supported by NRF-2008-C00043 and NRF-2009-0080533, 0450-20090014. The research by H. Lee was partially supported by Seoul R & D Program WR080951.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Douglas, C., Kim, I., Lee, H. et al. Higher-order schemes for the Laplace transformation method for parabolic problems. Comput. Visual Sci. 14, 39 (2011). https://doi.org/10.1007/s00791-011-0156-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00791-011-0156-6

Keywords

Search

Navigation