Skip to main content
SpringerLink
Log in

The impact of parameter selection on the performance of an automatic adaptive code for solving reaction–diffusion equations in three dimensions

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The performance of automatic codes for solving reaction–diffusion systems is controlled by a variety of parameters. Some of them are invisible to the user while others can be modified. For the latter default values are available. The effects of four such parameters are studied for a code that couples the method-of-lines in time with a high-order h-refinement finite element strategy in space. The key parameters considered are the ratio of the temporal error tolerance to the spatial error tolerance, two parameters governing convergence of the iterative methods for the linear and nonlinear systems, and the number of time steps between regridding. These parameters are typical in method-of-lines based codes. Computations on a model problem demonstrate that both small and large temporal to spatial error ratios lead to performance degradation as does less frequent regridding. They also show that insufficient convergence in the nonlinear solver can reduce the reliability of the spatial error estimates.

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. Adjerid, S., Belguendouz, B., Flaherty, J.E.: A posteriori finite element error estimation for diffusion problems. SIAM J. Sci. Comput. 21, 728–746 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Akrivis, G., Crouzeix, M., Makridakis, C.: Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comput. 67, 457–477 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Babuška, I., Feistauer, M., Šolin, P.: On one approach to a posteriori error estimates for evolution problems solved by the method of lines. Numer. Math. 89, 225–256 (2001)

    Article  MathSciNet  Google Scholar 

  5. Babuška, I., Ohnimus, S.: A posteriori error estimation for the semidiscrete finite element of parabolic differential equations. Comput. Methods Appl. Mech. Eng. 190, 4691–4712 (2001)

    Article  Google Scholar 

  6. Bank, R.E.: The efficient implementation of local mesh refinement algorithms. In: Babuška, I., Chandra, J., Flaherty, J.E. (eds.) Adaptive Computational Methods for Partial Differential Equations, pp. 74–81. SIAM, Philadelphia (1983)

    Google Scholar 

  7. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial Value Problems in Differential-Algebraic Equations. North Holland, New York (1989)

    MATH  Google Scholar 

  8. Brown, P.N., Hindmarsh, A.C., Petzold, L.R.: Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 1467–1488 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brown, P.N., Byrne, G.D., Hindmarsh, A.C.: VODE, a variable-coefficient ODE solver. SIAM J. Sci. Statist. Comput. 10, 1038–1051 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cohen, S.D., Hindmarsh, A.C.: CVODE user guide. Technical Report UCRL-MA-118618, pp. 1–91 (1994)

  11. Enright, W.H., Hull, T.E., Lindberg, B.: Comparing numerical methods for stiff systems of ODEs. BIT 15, 10–48 (1975)

    Article  MATH  Google Scholar 

  12. Flaherty, J.E., Moore, P.K.: Integrated space-time adaptive hp-refinement methods for parabolic systems. Appl. Numer. Math. 16, 317–341 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Flaherty, J.E., Wang, Y.: Experiments with an adaptive h-, p-, and r-refinement finite element method for parabolic systems. In: Byrne, G.D., Schiesser, W.E. (eds.) Recent Developments in Numerical Methods and Software for ODEs/DAEs/PDEs, pp. 55–80. World Scientific, Singapore (1992)

    Google Scholar 

  14. de Frutos, J., Novo, J.: Element-wise a posteriori estimates based on hierarchical bases for non-linear parabolic problems. Int. J. Numer. Methods Eng. 63, 1146–1173 (2005)

    Article  MATH  Google Scholar 

  15. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, Berlin (1991)

    MATH  Google Scholar 

  16. Hindmarsh, A.C.: ODEPACK, a systemized collection of ODE solvers. In: Stepleman, R.S., et al. (eds.) Scientific Computing, pp. 55–64. IMACS/North Holland, Amsterdam (1983)

    Google Scholar 

  17. Hindmarsh, A.C.: Brief description of ODEPACK - a systemized collection of ODE solvers. http://www.netlib.org/odepack/opkd-sum (2001)

  18. Lang, J.: Adaptive FEM for reaction–diffusion equations. Appl. Numer. Math. 26, 105–116 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lawson, J., Berzins, M., Dew, P.M.: Balancing space and time errors in the method of lines for parabolic equations. SIAM J. Sci. Statist. Comput. 12, 573–594 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  20. Makridakis, C., Nochetto, R.H.: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41, 1585–1594 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Moore, P.K.: Effects of basis selection and h-refinement on error estimator reliability and solution efficiency for higher-order methods in three space dimensions. Int. J. Numer. Anal. Mod. 3, 21–51 (2006)

    MATH  Google Scholar 

  22. Moore, P.K.: Implicit interpolation error-based error estimation for reaction–diffusion equations in two space dimensions. Comput. Methods Appl. Mech. Eng. 192, 4379–4401 (2003)

    Article  MATH  Google Scholar 

  23. Moore, P.K.: An incomplete assembly with thresholding algorithm for systems of reaction–diffusion equations in three dimensions: IAT for reaction–diffusion systems. J. Comput. Phys. 189, 130–158 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Moore, P.K.: Applications of Lobatto polynomials to an adaptive finite element method: a posteriori error estimates for hp-adaptivity and grid-to-grid interpolation. Numer. Math. 94, 367–401 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Moore, P.K.: An adaptive finite element method for parabolic differential systems: some algorithmic considerations in solving in three space dimensions. SIAM J. Sci. Comput. 21, 1567–1586 (2000)

    Article  MATH  Google Scholar 

  26. Moore, P.K., Dillon, R.H.: A comparison of preconditioners in the solution of parabolic systems in three space dimensions using DASPK and a higher order finite element method. Appl. Numer. Math. 20, 117–128 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Moore, P.K.: Comparison of adaptive methods for one-dimensional parabolic systems. Appl. Numer. Math. 16, 471–488 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ruuth, S.: Implicit-explicit methods for reaction–diffusion problems in pattern formation. J. Math. Biol. 34, 148–176 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Saad, Y.: ILUT: a dual threshold incomplete LU factorization. Numer. Linear Algebra Appl. 1, 387–402 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Schiesser, W.E.: The Numerical Method of Lines, Integration of Partial Differential Equations. Academic, San Diego (1991)

    MATH  Google Scholar 

  32. Schmidt, A., Siebert, K.G.: Design of Adaptive Finite Element Software. The Finite Element Toolbox ALBERTA. Springer, Berlin (2005)

    MATH  Google Scholar 

  33. Schönauer, W., Schnepf, E., Raith, K.: Numerical engineering: experiences in designing PDE software with selfadaptive variable stepsize/variable order difference methods. In: Böhmer, K., Stetter, H.J. (eds.) Defect Correction Methods, Theory and Applications, Computing, Supplementum Nr. 5, pp. 227–242. Springer, Berlin (1984)

    Google Scholar 

  34. Szabó, B., Babuška, I.: Finite Element Analysis. Wiley/Interscience, New York (1991)

    MATH  Google Scholar 

  35. Wang, R., Keast, P., Muir, P.: A high-order global spatially adaptive collocation method for 1-D parabolic PDEs. Appl. Numer. Math. 50, 239–250 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  36. Weiser, A.: Local-mesh, local-order, adaptive finite element methods with a posteriori error estimators for elliptic partial differential equations. Technical Report 213, Department of Computer Science, Yale University, New Haven (1981)

  37. Vande Wouwer, A., Saucez, Ph., Schiesser, W.E. (eds.): Adaptive Method of Lines. Chapman & Hall/CRC, Boca Raton (2001)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA

    Peter K. Moore

Authors
  1. Peter K. Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter K. Moore.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moore, P.K. The impact of parameter selection on the performance of an automatic adaptive code for solving reaction–diffusion equations in three dimensions. Numer Algor 46, 121–139 (2007). https://doi.org/10.1007/s11075-007-9131-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-007-9131-1

Keywords

Search

Navigation