Timestep Control for the Numerical Solutions of Initial-Boundary-Value Problems

  • H. Tadjeran
  • K. Gustafson
  • J. Gary

Abstract

The process of dynamic time-step selection for fixed spatial resolution based on the idea of balancing different local truncation errors will be presented. Some theoretical justification and numerical results will also be given. The application of iterative improvement to ADI methods will also be discussed.

Keywords

Boulder 

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References

  1. 1.
    Ames, W.F., Numerical Methods for Partial Differential Equations, 2nd Edition, Academic Press, New York, 1977.MATHGoogle Scholar
  2. 2.
    Douglas, Jr., J., “A survey of numerical methods for parabolic differential equations,” in Advances in Computers, Vol. 2, edited by F. Alt, New York, Academic Press, 1961, pp. 1–54.Google Scholar
  3. 3.
    Gary, J.M., “On the optimal time step and computational efficiency of difference schemes for PDE,” J. of Computational Physics, Vol. 16 (1974), pp. 298–303.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gary, J.M., “The method of lines applied to a simple hyperbolic equation,” J. of Computational Physics, Vol. 22 (1976), pp. 131–149.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations, Englewood Cliffs, N.J., Prentice Hall, 1971.MATHGoogle Scholar
  6. 6.
    Lindberg, B., “Error estimation and iterative improvement for the numerical solution of operator equations,” UIUCDSD-R-76- 820, Dept. of Computer Science, University of Illinois at Urbana-Champaign (July 1976).Google Scholar
  7. 7.
    Shampine, L.F., and M.K. Gordon, Computer Solution of Ordinary Differential Equations, San Francisco, W.H. Freeman and Company, 1975.MATHGoogle Scholar
  8. 8.
    Shampine, L.F., H.A. Watts, and S.M. Davenport, “Solving non-stiff ordinary differential equations— the state of the art,” SIAM Review, Vol. 18 (1976), pp. 376–410.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Swartz, B., and B. Wendroff, “The relative efficiency of finite difference and finite element methods, I: Hyperbolic problems and splines, ” SIAM J. of Numerical Analysis, Vol. 11 (1974), pp. 979–993.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Tadjeran, Hamid, Ph.D. Dissertation, University of Colorado at Boulder (1980).Google Scholar
  11. 11.
    Warming, R.F., and R.M. Bean, “Factored, A-stable, linear multistep methods — an alternative to the method of lines for multidimensions,” Conference Working Paper, 1979 SIGNUM Meeting on Numerical ODEs, Champaign Illinois, ( April, 1979 ).Google Scholar
  12. 12.
    Ulam, S., A Collection of Mathematical Problems, Wiley-Interscience, New York, 1960.MATHGoogle Scholar
  13. 13.
    Ulam, S., How to formulate mathematically problems of rate of evolution, Mathematical Challenges to the Neo-Darwinian Interpretation of Evolution, Symposium Held at the Wistar Institute of Anatomy and Biology, 1966, Ed.: P. Moorhead and M. Kaplan, Wistar Institute Symposium Monograph no. 5, Philadelphia, 1967, 21–23.Google Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • H. Tadjeran
    • 1
  • K. Gustafson
    • 1
  • J. Gary
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of ColoradoBoulderUSA

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