Journal of Scientific Computing

, Volume 26, Issue 1, pp 45–66 | Cite as

Stability Analysis of Difference Methods for Parabolic Initial Value Problems



A decomposition of the numerical solution can be defined by the normal mode representation, that generalizes further the spatial eigenmode decomposition of the von Neumann analysis by taking into account the boundary conditions which are not periodic. In this paper we present some new theoretical results on normal mode analysis for a linear and parabolic initial value problem. Furthermore we suggest an algorithm for the calculation of stability regions based on the normal mode theory.


Convection–diffusion finite differences stability normal mode analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carpenter, M. H., Gottlieb, D., Abarbanel, S. 1993The stability of numerical boundary treatments for compact high-order finite-difference schemesJ. Comput. Phys.108272295CrossRefMathSciNetGoogle Scholar
  2. 2.
    Gautschi, W. 1962On inverses of Vandermonde and confluent Vandermonde matricesNumerische Mathematik4117123CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Gautschi, W. 1963On inverses of Vandermonde and confluent Vandermonde matrices IINumerische Mathematik5425430CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Gautschi, W. 1978On inverses of Vandermonde and confluent Vandermonde matrices IIINumerische Mathematik29445450CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Godunov, S. K. and Ryabenkii, V. S. (1963). Spectral criteria for the stability of boundary problems for non-self-adjoint difference equations. Uspekhi Mat. Nauk. 18(3) (In Russian).Google Scholar
  6. 6.
    Goldberg, M., Tadmor, E. 1978Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problemsMath. Comput.3210971107MathSciNetGoogle Scholar
  7. 7.
    Gustafsson, B., Kreiss, H.-O., Sundstrom, A. 1972Stability theory of difference approximations for mixed initial boundary value problems, IIMath. Comput.26649686MathSciNetGoogle Scholar
  8. 8.
    Gustafsson, B., Kreiss, H.-O., Oliger, J. 1995Time-dependent Problems and Difference MethodsWiley-InterscienceNew YorkGoogle Scholar
  9. 9.
    Kreiss, H.-O. 1968Stability theory for difference approximations of mixed initial boundary value problemsI. Math. Comput.22703714MATHMathSciNetGoogle Scholar
  10. 10.
    Leonard, B. P. 1979A stable and accurate convective modelling procedure based on quadratic upstream interpolationComput. Methods Applied Mechanics and Engineering195998MATHGoogle Scholar
  11. 11.
    Michelson, D. 1983Stability theory of difference approximations for multidimensional initial-boundary value problemsMath. Comput.40145MATHMathSciNetGoogle Scholar
  12. 12.
    Oliger, J. 1974Fourth order difference methods for the initial boundary-value problem for hyperbolic equationsMath. Comput.281525MATHMathSciNetGoogle Scholar
  13. 13.
    Oliger, J. 1976Hybrid difference methods for the initial boundary-value problem for hyperbolic equationsMath. Comput.30724738MATHMathSciNetGoogle Scholar
  14. 14.
    Osher, S. 1969Stability of difference approximations of dissipative type for mixed initial-boundary value problemsMath. Comput.23335340MATHMathSciNetGoogle Scholar
  15. 15.
    Otto, K., Thuné, M. 1989Stability of a Runge-Kutta method for the Euler equations on a substructured domainSIAM J. Scien Stat. Comput.10154174Google Scholar
  16. 16.
    Richtmyer, R. D., Morton, K. W. 1967Difference Methods for Initial-value Problems2Wiley-InterscienceNew YorkGoogle Scholar
  17. 17.
    Sloan, D. M. 1983Boundary conditions for a fourth order hyperbolic difference schemeMath. Comput.41111MATHMathSciNetGoogle Scholar
  18. 18.
    Sousa, E. (2001). Finite Differences for the Convection-Diffusion Equation: On Stability and Boundary Conditions. Ph.D. Thesis, Oxford University.Google Scholar
  19. 19.
    Sousa, E. 2001A Godunov–Ryabenkii instability for a Quickest schemeLecture Notes Comput. Sci.1988732740MATHGoogle Scholar
  20. 20.
    Sousa, E., Sobey, I. J. 2002On the influence of boundary conditionsAppl. Numerical Math.41325344MathSciNetGoogle Scholar
  21. 21.
    Strikwerda, J. 1980Initial boundary value problems for the method of linesJ. Comput. Phys.3494107CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Strikwerda, J. 1989Finite Difference Schemes and Partial Differential EquationsWadsworth & BrooksCaliforniaGoogle Scholar
  23. 23.
    Thuné, M. 1986Automatic GKS stability analysisSIAM J. Sci. Stat. Comput.7959977CrossRefMATHGoogle Scholar
  24. 24.
    Thuné, M. 1990A numerical algorithm for stability analysis of difference methods for hyperbolic systemsSIAM J. Sci. Stat. Comput.116381CrossRefMATHGoogle Scholar
  25. 25.
    Trefethen, L. N. 1983Group velocity interpretation of the stability theory of Gustafsson, Kreiss and SundstromJ. Comput. Phys.49199217CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Trefethen, L. N. 1984Instability of difference models for hyperbolic initial boundary value problemsComm. Pure Appl. Math.37329367MATHMathSciNetGoogle Scholar
  27. 27.
    Varah, J. M. 1971Stability of difference approximations to the mixed initial boundary value problems for parabolic systemsSIAM J. Numerical Anal.8598615MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal

Personalised recommendations