Stability of Numerical Methods for solving Differential Equations

  • Manfred R. Trummer
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 121)


We introduce some of the stability concepts for finite-difference and spectral discretizations of partial differential equations (PDEs). Many of the matrices which occur in such discretizations are non-normal, and we give a few examples for which classical eigenvalue analysis fails to give the correct stability results. This is related to the concept of Hurwitz stability radii.


Stability Region Upwind Scheme Left Half Plane Compute Eigenvalue Implicit Formula 
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  1. [1]
    C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Series of Comput. Physics, Springer, Heidelberg, Berlin, New York, 1988.Google Scholar
  2. [2]
    D. Hinrichsen and A. J. Pritchard, Real and complex stability radii: a survey, in D. Hinrichsen and B. Mårtensson, editors, Control of Uncertain Systems, Birkhäuser, Boston, 1990.Google Scholar
  3. [3]
    R. Jeltsch, Stability of time discretization, Hurwitz determinants and order stars, this proceedings.Google Scholar
  4. [4]
    R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, Interscience, J. Wiley and Sons, 1967.zbMATHGoogle Scholar
  5. [5]
    L. N. Trefethen, Lax-stability vs. eigenvalue stability of spectral methods, in Numerical Methods for Fluid Dynamics III, K. W. Morton and M. J. Baines, eds., Clarendon Press, Oxford, 1988.Google Scholar
  6. [6]
    L. N. Trefethen, Spectra and Pseudospectra: The Behavior of Non-Normal Matrices and Operators, in preparation.Google Scholar
  7. [7]
    L. N. Trefethen and M. R. Trummer, An instability phenomenon in spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008–1023.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1996

Authors and Affiliations

  • Manfred R. Trummer
    • 1
  1. 1.Department of Mathematics & Statistics and Centre for Experimental & Constructive MathematicsSimon Fraser UniversityBurnabyCanada

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