BIT Numerical Mathematics

, Volume 36, Issue 4, pp 635–652 | Cite as

Extending convergence theory for nonlinear stiff problems part I

  • W. Auzinger
  • R. Frank
  • G. Kirlinger


Existing convergence concepts for the analysis of discretizations of nonlinear stiff problems suffer from considerable drawbacks. Our intention is to extend the convergence theory to a relevant class of nonlinear problems, where stiffness is axiomatically characterized in natural geometric terms.

Our results will be presented in a series of papers. In the present paper (Part I) we motivate the need for such an extension of the existing theory, and our approach is illustrated by means of a convergence argument for the Implicit Euler scheme.

Key words

Stiff differential equations convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. Auzinger, R. Frank, and G. Kirlinger,Extending convergence theory for nonlinear stiff problems. Part I, Report 116/94, Institute for Applied and Numerical Mathematics, Vienna University of Technology (extended version of the present paper).Google Scholar
  2. 2.
    W. Auzinger, R. Frank, and G. Kirlinger,A note on convergence concepts for stiff problems, Computing 44 (1990), pp. 197–208.Google Scholar
  3. 3.
    K. Dekker and J. G. Verwer,Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, 1984.Google Scholar
  4. 4.
    R. Frank and C. W. Ueberhuber,Iterated defect-correction for the efficient solution of stiff systems of ordinary differential equations, BIT 17 (1977), pp. 146–159.Google Scholar
  5. 5.
    R. Frank, J. Schneid, and C.W. Ueberhuber,The concept of B-convergence, SIAM J. Numer. Anal. 18 (1981), pp. 753–780.Google Scholar
  6. 6.
    R. Frank, J. Schneid, and C. W. Ueberhuber,Stability properties of implicit Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), pp. 497–515.Google Scholar
  7. 7.
    R. Frank, J. Schneid, and C. W. Ueberhuber,Order results for implicit Runge-Kutta methods applied to stiff systems, SIAM J. Numer. Anal. 22 (1985), pp. 515–534.Google Scholar
  8. 8.
    E. Hairer, C. Lubich, M. Roche,Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations, BIT 28 (1988), pp. 678–700.Google Scholar
  9. 9.
    E. Hairer and G. Wanner,Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, 1991.Google Scholar
  10. 10.
    Ch. Lubich,On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math. 58 (1991), pp. 839–853; Erratum, 61 (1992), pp. 277–279.Google Scholar
  11. 11.
    K. Nipp and D. Stoffer,Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type, Numer. Math., 70 (1995), pp. 245–257.Google Scholar
  12. 12.
    A. Prothero and A. Robinson,On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp. 28 (1974), pp. 145–162.Google Scholar

Copyright information

© BIT Foundation 1996

Authors and Affiliations

  • W. Auzinger
    • 1
  • R. Frank
    • 1
  • G. Kirlinger
    • 1
  1. 1.Institut für Angewandte und Numerische MathematikTechnische Universität WienWienAustria

Personalised recommendations