Summary
The structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems. The point is that, in general, the global error contains nonsmooth (oscillating) terms at the dominanth 2-level. However, it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor ɛ (where-1/ɛ is the magnitude of the stiff eigenvalues). In these cases a “full” asymptotic error expansion exists in thestrongly stiff case (ε sufficiently small compared to the stepsizeh). The general case (where the oscillating error components areO(h 2) and notO(ɛh 2)) and applications of our results (extrapolation and defect correction algorithims) will be studied in separate papers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auzinger, W., Frank, R., Macsek, F.: Asymptotic error expansions for stiff equations. Part 1: The strongly stiff case. Report Nr. 67/68, Institut fuer Angewandte und numerische Mathematik, TU Wien 1986
Auzinger, W., Frank, R., Macsek, F.: Asymptotic error expansions for stiff equations. Part 2: The mildly stiff case. Report Nr. 68/86. Institut fuer Angewandte und Numerische Mathematik, TU Wien 1986
Auzinger, W., Frank, R., Macsek, F.: Asymptotic error expansions for stiff equations: The implicit Euler scheme. Report Nr. 72/87, Institut fuer Angewandte und Numerische Mathematik, TU Wien 1987. SIAM J. Numer. Anal. (to appeart)
Auzinger, W., Frank, R.: Asymptotic error expansions for stiff equations: An analysis for the implicit midpoint and tranpezoidal rules in the strongly stiff case. Report Nr. 73/88, Institut fuer Angewandte und Numerische Mathematik, TU Wien 1988. (Estended version of the present paper)
Auzinger, W., Frank, R.: Asymptotic error expansions for stiff equations: The implicit midpoint rule, Report Nr. 77/88, Institut fuer Angewandte und Numerische Mathematik, TU Wien 1988
Auzinger, W. Frank, R., Kirlinger, G.: Asymptotic error expansions for stiff equations: Applications. Report Nr. 78/89, Institut fuer Angewandte und Numerische Mathematik, TU Wien 1989
Bader, G., Deuflhard, P.: A semi-implicit midpoint rule for stiff systems of ordinary differential equations. Numer. Math.41, 373–398 (1983)
Dahlquist, G., Lindberg, B.: On some implicit one-step methods for stiff differential equations. Dept. of Computer Sciences, Royal Institute of Technology. Report TRITA-NA-7302, 1973
Dekker, K., Verwer, J.G.: Stability of Runge-Kutta methods for stiff nonlinear differential equations. CWI Monography 2, North-Holland 1984
Frank, R., Schneid, J., Ueberhuber, C.W.: The concept ofB-convergence. SIAM J. Numer. Anal.18, 753–780 (1981)
Gragg, W.B.: Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. Thesis, UCLA 1963
Hairer, E., Lubich, C.: Extrapolation at stiff differential equations. Numer. Math.52, 377–400 (1988)
Kraaijevanger, J.F.B.M.:B-Convergence of the implicit midpoint rule and the trapezoidal rule. BIT25, 652–666 (1985)
O'Malley Jr., R.E.: Introduction to singular perturbations. Academic Press, New York 1974
Stetter, H.J.: Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations. Numer. Math.7, 18–31 (1965)
Van Veldhuizen, M.: Asymptotic expansions of the global error for the implicit midpoint rule (stiff case). Computing33, 185–192 (1984)
Van Veldhuizen, M.:D-Stability. SIAM J. Numer. Anal.18, 45–64 (1981)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Auzinger, W., Frank, R. Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case. Numer. Math. 56, 469–499 (1989). https://doi.org/10.1007/BF01396649
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01396649