Abstract
The solution of quasilinear-implicit ODEs using Rosenbrock type methods may suffer from stability problems despite stability properties such as A-stability or L-stability, respectively. These problems are caused by inexact computation of artificial introduced components (transformation to DAE system). The paper investigates the source of the numerical difficulties and shows modifications to overcome them.
Zusammenfassung
Bei der Lösung quasilinear-impliziter ODEs mittels Rosenbrock-Typ-Methoden können trotz guter Stabilitätseigenschaften (A- bzw. L-Stabilität) des Grundverfahrens Stabilitätsprobleme auftreten. Diese Schwierigkeiten sind auf Ungenauigkeiten bei der Berechnung künstlich eingeführter Komponenten (Überführung in DAEs) zurückzuführen. Die Arbeit untersucht die Ursachen für diese Effekte und zeigt Möglichkeiten, diese zu überwinden.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Denk, G.: Die numerische Integration von Algebro-Differentialgleichungen bei der Simulation elektrischer Schaltkreise mit SPICE2. Tech. Report TUM-M8809, Technische Universität München, 1988.
Deuflhard, P., Novak, U.: Extrapolation integrators for quasilinear implicit ODEs. In: Large-scale scientific computing (Deuflhard, P., Engquist, B., eds.), pp. 37–50. Basel: Birkhäuser, 1987.
Günther, M., Rentrop, P.: Suitable one-step methods, for quasilinear-implicit ODE's. Tech. Report TUM-M9405, Technische Universität München, 1994.
Hairer, E., Wanner, G.: Solving ordinary differential equations II. Berlin Heidelberg New York Tokyo: Springer, 1991.
Lubich, Ch., Roche, M.: Rosenbrock methods for differential algebraic systems with solution-dependent singular matrix multiplying the derivative. Computing43, 325–342 (1990).
Roche, M.: Rosenbrock methods for differential algebraic systems. Numer. Math.52, 45–63 (1988).
Saad, Y.: Krylov subspace methods for solving large unsymmetric linear systems. Math. Comp.37, 105–126 (1981).
Walker, H. F.: Implementation of the GMRES method using Householder transformations. SIAM J. Sci. Stat. Comput.9, 152–163 (1988).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Büttner, M., Weiner, R. & Strehmel, K. A note on stability investigations for Rosenbrock-type methods for quasilinear-implicit differential equations. Computing 56, 47–59 (1996). https://doi.org/10.1007/BF02238291
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02238291