Abstract
This paper studies partitioned linearly implicit Runge-Kutta methods as applied to approximate the smooth solution of a perturbed problem with stepsizes larger than the stiffness parameterε. Conditions are supplied for construction of methods of arbitrary order. The local and global error are analyzed and the limiting caseε → 0 considered yielding a partitioned linearly implicit Runge-Kutta method for differential-algebraic equations of index one. Finally, some numerical experiments demonstrate our theoretical results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Deuflhard, P., E. Hairer and J. Zugck,One-step and extrapolation methods for differential-algebraic systems, Numer. Math. 51, 501–516, 1987.
Griepentrog, E. and R. März,Differential-algebraic Equations and their Numerical Treatment, BSB B.G. Teubner, Leipzig 1986.
Hairer, E., Bader, G. and Ch. Lubich,On the stability of semi-implicit methods for ordinary differential equations, BIT 22 (1982), 211–232.
Hairer, E. and Ch. Lubich,Convergence of one-step methods at stiff differential equations, University of Geneva, 1987.
Hairer, E., Ch. Lubich and M. Roche,Error of Rosenbrock methods for stiff problems studied via differential-algebraic equations, BIT 29 (1989), 77–90.
Hairer, E., S. P. Nørsett and G. Wanner,Solving Ordinary Differential Equations I, Springer-Verlag, Berlin-Heidelberg, 1987.
Hundsdorfer, W. H.,The Numerical Solution of Nonlinear Stiff Initial Value Problems, Centrum voor Wiskunde en Informatica, Amsterdam 1984.
O'Malley, R. E.,Introduction to Singular Perturbations, Academic Press, New York and London, 1974.
Rentrop, P. and G. Steinebach,The numerical solution of implicit ordinary differential equations arising in vehicle dynamic, In:Numerical Treatment of Differential Equaitions, ed. by K. Strehmel, Teubner-Texte zur Mathematik, Leipzig 1988.
Roche, M.,Rosenbrock methods for differential-algebraic equations, Numer. Math., 52 (1988), 45–63.
Strehmel, K. and R. Weiner,Partitioned adaptive Runge-Kutta methods and their stability, Numer. Math. 45 (1984), 283–300.
Strehmel, K. and R. Weiner,B-convergence results for linearly implicit one step methods, BIT 27 (1987), 264–281.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Strehmel, K., Weiner, R. & Dannehl, I. On error behaviour of partitioned linearly implicit runge-kutta methods for stiff and differential algebraic systems. BIT 30, 358–375 (1990). https://doi.org/10.1007/BF02017354
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02017354