Abstract
This paper describes an implementation of multistep collocation methods, which are applicable to stiff differential problems, singular perturbation problems, and D.A.E.s of index 1 and 2.
These methods generalize one-step implicit Runge-Kutta methods as well as multistep one-stage BDF methods. We give numerical comparisons of our code with two representative codes for these methods, RADAU5 and LSODE.
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References
T. A. Bickart,An efficient solution process for implicit Runge-Kutta methods. SIAM J. Numer. Anal., vol. 14, 1022–1027, 1977.
K. Burrage, J. C. Butcher & F. H. Chipman,An implementation of singly-implicit Runge-Kutta methods. BIT, vol. 20, pp. 326–340, 1980.
J. C. Butcher,On the implementation of implicit Runge-Kutta methods. BIT, vol. 16, pp. 237–240, 1976.
W. H. Enright, T. E. Hull & B. Lindberg,Comparing numerical methods for stiff systems of ODEs. BIT, vol. 15, pp. 10–48, 1975.
J. Field & R. M. Noyes,Oscillations in chemical systems. IV: Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys., vol. 60, pp. 1877–1884, 1974.
A. Guillou & J. L. Soulé,La résolution numérique des problèmes différentiels aux conditions initiales par des méthodes de collocation. R.A.I.R.O., vol. R-3, pp. 17–44, 1969.
E. Hairer, Ch. Lubich and M. Roche,The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer Lecture Notes in Mathematics 1409, 1989.
E. Hairer, S. P. Nørsett and G. Wanner,Solving Ordinary Differential Equations I. Nonstiff Problems. Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987.
E. Hairer and G. Wanner,Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1991.
W. H. Hundsdorfer,On the Error of General Linear Methods for Stiff Dissipative Differential Equations. Submitted to IMA J. of Numer. Anal., 1991.
V. I. Krylov,Approximate Calculation of Integrals. Mac Millan 1959.
I. Lie,Multistep collocation for stiff systems. Ph.,D. thesis, the Norwegian Institute of Thechnology, Dept. of Numerical Mathematics, Trondheim, Norway, 1985.
I. Lie,Local error estimation for multistep collocation methods, BIT vol. 30, pp. 126–144, 1990.
I. Lie,The stability function for multistep collocation methods. Numer. Math., vol. 57, pp. 779–787, 1989.
I. Lie and S. P. Nørsett,Superconvergence of multistep collocation, Math. Comp. vol. 52, pp. 65–79.
L. F. Shampine,Implementation of implicit formulas for the solution of ODEs. SIAM J. Sci. Stat. Comput., vol. 1, pp. 103–118, 1980.
L. F. Shampine and M. K. Gordon,Computer Solution of Ordinary Differential Equations: The Initial Value Problem. W. H. Freeman 1975.
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Schneider, S. Numerical experiments with a multistep Radau method. BIT 33, 332–350 (1993). https://doi.org/10.1007/BF01989754
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DOI: https://doi.org/10.1007/BF01989754