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Rosenbrock methods for differential algebraic equations. (English) Zbl 0613.65076
This paper deals with the numerical solution of differential algebraic equations (DAE) of index one. It begins with the development of a general theory on the Taylor expansion for the exact solutions of these problems, which extends the well known theory of Butcher for first order ordinary differential equations to DAE’s of index one. As an application, we obtain Butcher-type results for Rosenbrock methods applied to DAE’s of index one and we characterize numerical methods as applications of certain sets of trees. We derive convergent embedded methods of order 4(3) which require 4 or 5 evaluations of the functions, 1 evaluation of the Jacobian and 1 LU factorization per step.

65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
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[1] Rheinboldt, W.C.: Differential-Algebraic Systems as Differential Equations on Manifolds. Math. Comput.43, 473-482 (1984) · Zbl 0581.65058 · doi:10.1090/S0025-5718-1984-0758195-5
[2] Gear, C.W., Petzold, L.: ODE Methods for the Solution of Differential Algebraic Systems. SIAM J. Numer. Anal21, 716-728 (1985) · Zbl 0557.65053 · doi:10.1137/0721048
[3] Petzold, L.: Differential/Algebraic Equations are not ODE’s. SIAM J. Stat. Sci. Comput.3, 367-384 (1982) · Zbl 0482.65041 · doi:10.1137/0903023
[4] Petzold, L.: Order results for implicit Runge-Kutta methods applied to Differential/Algebraic systems. SIAM J. Numer. Anal.23, 837-852 (1986) · Zbl 0635.65084 · doi:10.1137/0723054
[5] Petzold, L.: A description of DASSL. In: A Differential/Algebraic System Solver. R.S. Stepleman (ed.). Proc. IMACS Trans. on Scientific Computation1 (1982)
[6] März, R.: Multisteps methods for initial value problems in implicit differential-algebraic equations. Numer. Math.12, 107-123 (1984) · Zbl 0533.65043
[7] März, R.: On numerical integration methods for implicit ordinary differential equations and differential-algebraic equations. Proc. Kolloquium ?Numerische Behandlung von Differentialgleichungen? pp. 1-15. Wiss. Beitr. Univ. Jena, 1983
[8] Griepentrog, E., März, R.: Differential-Algebraic Equations and Their Numerical Treatment. Teubner Texte zur Mathematik, 88, Leipzig: Teubner 1986 · Zbl 0629.65080
[9] Deuflhard, P., Hairer, E., Zugck, J.: One-step and Extrapolation Methods for Differential-Algebraic Systems. Numer. Math.51, 501-516 (1987) · Zbl 0635.65083 · doi:10.1007/BF01400352
[10] Gallun, S., Holland, C.: Gear’s procedure for the simultaneous solution of differential and algebraic equations with application to unsteady state distillation problems. Comp. Chem. Eng.6, 231-244 (1982) · doi:10.1016/0098-1354(82)80014-8
[11] Feng, A., Holland, C., Gallun, S.: Development and comparison of generalized semi-implicit Runge-Kutta method with Gear’s method for systems of coupled differential and algebraic equations. Comp. Chem. Eng.8, 51-59 (1984) · doi:10.1016/0098-1354(84)80015-0
[12] Miranker, W.L.: Numerical Methods for Stiff Equations and Singular Perturbation Problems. Dordrecht Reidel 1981 · Zbl 0454.65051
[13] Kaps, P., Wanner, G.: A Study of Rosenbrock-Type Methods and High Order. Numer. Math.38, 279-298 (1981) · Zbl 0469.65047 · doi:10.1007/BF01397096
[14] Kaps, P., Rentrop, P.: Generalized Runge-Kutta Methods of order four with step-size control for stiff ODE’s. Numer. Math.33, 55-68 (1979) · Zbl 0436.65047 · doi:10.1007/BF01396495
[15] Verwer, J.G.: Instructive experiments with some Runge-Kutta-Rosenbrock methods. Comp. Comp. Math. Appl.8, 217-229 (1982) · Zbl 0481.65037 · doi:10.1016/0898-1221(82)90045-1
[16] Hairer, E., Norsett, S., Wanner, G.: Solving Ordinary Differential Equations, Vol. 1. Berlin, Heidelberg, New York, Tokyo, Springer 1986
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