Half-explicit Runge-Kutta methods for semi-explicit differential- algebraic equations of index 1. (English) Zbl 0791.65055
For the numerical solution of non-stiff semi-explicit differential- algebraic equations (DAEs) of index 1 half-explicit Runge-Kutta methods (HERK) are considered that combine an explicit Runge-Kutta method for the differential part with a simplified Newton method for the (approximate) solution of the algebraic part of the DAE. Two principles for the choice of the initial guesses and the number of Newton steps at each stage are given that allow to construct HERK of the same order as the underlying explicit Runge-Kutta method. Numerical tests illustrate the efficiency of these methods.
Reviewer: M.Arnold (Rostock)
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
non-stiff semi-explicit differential-algebraic equations of index 1; half-explicit Runge-Kutta methods; numerical tests; Newton methodReferences:
| [1] | Arnold, M. (1990): Numerische Behandlung von semi-expliziten Algebrodifferentialgleichungen vom Index 1 mit linear-impliziten Verfahren. PhD thesis, Martin-Luther-Universit?t Halle, Sektion Mathematik · Zbl 0741.65054 |
| [2] | Butcher, J.C. (1987): The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods. Wiley, Chichester New York · Zbl 0616.65072 |
| [3] | Deuflhard, P., Hairer, E., Zugck, J. (1987): One-step and extrapolation methods for differential-algebraic systems. Numer. Math.51, 501-516 · Zbl 0635.65083 · doi:10.1007/BF01400352 |
| [4] | Dormand, J.R., Prince, P.J. (1980): A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math.6, 19-26 · Zbl 0448.65045 · doi:10.1016/0771-050X(80)90013-3 |
| [5] | Griepentrog, E., M?rz, R. (1986): Differential-algebraic equations and their numerical treatment. Teubner Texte zur Mathematik. Teubner, Leipzig · Zbl 0629.65080 |
| [6] | Gear, C.W., Petzold, L.R. (1984): ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal.21, 716-728 · Zbl 0557.65053 · doi:10.1137/0721048 |
| [7] | Hairer, E., Lubich, Ch., Roche, M. (1989): The numerical solution of differential-algebraic systems by Runge-Kutta methods. Lecture Notes in Mathematics, 1409. Springer, Berlin Heidelberg New York · Zbl 0657.65093 |
| [8] | Hairer, E., N?rsett, S.P., Wanner, G. (1978): Solving ordinary differential equations.I. Nonstiff Problems. Springer, Berlin Heidelberg New York · Zbl 0789.65048 |
| [9] | Hairer, E., Wanner, G. (1974): On the Butcher group and general multivalue methods. Computing,13, 1-15 · Zbl 0293.65050 · doi:10.1007/BF02268387 |
| [10] | Hairer, E., Wanner, G. (1991): Solving ordinary differential equations.II. Stiff and differential-algebraic problems. Springer-Verlag, Berlin Heidelberg New York · Zbl 0729.65051 |
| [11] | Prince, P.J., Dormand, J.R. (1981): High order embedded Runge-Kutta formulae. J. Comput. Appl. Math.7, 67-75 · Zbl 0449.65048 · doi:10.1016/0771-050X(81)90010-3 |
| [12] | Petzold, L.R. (1986): Order results for implicit Runge-Kutta methods applied to differential/algebraic systems. SIAM J. Numer. Anal.23, 837-852 · Zbl 0635.65084 · doi:10.1137/0723054 |
| [13] | Rentrop, P., Roche, M., Steinebach, G. (1989): The application of Rosenbrock-Wanner type methods with stepsize control in differential-algebraic equations. Numer. Math.55, 545-563 · Zbl 0672.65046 · doi:10.1007/BF01398915 |
| [14] | Rentrop, P., Steinebach, G. (1988): The numerical solution of ordinary differential equations arising in vehicle dynamic. In K. Strehmel, ed., Numerical treatment of differential equations. Proceedings of the fourth seminar ?NUMDIFF-4?, Halle 1987, Leipzig. Teubner-Texte zur Mathematik, Teubner, Leipzig |
| [15] | Roche, M. (1988): Rosenbrock methods for differential-algebraic systems. Numer. Math.52, 45-63 · Zbl 0613.65076 · doi:10.1007/BF01401021 |
| [16] | Strehmel, K., Weiner, R., Dannehl, I. (1990): On error behaviour of partitioned linearly implicit Runge-Kutta methods for stiff and differential-algebraic systems. BIT,30, 358-375 · Zbl 0702.65073 · doi:10.1007/BF02017354 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
