Runge-Kutta methods applied to fully implicit differential-algebraic equations of index 1

Kværnø, Anne

doi:10.1090/S0025-5718-1990-1010600-8

Runge-Kutta methods applied to fully implicit differential-algebraic equations of index $1$
HTML articles powered by AMS MathViewer

by Anne Kværnø PDF

Math. Comp. 54 (1990), 583-625 Request permission

Abstract:

In this paper we study the order of Runge-Kutta methods applied to differential-algebraic equations of index one. We derive general order conditions for the local order ${k_L}$, and give a convergence result, which shows that the order ${k_G}$ of the global error satisfies ${k_G} \geq {k_L} - 1$. We also describe some numerical experiments, which are in agreement with our results.

References

The numerical solution of higher index differential/algebraic equations by implicit Runge-Kutta methods

Kevin Burrage and Linda Petzold, On order reduction for Runge-Kutta methods applied to differential/algebraic systems and to stiff systems of ODEs, SIAM J. Numer. Anal. 27 (1990), no. 2, 447–456. MR 1043615, DOI 10.1137/0727027
J. C. Butcher, The numerical analysis of ordinary differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. MR 878564
J. R. Cash, Semi-implicit Runge-Kutta procedures with error estimates for the numerical integration of stiff systems of ordinary differential equations, J. Assoc. Comput. Mach. 23 (1976), no. 3, 455–460. MR 471312, DOI 10.1145/321958.321966
F. H. Chipman, $A$-stable Runge-Kutta processes, Nordisk Tidskr. Informationsbehandling (BIT) 11 (1971), 384–388. MR 295582, DOI 10.1007/bf01939406
C. W. Gear, Differential-algebraic equation index transformations, SIAM J. Sci. Statist. Comput. 9 (1988), no. 1, 39–47. MR 922863, DOI 10.1137/0909004
Ernst Hairer, Christian Lubich, and Michel Roche, The numerical solution of differential-algebraic systems by Runge-Kutta methods, Lecture Notes in Mathematics, vol. 1409, Springer-Verlag, Berlin, 1989. MR 1027594, DOI 10.1007/BFb0093947

On the consistent initialization of differential-algebraic equations

Order conditions for Runge-Kutta methods applied to differential-algebraic systems of index

Per Lötstedt and Linda Petzold, Numerical solution of nonlinear differential equations with algebraic constraints. I. Convergence results for backward differentiation formulas, Math. Comp. 46 (1986), no. 174, 491–516. MR 829621, DOI 10.1090/S0025-5718-1986-0829621-X

Semi explicit Runge-Kutta methods

Syvert P. Nørsett and Per G. Thomsen, Local error control in SDIRK-methods, BIT 26 (1986), no. 1, 100–113. MR 833835, DOI 10.1007/BF01939366
L. R. Petzold, Order results for implicit Runge-Kutta methods applied to differential/algebraic systems, SIAM J. Numer. Anal. 23 (1986), no. 4, 837–852. MR 849286, DOI 10.1137/0723054
Michel Roche, Rosenbrock methods for differential algebraic equations, Numer. Math. 52 (1988), no. 1, 45–63. MR 918316, DOI 10.1007/BF01401021

Implicit Runge-Kutta methods for differential algebraic equations