Analyzing the stability behaviour of solutions and their approximations in case of index-2 differential-algebraic systems

März, Roswitha; Rodríguez-Santiesteban, Antonio

doi:10.1090/S0025-5718-01-01408-9

Analyzing the stability behaviour of solutions and their approximations in case of index-$2$ differential-algebraic systems
HTML articles powered by AMS MathViewer

by Roswitha März and Antonio R. Rodríguez-Santiesteban;

Math. Comp. 71 (2002), 605-632

DOI: https://doi.org/10.1090/S0025-5718-01-01408-9

Published electronically: December 5, 2001

PDF | Request permission

Abstract:

When integrating regular ordinary differential equations numerically, one tries to match carefully the dynamics of the numerical algorithm with the dynamical behaviour of the true solution. The present paper deals with linear index-$2$ differential-algebraic systems. It is shown how knowledge pertaining to (numerical) regular ordinary differential equations applies provided a certain subspace which is closely related to the tangent space of the constraint manifold remains invariant.

References

K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical solution of initial value problems in differential-algebraic equations, North-Holland Publishing Co., New York, 1989. MR 1101809
Roswitha März, Numerical methods for differential algebraic equations, Acta numerica, 1992, Acta Numer., Cambridge Univ. Press, Cambridge, 1992, pp. 141–198. MR 1165725, DOI 10.1017/s0962492900002269
E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, 2nd ed., Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1993. Nonstiff problems. MR 1227985
Eberhard Griepentrog and Roswitha März, Differential-algebraic equations and their numerical treatment, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 88, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1986. With German, French and Russian summaries. MR 881052
M. Hanke and R. März: On the asymptotics in the case of differential-algebraic equations. Talk given in Oberwolfach, October 1995.
R. Sudarsan and S. Sathiya Keerthi, An efficient approach for the numerical simulation of multibody systems, Appl. Math. Comput. 92 (1998), no. 2-3, 195–218. MR 1625517, DOI 10.1016/S0096-3003(97)10041-8
D. Estévez Schwarz and C. Tischendorf: Structural analysis for electric circuits and consequences for MNA. Intern. J. of Circuit Theory and Applications 18, 131-162, 2000.
Roswitha März, Index-$2$ differential-algebraic equations, Results Math. 15 (1989), no. 1-2, 149–171. MR 979449, DOI 10.1007/BF03322452
K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monographs, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR 774402
Inmaculada Higueras and Berta García-Celayeta, Logarithmic norms for matrix pencils, SIAM J. Matrix Anal. Appl. 20 (1999), no. 3, 646–666. MR 1685047, DOI 10.1137/S0895479897325955
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
E. Izquierdo Macana: Numerische Approximation von Algebro-Differentialgleichungen mit Index $2$ mittels impliziter Runge-Kutta-Verfahren. Doctoral thesis, Humboldt-Univ., Fachbereich Mathematik, Berlin, 1993.
Uri M. Ascher and Linda R. Petzold, Projected implicit Runge-Kutta methods for differential-algebraic equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 1097–1120. MR 1111456, DOI 10.1137/0728059