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Stability preserving integration of index-1 DAEs. (English) Zbl 1041.65065
The paper is concerned with differential-algebraic equations (DAEs) of index 1 and their numerical integration. The main topic is the investigation of conditions under which certain qualitative properties of the DAE, in particular contractivity or dissipativity, are properly reflected by their numerical approximations without stepsize restriction, when numerical schemes with suitable stability properties are used. For this purpose, the DAE is required to have a “properly stated leading term” and to be “numerically qualified”. But also for not numerically qualified DAE’s, equivalent reformulations can help to obtain the desired properties, as is illustrated in several examples.

65L80 Numerical methods for differential-algebraic equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
Full Text: DOI
[1] Ascher, U.; Petzold, L.R., Stability of computational methods for constrained dynamic systems, SIAM J. sci. comput., 14, 95-120, (1993) · Zbl 0773.65044
[2] Balla, K.; März, R., A unified approach to linear differential algebraic equations and their adjoints, Z. anal. anwendungen, 21, 3, 783-802, (2002) · Zbl 1024.34002
[3] Brenan, K.E.; Campbell, S.L.; Petzold, L.R., Numerical solution of initial value problems in differential algebraic equations, (1989), North-Holland Amsterdam · Zbl 0699.65057
[4] Clark, K.D., Difference methods for the numerical solution of time varying singular systems of differential equations, SIAM J. alg. disc. math., 7, 2, 236-246, (1986) · Zbl 0594.34003
[5] Estévez Schwarz, D.; Tischendorf, C., Structural analysis of electric circuits and consequences for MNA, Internat. J. circ. theor. appl., 28, 131-162, (2000) · Zbl 1054.94529
[6] Garcia-Celayeta, B.; Higueras, I., Runge – kutta methods for DAEs—A new approach, J. comput. appl. math., 111, 1-2, 49-61, (1999) · Zbl 0948.65081
[7] Griepentrog, E.; März, R., Differential – algebraic equations and their numerical treatment, (1986), Teubner Leipzig · Zbl 0629.65080
[8] Günther, M., Simulating digital circuits numerically—A charge-oriented ROW approach, Numer. math., 79, 203-212, (1998) · Zbl 0905.65089
[9] Hanke, M.; Izquierdo Macana, E.; März, R., On asymptotics in case of linear index-2 differential – algebraic equations, SIAM J. numer. anal., 35, 1326-1346, (1998) · Zbl 0946.65067
[10] Hanke, M.; März, R., On asymptotics in case of daes, Z. angew. math. mech. suppl., 76, 1, 99-102, (1996) · Zbl 0900.65247
[11] I. Higueras, R. März, Differential algebraic equations with properly stated leading term, Preprint 2000-20, Humboldt Univ. of Berlin, Institute of Mathematics, 2000, Comput. Math. Appl., to appear · Zbl 1068.34005
[12] März, R., On initial value problems in differential – algebraic equations and their numerical treatment, Computing, 35, 13-37, (1985) · Zbl 0554.65050
[13] März, R., Differential algebraic systems anew, Appl. numer. math., 42, 315-335, (2002) · Zbl 1005.65080
[14] März, R.; Rodriguez Santiesteban, A.R., Analyzing the stability behaviour of solutions and their approximations in case of index-2 differential – algebraic systems, Math. comput., 71, 238, 605-632, (2001) · Zbl 1002.65091
[15] A.R. Rodriguez Santiesteban, Asymptotische Stabilität von Index-2-Algebro-Differentialgleichungen, Ph.D. Thesis, Humboldt Univ. of Berlin, Institute of Mathematics, Berlin, 2001 · Zbl 1022.65086
[16] Stuart, A.M.; Humphries, A.R., Dynamical systems and numerical analysis, (1998), Cambridge University Press Cambridge · Zbl 0913.65068
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