Abstract
The computation of consistent initial values for differential–algebraic equations (DAEs) is essential for starting a numerical integration. Based on the tractability index concept a method is proposed to filter those equations of a system of index-2 DAEs, whose differentiation leads to an index reduction. The considered equation class covers Hessenberg-systems and the equations arising from the simulation of electrical networks by means of Modified Nodal Analysis (MNA). The index reduction provides a method for the computation of the consistent initial values. The realized algorithm is described and illustrated by examples.
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References
P. Amodio and F. Mazzia, An algorithm for the computation of consistent initial values for differential-algebraic equations, Numer. Algorithms 19 (1998) 13–23.
K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (North-Holland, New York, 1989).
S.L. Campbell and C.W. Gear, The index of general nonlinear DAEs, Numer. Math. 72 (1995) 173-196.
D. Estévez Schwarz, Topological analysis for consistent initialization in circuit simulation, Preprint 99-3, Humboldt-Universität, Berlin (1999).
D. Estévez Schwarz, Consistent initial values for DAE system in circuit simulation, Preprint 99-5, Humboldt-Universität, Berlin (1999).
D. Estévez Schwarz, Consistent initialization for index-2 differential algebraic equations and its application to circuit simulation, Ph.D. thesis, Humboldt-Universität, Berlin (2000) in preparation.
D. Estévez Schwarz and C. Tischendorf, Structural analysis for electric circuits and consequences for MNA, Internat. J. Circuit Theory Appl. (1999) to appear.
D. Estévez Schwarz, U. Feldmann, R. März, S. Sturtzel and C. Tischendorf, Finding beneficial DAE structures in circuit simulation, Preprint 00-7, Humboldt-Universität, Berlin (2000).
C.W. Gear, Differential-algebraic equation index transformations, SIAM J. Sci. Statist. Comput. 9 (1988) 39–47.
C.W. Gear, G.K. Gupta and B.J. Leimkuhler, Automatic integration of Euler-Lagrange equations with constraints, J. Comput. Appl. Math. 12/13 (1985) 77–90.
E. Griepentrog, Index reduction methods for differential-algebraic equations, in: Seminarbericht 921, eds. E. Griepentrog, M. Hanke and R. März (Humboldt-Universität, Berlin, 1992) pp. 14–29.
E. Griepentrog and R. März, Differential-Algebraic Equations and Their Numerical Treatment, Teubner-Texte zur Mathematik, Vol. 88 (Teubner, Leipzig, 1986).
M. Günther and U. Feldmann, CAD-based electric-circuitmodeling in industry, I. Mathematical structure and index of network equations, Surveys Math. Indust. 8 (1999) 97–129.
E. Hairer, Ch. Lubich and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics, Vol. 1409 (Springer, New York, 1989).
B. Hansen, Computing consistent initial values for nonlinear index-2 differential-algebraic equations, in: Seminarbericht 92-1, eds. E. Griepentrog, M. Hanke and R. März (Humboldt-Universität, Berlin, 1992) pp. 142–157.
R. Lamour, A well-posed shooting method for transferable DAEs, Numer. Mathematik 59 (1991) 815–829. D. Estévez Schwarz, R. Lamour / Consistent initial values 75
R. Lamour, A shooting method for fully implicit index-2 differential-algebraic equations, SIAM J. Sci. Comput. 18 (1997) 94–114.
B. Leimkuhler, Approximation methods for the consistent initialization of differential-algebraic equations, University of Illinois (1988).
B. Leimkuhler, L.R. Petzold and C.W. Gear, Approximation methods for the consistent initialization of differential-algebraic equations, SIAM J. Numer. Anal. 28 (1991) 205–226.
R. März, Numerical methods for differential-algebraic equations, Acta Numer. (1992) 141–198.
R. März, Analysis and numerics of DAEs, Special lecture, Humboldt-Universität, Berlin (1998).
R. März, EXTRA-ordinary differential equations: Attempts to an analysis of differential-algebraic systems, in: Progress in Mathematics, Vol. 168 (Birkhäuser, Boston, MA, 1998) pp. 313–334.
R. März and A.R. Rodríguez Santiesteban, Analyzing the stability behaviour of DAE solutions and their approximations, Preprint 99-2, Humboldt-Universität, Berlin (1999).
R. März and C. Tischendorf, Recent results in solving index-2 differential-algebraic equations in circuit simulation, SIAM J. Sci. Statist. Comput. 18(1) (1997) 139–159.
C.C. Pantelides, The consistent initialization of differential-algebraic systems, SIAM J. Sci. Statist. Comput. 9 (1988) 213–231.
C. Tischendorf, Solution of index-2 differential-algebraic equations and its application in circuit simulation, Ph.D. thesis, Humboldt-Universität zu Berlin (1996).
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Estévez Schwarz, D., Lamour, R. The computation of consistent initial values for nonlinear index-2 differential–algebraic equations. Numerical Algorithms 26, 49–75 (2001). https://doi.org/10.1023/A:1016696413810
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DOI: https://doi.org/10.1023/A:1016696413810