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On the computation of impasse points of quasilinear differential-algebraic equations

Authors: Patrick J. Rabier and Werner C. Rheinboldt
Journal: Math. Comp. 62 (1994), 133-154
MSC: Primary 65L05; Secondary 34A09, 34A47, 58F14
MathSciNet review: 1208224
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Abstract: We present computational algorithms for the calculation of impasse points and higher-order singularities in quasi-linear differential-algebraic equations. Our method combines a reduction step, transforming the DAE into a singular ODE, with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory. Two numerical examples with physical relevance are given.

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  • [1] R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, 2nd ed., Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988. MR 960687
  • [2] Richard P. Brent, Some efficient algorithms for solving systems of nonlinear equations, SIAM J. Numer. Anal. 10 (1973), 327–344. Collection of articles dedicated to the memory of George E. Forsythe. MR 0331764
  • [3] 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
  • [4] George D. Byrne and Alan C. Hindmarsh, Stiff ODE solvers: a review of current and coming attractions, J. Comput. Phys. 70 (1987), no. 1, 1–62. MR 888931, 10.1016/0021-9991(87)90001-5
  • [5] L. O. Chua, Introduction to nonlinear networks, McGraw-Hill, New York, 1969.
  • [6] Leon O. Chua and An-Chang Deng, Impasse points. I. Numerical aspects, Internat. J. Circuit Theory Appl. 17 (1989), no. 2, 213–235. MR 991519, 10.1002/cta.4490170207
  • [7] 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
  • [8] F. A. Potra and W. C. Rheinboldt, Differential-geometric techniques for solving differential-algebraic equations, Real-Time Integration Methods for Mechanical System Simulation (E. J. Haug and R. C. Deyo, eds.), Springer-Verlag, New York, 1991, pp. 155-192.
  • [9] Patrick J. Rabier, Implicit differential equations near a singular point, J. Math. Anal. Appl. 144 (1989), no. 2, 425–449. MR 1027045, 10.1016/0022-247X(89)90344-2
  • [10] Patrick J. Rabier and Werner C. Rheinboldt, A general existence and uniqueness theory for implicit differential-algebraic equations, Differential Integral Equations 4 (1991), no. 3, 563–582. MR 1097919
  • [11] -, A geometric treatment of implicit differential-algebraic equations, Tech. Report TR-ICMA-162, Inst. Comput. Math. Appl., Univ. of Pittsburgh, June 1991; J. Differential Equations (in press).
  • [12] -, On impasse points of quasilinear differential-algebraic equations, Tech. Report TR-ICMA-171, Inst. Comput. Math. Appl., Univ. of Pittsburgh, April 1992; J. Math. Anal. Appl. (in press).
  • [13] Werner C. Rheinboldt, Numerical analysis of parametrized nonlinear equations, University of Arkansas Lecture Notes in the Mathematical Sciences, 7, John Wiley & Sons, Inc., New York, 1986. A Wiley-Interscience Publication. MR 815107
  • [14] Werner C. Rheinboldt, On the computation of multidimensional solution manifolds of parametrized equations, Numer. Math. 53 (1988), no. 1-2, 165–181. MR 946374, 10.1007/BF01395883
  • [15] Werner C. Rheinboldt, On the existence and uniqueness of solutions of nonlinear semi-implicit differential-algebraic equations, Nonlinear Anal. 16 (1991), no. 7-8, 647–661. MR 1097322, 10.1016/0362-546X(91)90172-W
  • [16] G. W. Stewart, Introduction to matrix computations, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Computer Science and Applied Mathematics. MR 0458818
  • [17] Floris Takens, Constrained equations; a study of implicit differential equations and their discontinuous solutions, Mathematisch Institut, Rijksuniversiteit, Groningen, 1975. Report ZW-75-03. MR 0478236

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Article copyright: © Copyright 1994 American Mathematical Society