Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the computation of impasse points of quasilinear differential-algebraic equations
HTML articles powered by AMS MathViewer

by Patrick J. Rabier and Werner C. Rheinboldt PDF
Math. Comp. 62 (1994), 133-154 Request permission

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.
References
  • 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, DOI 10.1007/978-1-4612-1029-0
  • Richard P. Brent, Some efficient algorithms for solving systems of nonlinear equations, SIAM J. Numer. Anal. 10 (1973), 327–344. MR 331764, DOI 10.1137/0710031
  • 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
  • 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, DOI 10.1016/0021-9991(87)90001-5
    • L. O. Chua, Introduction to nonlinear networks, McGraw-Hill, New York, 1969.
  • 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, DOI 10.1002/cta.4490170207
  • 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
    • 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.
  • Patrick J. Rabier, Implicit differential equations near a singular point, J. Math. Anal. Appl. 144 (1989), no. 2, 425–449. MR 1027045, DOI 10.1016/0022-247X(89)90344-2
  • 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
    • —, 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). —, 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).
  • Werner C. Rheinboldt, Numerical analysis of parametrized nonlinear equations, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 7, John Wiley & Sons, Inc., New York, 1986. A Wiley-Interscience Publication. MR 815107
  • Werner C. Rheinboldt, On the computation of multidimensional solution manifolds of parametrized equations, Numer. Math. 53 (1988), no. 1-2, 165–181. MR 946374, DOI 10.1007/BF01395883
  • 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, DOI 10.1016/0362-546X(91)90172-W
  • G. W. Stewart, Introduction to matrix computations, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0458818
  • Floris Takens, Constrained equations; a study of implicit differential equations and their discontinuous solutions, Rijksuniversiteit te Groningen, Mathematisch Instituut, Groningen, 1975. Report ZW-75-03. MR 0478236
Similar Articles
Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Math. Comp. 62 (1994), 133-154
  • MSC: Primary 65L05; Secondary 34A09, 34A47, 58F14
  • DOI: https://doi.org/10.1090/S0025-5718-1994-1208224-6
  • MathSciNet review: 1208224