Numerical solution of nonlinear differential equations with algebraic constraints. I. Convergence results for backward differentiation formulas

Authors:
Per Lötstedt and Linda Petzold

Journal:
Math. Comp. **46** (1986), 491-516

MSC:
Primary 65L05; Secondary 65L07

DOI:
https://doi.org/10.1090/S0025-5718-1986-0829621-X

MathSciNet review:
829621

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Abstract: In this paper we investigate the behavior of numerical ODE methods for the solution of systems of differential equations coupled with algebraic constraints. Systems of this form arise frequently in the modelling of problems from physics and engineering; we study some particular examples from electrical networks, fluid dynamics and constrained mechanical systems. We show that backward differentiation formulas converge with the expected order of accuracy for these systems.

**[1]**Adi Ben-Israel and Thomas N. E. Greville,*Generalized inverses: theory and applications*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0396607****[2]**F. H. Branin, Jr., "Computer methods of network analysis,"*Proc. IEE-E*, v. 55, 1967, pp. 1787-1801.**[3]**F. H. Branin, Jr., "The network concept as a unifying principle in engineering and the physical sciences," in*Problem Analysis in Science and Engineering*(F. H. Branin, Jr. and K. Huseyin, eds.), Academic Press, New York, 1977.**[4]**K. Brenan,*Stability and Convergence of Difference Approximations for Higher Index Differential-Algebraic Systems with Applications in Trajectory Control*, Ph. D. Thesis, University of California at Los Angeles, 1983.**[5]**D. A. Calahan,*Computer-Aided Network Design*, rev. ed., McGraw-Hill, New York, 1972.**[6]**Stephen L. Campbell,*The numerical solution of higher index linear time varying singular systems of differential equations*, SIAM J. Sci. Statist. Comput.**6**(1985), no. 2, 334–348. MR**779409**, https://doi.org/10.1137/0906024**[7]**Stephen L. Campbell,*Consistent initial conditions for linear time varying singular systems*, Frequency domain and state space methods for linear systems (Stockholm, 1985) North-Holland, Amsterdam, 1986, pp. 313–318. MR**924216****[8]**S. L. Campbell,*Singular systems of differential equations*, Research Notes in Mathematics, vol. 40, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. MR**569589****[9]**Stephen L. Campbell and Linda R. Petzold,*Canonical forms and solvable singular systems of differential equations*, SIAM J. Algebraic Discrete Methods**4**(1983), no. 4, 517–521. MR**721621**, https://doi.org/10.1137/0604051**[10]**Richard W. Cottle,*Manifestations of the Schur complement*, Linear Algebra and Appl.**8**(1974), 189–211. MR**0354727****[11]**R. Courant and K. O. Friedrichs,*Supersonic Flow and Shock Waves*, Interscience Publishers, Inc., New York, N. Y., 1948. MR**0029615****[12]**J. Dieudonné,*Foundations of modern analysis*, Academic Press, New York-London, 1969. Enlarged and corrected printing; Pure and Applied Mathematics, Vol. 10-I. MR**0349288****[13]**C. W. Gear, "Simultaneous numerical solution of differential/algebraic equations,"*IEEE Trans. Circuit Theory*, CT-18, 1971, pp. 89-95.**[14]**C. W. Gear and L. R. Petzold,*ODE methods for the solution of differential/algebraic systems*, SIAM J. Numer. Anal.**21**(1984), no. 4, 716–728. MR**749366**, https://doi.org/10.1137/0721048**[15]**P. M. Gresho, R. L. Lee & R. L. Sani, "On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions,"*Recent Advances in Numerical Methods in Fluids*, vol. 1, Pineridge, Swansea, 1980.**[16]**P. M. Gresho & C. D. Upson,*Current Progress in Solving the Time-Dependent, Incompressible Navier-Stokes Equations in Three-Dimensions by*(*Almost*)*the FEM*, UCRL-87445, Lawrence Livermore National Laboratory, Livermore, California, 1982.**[17]**G. D. Hachtel, R. K. Brayton & F. G. Gustavson, "The sparse tableau approach to network analysis and deisgn,"*IEEE Trans. Circuit Theory*, CT-18, 1971, pp. 101-113.**[18]**Peter Henrici,*Error propagation for difference method*, John Wiley and Sons, Inc., New York-London, 1963. MR**0154416****[19]**E. S. Kuh & R. A. Rohrer, "The state-variable approach to network analysis,"*Proc. IEE-E*, v. 53, 1965, pp. 672-686.**[20]**W. Liniger, "Multistep and one-leg methods for implicit mixed differential algebraic systems,"*IEEE Trans. Circuits and Systems*, CAS-26, 1979, pp. 755-762.**[21]**Per Lötstedt,*Mechanical systems of rigid bodies subject to unilateral constraints*, SIAM J. Appl. Math.**42**(1982), no. 2, 281–296. MR**650224**, https://doi.org/10.1137/0142022**[22]**P. Lötstedt & L. R. Petzold,*Numerical Solution of Nonlinear Differential Equations with Algebraic Constraints*, SAND 83-8877, Sandia National Laboratories, Livermore, California, 1983.**[23]**J. F. Painter,*Solving the Navier-Stokes Equations with LSODI and the Method of Lines*, Report UCID-19262, Lawrence Livermore National Laboratory, Livermore, California, 1981.**[24]**B. Paul, "Analytical dynamics of mechanisms--A computer oriented overview,"*Mech. Mach. Theory*, v. 10, 1975, pp. 481-507.**[25]**Linda Petzold,*Differential/algebraic equations are not ODEs*, SIAM J. Sci. Statist. Comput.**3**(1982), no. 3, 367–384. MR**667834**, https://doi.org/10.1137/0903023**[26]**Linda Petzold and Per Lötstedt,*Numerical solution of nonlinear differential equations with algebraic constraints. II. Practical implications*, SIAM J. Sci. Statist. Comput.**7**(1986), no. 3, 720–733. MR**848560**, https://doi.org/10.1137/0907049**[27]**R. F. Sincovec, B. Dembart, M. A. Epton, A. M. Erisman, J. W. Manke & E. L. Yip,*Solvability of Large-Scale Descriptor Systems*, Report, Boeing Computer Services Company, Seattle, Washington, 1979.**[28]**Jens Wittenburg,*Dynamics of systems of rigid bodies*, B. G. Teubner, Stuttgart, 1977. Leitfäden der angewandten Mathematik und Mechanik, Band 33. MR**0471497**

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0829621-X

Article copyright:
© Copyright 1986
American Mathematical Society