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), 491516
MSC:
Primary 65L05; Secondary 65L07
MathSciNet review:
829621
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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
BenIsrael and Thomas
N. E. Greville, Generalized inverses: theory and applications,
WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1974.
Pure and Applied Mathematics. MR 0396607
(53 #469)
 [2]
F. H. Branin, Jr., "Computer methods of network analysis," Proc. IEEE, v. 55, 1967, pp. 17871801.
 [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 DifferentialAlgebraic Systems with Applications in Trajectory Control, Ph. D. Thesis, University of California at Los Angeles, 1983.
 [5]
D. A. Calahan, ComputerAided Network Design, rev. ed., McGrawHill, 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
(86e:65091), http://dx.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), NorthHolland, Amsterdam, 1986,
pp. 313–318. MR 924216
(88k:93066)
 [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
(81g:34003)
 [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
(85a:34056), http://dx.doi.org/10.1137/0604051
 [10]
Richard
W. Cottle, Manifestations of the Schur complement, Linear
Algebra and Appl. 8 (1974), 189–211. MR 0354727
(50 #7204)
 [11]
R.
Courant and K.
O. Friedrichs, Supersonic Flow and Shock Waves, Interscience
Publishers, Inc., New York, N. Y., 1948. MR 0029615
(10,637c)
 [12]
J.
Dieudonné, Foundations of modern analysis, Academic
Press, New YorkLondon, 1969. Enlarged and corrected printing; Pure and
Applied Mathematics, Vol. 10I. MR 0349288
(50 #1782)
 [13]
C. W. Gear, "Simultaneous numerical solution of differential/algebraic equations," IEEE Trans. Circuit Theory, CT18, 1971, pp. 8995.
 [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 (85i:65112), http://dx.doi.org/10.1137/0721048
 [15]
P. M. Gresho, R. L. Lee & R. L. Sani, "On the timedependent solution of the incompressible NavierStokes 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 TimeDependent, Incompressible NavierStokes Equations in ThreeDimensions by (Almost) the FEM, UCRL87445, 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, CT18, 1971, pp. 101113.
 [18]
Peter
Henrici, Error propagation for difference method, John Wiley
and Sons, Inc., New YorkLondon, 1963. MR 0154416
(27 #4365)
 [19]
E. S. Kuh & R. A. Rohrer, "The statevariable approach to network analysis," Proc. IEEE, v. 53, 1965, pp. 672686.
 [20]
W. Liniger, "Multistep and oneleg methods for implicit mixed differential algebraic systems," IEEE Trans. Circuits and Systems, CAS26, 1979, pp. 755762.
 [21]
Per
Lötstedt, Mechanical systems of rigid bodies subject to
unilateral constraints, SIAM J. Appl. Math. 42
(1982), no. 2, 281–296. MR 650224
(83i:49020), http://dx.doi.org/10.1137/0142022
 [22]
P. Lötstedt & L. R. Petzold, Numerical Solution of Nonlinear Differential Equations with Algebraic Constraints, SAND 838877, Sandia National Laboratories, Livermore, California, 1983.
 [23]
J. F. Painter, Solving the NavierStokes Equations with LSODI and the Method of Lines, Report UCID19262, Lawrence Livermore National Laboratory, Livermore, California, 1981.
 [24]
B. Paul, "Analytical dynamics of mechanismsA computer oriented overview," Mech. Mach. Theory, v. 10, 1975, pp. 481507.
 [25]
Linda
Petzold, Differential/algebraic equations are not ODEs, SIAM
J. Sci. Statist. Comput. 3 (1982), no. 3,
367–384. MR
667834 (83i:65066), http://dx.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 (87j:65079), http://dx.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 LargeScale 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
(57 #11227)
 [1]
 A. BenIsrael and T. N. E. Greville, Generalized Inverses: Theory and Applications. Wiley, New York, 1974. MR 0396607 (53:469)
 [2]
 F. H. Branin, Jr., "Computer methods of network analysis," Proc. IEEE, v. 55, 1967, pp. 17871801.
 [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 DifferentialAlgebraic Systems with Applications in Trajectory Control, Ph. D. Thesis, University of California at Los Angeles, 1983.
 [5]
 D. A. Calahan, ComputerAided Network Design, rev. ed., McGrawHill, New York, 1972.
 [6]
 S. L. Campbell, "The numerical solution of higher index linear time varying singular systems of differential equations," SIAM J. Sci. Statist. Comput., v. 6, 1985, pp. 334348. MR 779409 (86e:65091)
 [7]
 S. L. Campbell, Explicit Methods for Solving Singular Differential Equation Systems, North Carolina State University, Raleigh, North Carolina, 1984. (Preprint.) MR 924216 (88k:93066)
 [8]
 S. L. Campbell, Singular Systems of Differential Equations, Pitman, San Francisco, 1979. MR 569589 (81g:34003)
 [9]
 S. L. Campbell & L. R. Petzold, "Canonical forms and solvable singular systems of differential equations," SIAM J. Algebraic Discrete Methods, v. 4, 1983, pp. 517521. MR 721621 (85a:34056)
 [10]
 R. W. Cottle, "Manifestations of the Schur complement," Linear Algebra Appl., v. 8, 1974, pp. 189211. MR 0354727 (50:7204)
 [11]
 R. Courant & K. O. Friedrichs, Supersonic Flow and Shock Waves, SpringerVerlag, Berlin and New York, 1948. MR 0029615 (10:637c)
 [12]
 J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1969. MR 0349288 (50:1782)
 [13]
 C. W. Gear, "Simultaneous numerical solution of differential/algebraic equations," IEEE Trans. Circuit Theory, CT18, 1971, pp. 8995.
 [14]
 C. W. Gear & L. R. Petzold, "ODE methods for the solution of differential/algebraic systems," SIAM J. Numer. Anal., v. 21, no. 4, 1984, pp. 367384. MR 749366 (85i:65112)
 [15]
 P. M. Gresho, R. L. Lee & R. L. Sani, "On the timedependent solution of the incompressible NavierStokes 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 TimeDependent, Incompressible NavierStokes Equations in ThreeDimensions by (Almost) the FEM, UCRL87445, 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, CT18, 1971, pp. 101113.
 [18]
 P. Henrici, Error Propagation for Difference Methods, Wiley, New York, 1963. MR 0154416 (27:4365)
 [19]
 E. S. Kuh & R. A. Rohrer, "The statevariable approach to network analysis," Proc. IEEE, v. 53, 1965, pp. 672686.
 [20]
 W. Liniger, "Multistep and oneleg methods for implicit mixed differential algebraic systems," IEEE Trans. Circuits and Systems, CAS26, 1979, pp. 755762.
 [21]
 P. Lötstedt, "Mechanical systems of rigid bodies subject to unilateral constraints," SIAM J. Appl. Math., v. 42, 1982, pp. 281296. MR 650224 (83i:49020)
 [22]
 P. Lötstedt & L. R. Petzold, Numerical Solution of Nonlinear Differential Equations with Algebraic Constraints, SAND 838877, Sandia National Laboratories, Livermore, California, 1983.
 [23]
 J. F. Painter, Solving the NavierStokes Equations with LSODI and the Method of Lines, Report UCID19262, Lawrence Livermore National Laboratory, Livermore, California, 1981.
 [24]
 B. Paul, "Analytical dynamics of mechanismsA computer oriented overview," Mech. Mach. Theory, v. 10, 1975, pp. 481507.
 [25]
 L. Petzold, "Differential/algebraic equations are not ODEs," SIAM J. Sci. Statist. Comput., v. 3, no. 3, 1982, pp. 367384. MR 667834 (83i:65066)
 [26]
 L. R. Petzold & P. Lötstedt, "Numerical solution of nonlinear differential equations with algebraic constraints II: Practical implications," SIAM J. Sci. Statist. Comput. (To appear.) MR 848560 (87j:65079)
 [27]
 R. F. Sincovec, B. Dembart, M. A. Epton, A. M. Erisman, J. W. Manke & E. L. Yip, Solvability of LargeScale Descriptor Systems, Report, Boeing Computer Services Company, Seattle, Washington, 1979.
 [28]
 J. Wittenburg, Dynamics of Systems of Rigid Bodies, Teubner, Stuttgart, 1977. MR 0471497 (57:11227)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65L05,
65L07
Retrieve articles in all journals
with MSC:
65L05,
65L07
Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819860829621X
PII:
S 00255718(1986)0829621X
Article copyright:
© Copyright 1986
American Mathematical Society
