Progress on a general numerical method for nonlinear higher index DAEs II

A method has been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit Runge-Kutta may not be appropriate. This method is based on solving nonlinear DAE derivative arrays. This paper discusses progress on the implementation of this method, resolves some of the issues involved, and lists some remaining problems. Computational experience on two prescribed path control problems is presented showing that the approach should prove practical for many applications.

[1]  G. R. Walsh,et al.  Methods Of Optimization , 1976 .

[2]  J. Baumgarte.  Stabilization of constraints and integrals of motion in dynamical systems , 1972 .

[3]  P. Boggs.  The convergence of the Ben-Israel iteration for nonlinear least squares problems , 1976 .

[4]  C. D. Meyer,et al.  Generalized inverses of linear transformations , 1979 .

[5]  V. Klema.  LINPACK user's guide , 1980 .

[6]  E. Griepentrog,et al.  Differential-algebraic equations and their numerical treatment , 1986 .

[7]  Jack Dongarra,et al.  LINPACK Users' Guide , 1987 .

[8]  Stephen L. Campbell,et al.  A general form for solvable linear time varying singular systems of differential equations , 1987 .

[9]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[10]  C. W. Gear,et al.  Approximation methods for the consistent initialization of differential-algebraic equations , 1988 .

[11]  C. Pantelides.  The consistent intialization of differential-algebraic systems , 1988 .

[12]  Ernst Hairer,et al.  The numerical solution of differential-algebraic systems by Runge-Kutta methods , 1989 .

[13]  Florian A. Potra,et al.  Differential-Geometric Techniques for Solving Differential Algebraic Equations , 1990 .

[14]  S. Campbell,et al.  Differentiation of Constraints in Differential-Algebraic Equations∗ , 1991 .

[15]  W. Rheinboldt,et al.  A general existence and uniqueness theory for implicit differential-algebraic equations , 1991, Differential and Integral Equations.

[16]  Per Christian Hansen,et al.  Some Applications of the Rank Revealing QR Factorization , 1992, SIAM J. Sci. Comput..

[17]  Stephen L. Campbell,et al.  Uniqueness of completions for linear time varying differential algebraic equations , 1992 .

[18]  Richard J. Fateman,et al.  Automatic Differentiation of Algorithms: Theory, Implementation, and Application (Andreas Griewank and George F. Corliss, eds.) , 1993, SIAM Rev..

[19]  Stephen L. Campbell,et al.  Least squares completions for nonlinear differential algebraic equations , 1993 .

[20]  S. Campbell,et al.  Utilization of automatic differentiation in control algorithms , 1994, IEEE Trans. Autom. Control..

[21]  A. Griewank,et al.  Automatic differentiation of algorithms : theory, implementation, and application , 1994 .

[22]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.