Solving Differential-Algebraic Equations by Taylor Series (I): Computing Taylor Coefficients

This paper is one of a series underpinning the authors’ DAETS code for solving DAE initial value problems by Taylor series expansion. First, building on the second author’s structural analysis of DAEs (BIT, 41 (2001), pp. 364–394), it describes and justifies the method used in DAETS to compute Taylor coefficients (TCs) using automatic differentiation. The DAE may be fully implicit, nonlinear, and contain derivatives of order higher than one. Algorithmic details are given.Second, it proves that either the method succeeds in the sense of computing TCs of the local solution, or one of a number of detectable error conditions occurs.

[1]  David Barton,et al.  The Automatic Solution of Systems of Ordinary Differential Equations by the Method of Taylor Series , 1971, Computer/law journal.

[2]  Louis B. Rall,et al.  Automatic Differentiation: Techniques and Applications , 1981, Lecture Notes in Computer Science.

[3]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

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

[5]  G. Corliss,et al.  ATOMFT: solving ODEs and DAEs using Taylor series , 1994 .

[6]  C. W. Gear,et al.  The index of general nonlinear DAEs , 1995 .

[7]  Andreas Griewank,et al.  Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++ , 1996, TOMS.

[8]  J.J.B. de Swart,et al.  PSIDE: Parallel software for implicit differential equations , 1997 .

[9]  C. Bendtsen,et al.  TADIFF , A FLEXIBLE C + + PACKAGE FOR AUTOMATIC DIFFERENTIATION using Taylor series expansion , 1997 .

[10]  Nedialko S. Nedialkov,et al.  An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation , 1998, SCAN.

[11]  Nedialko S. Nedialkov,et al.  Validated solutions of initial value problems for ordinary differential equations , 1999, Appl. Math. Comput..

[12]  N. Nedialkov,et al.  Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation , 1999 .

[13]  Francesca Mazzia,et al.  Block-Boundary Value Methods for the Solution of Ordinary Differential Equations , 1999, SIAM J. Sci. Comput..

[14]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[15]  J. Pryce.  A Simple Structural Analysis Method for DAEs , 2001 .

[16]  Nedialko S. Nedialkov,et al.  Some recent advances in validated methods for IVPs for ODEs , 2002 .

[17]  Alexander Gofen.  The Taylor Center for PCs: Exploring, Graphing and Integrating ODEs with the Ultimate Accuracy , 2002, International Conference on Computational Science.

[18]  Francesca Mazzia,et al.  Test Set for Initial Value Problem Solvers , 2003 .

[19]  J. D. Pryce,et al.  Solving high-index DAEs by Taylor series , 1998, Numerical Algorithms.

[20]  Angel Jorba,et al.  A Software Package for the Numerical Integration of ODEs by Means of High-Order Taylor Methods , 2005, Exp. Math..

[21]  N. Nedialkov,et al.  Solving differential algebraic equations by Taylor Series(III): the DAETS Code , 2005 .

[22]  A. Volgenant,et al.  A shortest augmenting path algorithm for dense and sparse linear assignment problems , 1987, Computing.

[23]  Roberto Barrio,et al.  Performance of the Taylor series method for ODEs/DAEs , 2005, Appl. Math. Comput..

[24]  Roberto Barrio,et al.  VSVO formulation of the taylor method for the numerical solution of ODEs , 2005 .

[25]  N. Nedialkov,et al.  Solving differential algebraic equations by Taylor Series(III): the DAETS Code , 2008 .