Solving differential algebraic equations by Taylor Series(III): the DAETS Code

The authors have developed a Taylor series method for solving numerically an initial-value problem differential-algebraic equation (DAE) that can be of high index, high order, nonlinear, and fully implicit, BIT, 45 (2005), pp. 561–592. Numerical results have shown that this method is efficient and very accurate. Moreover, it is particularly suitable for problems that are of too high an index for present DAE solvers.This paper develops an effective method for computing a DAE’s System Jacobian, which is needed in the structural analysis of the DAE and computation of Taylor coefficients. Our method involves preprocessing of the DAE and code generation employing automatic differentiation. Theory and algorithms for preprocessing and code generation are presented.An operator-overloading approach to computing the System Jacobian is also discussed.

[1]  N. S. Mendelsohn,et al.  Coverings of Bipartite Graphs , 1958, Canadian Journal of Mathematics.

[2]  L. Watson.  A globally convergent algorithm for computing fixed points of C2 maps , 1979 .

[3]  P. M. Hanson,et al.  Controlling the defect in existing variable-order Adams codes for initial-value problems , 1983, TOMS.

[4]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

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

[6]  Griewank,et al.  On automatic differentiation , 1988 .

[7]  Alex Pothen,et al.  Computing the block triangular form of a sparse matrix , 1990, TOMS.

[8]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[9]  Sven Erik Mattsson,et al.  Index Reduction in Differential-Algebraic Equations Using Dummy Derivatives , 1993, SIAM J. Sci. Comput..

[10]  U. Ascher,et al.  Stabilization of DAEs and invariant manifolds , 1994 .

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

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

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

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

[15]  Walter M. Lioen,et al.  Test set for initial value problem solvers , 1998 .

[16]  Antonio Elipe,et al.  Automatic programming of recurrent power series , 1999 .

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

[18]  Paul I. Barton,et al.  Differential-Algebraic Equations of Index 1 May Have an Arbitrarily High Structural Index , 1999, SIAM J. Sci. Comput..

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

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

[21]  G. Reid,et al.  Differential Elimination–Completion Algorithms for DAE and PDAE , 2001 .

[22]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[23]  Peter Fritzson,et al.  Methods for Structural Analysis and Debugging of Modelica Models , 2002 .

[24]  Fixed points of C 2 maps , 2003 .

[25]  Andreas Griewank,et al.  On the Efficient Generation of Taylor Expansions for DAE Solutions by Automatic Differentiation , 2004, PARA.

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

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

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

[29]  N. Nedialkov,et al.  Solving Differential-Algebraic Equations by Taylor Series (I): Computing Taylor Coefficients , 2005 .

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

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

[32]  X. Yi.  On Automatic Differentiation , 2005 .