##
**Numerical solution of initial-value problems in differential-algebraic equations.**
*(English)*
Zbl 0699.65057

New York etc.: North-Holland. viii, 210 p. $ 44.95; Dfl. 110.00 (1989).

Many models of Applied Sciences, when described in their natural variables, are governed by implicit systems of ordinary differential equations \(F(t,y,y')=0\), where the Jacobian of F with respect to \(y'\) is singular. Equations of this type are usually referred to as differential- algebraic equations (DAE’s). The book under review is devoted to present the current status of the numerical solution of DAE’s.

The stated purpose of this monograph is “to bring together in a unified way the developments in the understanding of the mathematical structure of DAE’s, the analysis of numerical methods applied to DAE systems, the development of robust and efficient mathematical software and the formulation of DAE systems arising from problems in science and engineering”. This is indeed an ambiguous project, but in this book it is carried out with great success.

A couple of nice books on numerical solution of DAE’s have been published in the last few years but they are concerned with more restricted features of DAE’s. Thus, in the book of E. Griepentrog and R. März [Differential-algebraic equations and their numerical treatment (1986; Zbl 0629.65080)] the mathematical theory and the analysis of linear methods for DAE’s are studied in detail. On the other hand the book of E. Hairer, Ch. Lubich and M. Roche [The numerical solution of differential-algebraic systems by Runge-Kutta methods (1989; Zbl 0683.65050)] is devoted to the class of Runge-Kutta methods. In the book under review a general and up to day presentation of the state of the numerical solution of DAE’s has been attempted in two hundred pages. Clearly with this space constraint it is impossible to give a detailed account of all research on DAE’s, however due to the wise experience of the authors in DAE’s, they have been able to give a comprehensive account of the most relevant results in this field.

The book begins with an introductory Chapter in which several examples and applications are presented to show that some models are naturally formulated by using DAE’s. Moreover the basic types of DAE’s that will be considered in this book are introduced.

Chapter two deals with the mathematical theory of DAE’s. Several structural types which arise in the applications have been considered. First, the well understood theory of the linear constant coefficient DAE’s is recalled. Next, the linear time varying DAE’s, as intermediate step to some non linear types of DAE’s, are studied. On non linear DAE’s only those which can be written in Hessenberg form are considered. The concepts of solvability and index are defined for the above types of DAE’s.

Chapter three handles the linear multistep methods. Starting with an historical perspective to motivate why the backward differentiation methods (BDF) were proposed by C. W. Gear [Numerical initial value problems in ordinary differential equations (1971; Zbl 1145.65316)] to solve DAE’s, the authors study the convergence of BDF methods for several classes of DAE’s (Semi-Explicit Index One, Fully-Implicit Index One, Semi-Explicit Index Two and Hessenberg Index Three systems). Finally some convergence results for general linear multistep and one-leg methods are given.

In Chapter four one step methods, specifically implicit Runge-Kutta (IRK) and extrapolation methods, are treated. First the numerical analysis of general IRK methods when applied to linear constant coefficient DAE’s is considered. In this case a fairly complete theory of stability and convergence can be given. Then, more general DAE’s of index one and two are studied. Since recent researches point out a close relationship between stiff systems and DAE’s, a section to link the stiff order and DAE order of some IRK is included. The Chapter ends with a brief introduction to the extrapolation methods developed by P. Deuflhard, E. Hairer and J. Zugck [Numer. Math. 51, 501-516 (1987; Zbl 0635.65083)].

Chapter five is mainly devoted to describe the code DASSL designed by one of the authors (Petzold) to solve index one DAE’s. This code is based on variable stepsize variable order BDF formulas with order \(\leq 5\), using the fixed leading coefficient technique for stepsize changing by K. R. Jackson and R. Sacks-Davis [ACM Trans. Math. Software 6, 295- 318 (1980; Zbl 0434.65046)]. The algorithms and strategies that are relevant in DASSL are analyzed. It must be remarked that for anyone interested in using DASSL, the reading of this Chapter is highly recommended, since it can be very helpful to understand why some problems can not be successfully solved and to modify the code.

Finally, chapter six discusses several issues important to the numerical solution of some DAE’s which arise in applications. By means of some examples it is shown that problem formulation as well as the computation of consistent initial conditions have a strong influence on the performance of a numerical method.

In conclusion, the book under review gives a good insight into the numerical solution of DAE’s, collecting very recent results of the authors. It is highly recommended to any researcher with some experience in the numerical solution of ODE’s. Altogether, this monograph is an important contribution to the literature.

The stated purpose of this monograph is “to bring together in a unified way the developments in the understanding of the mathematical structure of DAE’s, the analysis of numerical methods applied to DAE systems, the development of robust and efficient mathematical software and the formulation of DAE systems arising from problems in science and engineering”. This is indeed an ambiguous project, but in this book it is carried out with great success.

A couple of nice books on numerical solution of DAE’s have been published in the last few years but they are concerned with more restricted features of DAE’s. Thus, in the book of E. Griepentrog and R. März [Differential-algebraic equations and their numerical treatment (1986; Zbl 0629.65080)] the mathematical theory and the analysis of linear methods for DAE’s are studied in detail. On the other hand the book of E. Hairer, Ch. Lubich and M. Roche [The numerical solution of differential-algebraic systems by Runge-Kutta methods (1989; Zbl 0683.65050)] is devoted to the class of Runge-Kutta methods. In the book under review a general and up to day presentation of the state of the numerical solution of DAE’s has been attempted in two hundred pages. Clearly with this space constraint it is impossible to give a detailed account of all research on DAE’s, however due to the wise experience of the authors in DAE’s, they have been able to give a comprehensive account of the most relevant results in this field.

The book begins with an introductory Chapter in which several examples and applications are presented to show that some models are naturally formulated by using DAE’s. Moreover the basic types of DAE’s that will be considered in this book are introduced.

Chapter two deals with the mathematical theory of DAE’s. Several structural types which arise in the applications have been considered. First, the well understood theory of the linear constant coefficient DAE’s is recalled. Next, the linear time varying DAE’s, as intermediate step to some non linear types of DAE’s, are studied. On non linear DAE’s only those which can be written in Hessenberg form are considered. The concepts of solvability and index are defined for the above types of DAE’s.

Chapter three handles the linear multistep methods. Starting with an historical perspective to motivate why the backward differentiation methods (BDF) were proposed by C. W. Gear [Numerical initial value problems in ordinary differential equations (1971; Zbl 1145.65316)] to solve DAE’s, the authors study the convergence of BDF methods for several classes of DAE’s (Semi-Explicit Index One, Fully-Implicit Index One, Semi-Explicit Index Two and Hessenberg Index Three systems). Finally some convergence results for general linear multistep and one-leg methods are given.

In Chapter four one step methods, specifically implicit Runge-Kutta (IRK) and extrapolation methods, are treated. First the numerical analysis of general IRK methods when applied to linear constant coefficient DAE’s is considered. In this case a fairly complete theory of stability and convergence can be given. Then, more general DAE’s of index one and two are studied. Since recent researches point out a close relationship between stiff systems and DAE’s, a section to link the stiff order and DAE order of some IRK is included. The Chapter ends with a brief introduction to the extrapolation methods developed by P. Deuflhard, E. Hairer and J. Zugck [Numer. Math. 51, 501-516 (1987; Zbl 0635.65083)].

Chapter five is mainly devoted to describe the code DASSL designed by one of the authors (Petzold) to solve index one DAE’s. This code is based on variable stepsize variable order BDF formulas with order \(\leq 5\), using the fixed leading coefficient technique for stepsize changing by K. R. Jackson and R. Sacks-Davis [ACM Trans. Math. Software 6, 295- 318 (1980; Zbl 0434.65046)]. The algorithms and strategies that are relevant in DASSL are analyzed. It must be remarked that for anyone interested in using DASSL, the reading of this Chapter is highly recommended, since it can be very helpful to understand why some problems can not be successfully solved and to modify the code.

Finally, chapter six discusses several issues important to the numerical solution of some DAE’s which arise in applications. By means of some examples it is shown that problem formulation as well as the computation of consistent initial conditions have a strong influence on the performance of a numerical method.

In conclusion, the book under review gives a good insight into the numerical solution of DAE’s, collecting very recent results of the authors. It is highly recommended to any researcher with some experience in the numerical solution of ODE’s. Altogether, this monograph is an important contribution to the literature.

Reviewer: M.Calvo

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

34-04 | Software, source code, etc. for problems pertaining to ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

65J99 | Numerical analysis in abstract spaces |