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Numerical methods for ordinary differential systems: the initial value problem. (English) Zbl 0745.65049
Chichester etc.: John Wiley & Sons. x, 293 p. (1991).
This replacement for the author’s earlier monograph “Computational methods in ordinary differential equations” (1973; Zbl 0258.65069) provides a contemporary introduction to the subject at the senior undergraduate or beginning graduate level. While much of the previous material re-appears, primarily in the first half, the new book consits of a complete reordering and rewriting of these selected topics together with new material. A variety of novel methods described in the previous version have been omitted, and many of the original detailed examples are now relegated to exercises. Hence, the original will remain a useful reference volume.
Chapter 1 contains most of the background material required in later chapters. Definitions of stability, consistency and convergence in Chapter 2 span a broad class of methods, although there remains some emphasis on the notation suitable for linear multistep methods. [The author’s definition of total stability for systems of differential equations is a restricted form of that in the monographs of W. Hahn, Stability of motion (1967; Zbl 0189.385) and of H. J. Stetter, Analysis of discretization methods for ordinary differential equations (1973; Zbl 0276.65001)]. Since most methods rely substantially on linearity for the major estimation components, this beginning is appropriate. Chapter 3 presents the formal aspects involving linear multistep methods, including their representation both in standard and backward difference forms. Chapter 4 applies these aspects to predictor- corrector methods, and introduces other practical considerations including implementation, stepsize control and variable step, variable order strategies.
While the derivation of low order Runge-Kutta methods is motivated by the traditional Taylor’s series expansions, Chapter 5 includes a working introduction to Butcher’s derivation using rooted trees [and most of the Butcher’s particular notation for this treatment; cf. J. C. Butcher, The numerical analysis of ordinary differential equations. Runge-Kutta and general linear methods (1987; Zbl 0616.65072)]. The treatment includes well-known results and examples of efficient explicit and implicit methods. Inclusion of an alternative formulation of order conditions by P. Albrecht serves to identify the role of linearity in the derivation of such methods.
The final two chapters are devoted to developments in the theory of linear and non-linear stability, and their implications for stiff problems. After a detailed analysis to identify the nature of stiffness, Chapter 6 describes traditional concepts of \(A\)- and \(L\)-stability, which lead to an illuminating introduction to the elegant theory of order stars developed by G. Wanner, E. Hairer and S. P. Nørsett [BIT 18, 475-489 (1978; Zbl 0444.65039)]. Both linear multistep and Runge- Kutta methods appropriate for treating stiff systems are considered. In considering non-linear stability, Chapter 7 shows that dissipative systems may be treated adequately by either \(G\)-stable multistep or algebraically stable Runge-Kutta methods. In spite of this theory favouring methods of the latter type because high order can be achieved, the author observes that “most real-life stiff problems continue to be solved by highly-tuned codes based on BDF schemes, which give excellent results for the vast majority of problems.”
This monograph provides a good introduction to and treatment of the relevant theory of methods for initial value problems which form the basic algorithms of the currently available most efficient codes. The notation and treatment will prepare the diligent reader for the study of many recent monographs that treat these topics at a more rigorous and advanced level.

65L05 Numerical methods for initial value problems
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems, general
34A34 Nonlinear ordinary differential equations and systems, general theory
34E13 Multiple scale methods for ordinary differential equations