CWI Monographs, 2. Amsterdam - New York - Oxford: North-Holland. IX, 307 p. $ 36.50; Dfl. 95.00 (1984).
In the analysis of numerical schemes for ordinary differential equations one has traditionally assumed that the differential system is Lipschitz continuous. However, for stiff problems it is more appropriate to assume that the system satisfies some one sided Lipschitz condition. Based on this assumption G. Dahlquist introduced in 1975 the concept of G- stability for linear multistep methods. In the same year, J. Butcher published the corresponding definition of B-stability for Runge-Kutta schemes. The book gives a detailed and thorough report of the development of the nonlinear stability analysis of Runge-Kutta schemes from Butcher’s B-stability definition up to Hairer and Türke’s equivalence of A- stability and B-stability. The book is lucidly written. The authors have succeeded almost completely to melt the different research papers to a uniform theory. However, on some occasions one would have liked a weeding out of some unimportant concepts such as for example AN-stability. The authors have stayed consequently within the title of the book. Hence, the beautiful rooted trees theory for the order conditions has been omitted while the linear absolute stability theory is kept to its bare minimum. This makes the book to specialized for a general course on Runge-Kutta schemes. However, it is definitely well suited for a special course or seminar on this somewhat limited topic. It is a very good book for any researcher in this field since it collects practically all relevant work done since 1975 in one volume. The subject is devided into ten chapters.
In the first two introductory chapters the phenomena of stiffness are explained and several interesting examples are given. The concepts of absolute stability for the linear equation, contractivity and the one- sided Lipschitz condition are introduced and some results and examples are given for illustration. In addition, the logarithmic norm is treated. In chapter 3 the Runge-Kutta schemes are introduced along with the simplifying order conditions. The methods based on high order quadrature and some of their linear stability properties are given. Then results concerning the order of diagonally- and singly-implicit methods are given. The beautiful order conditions using the W-transform of Hairer and Wanner are used to treat Runge-Kutta schemes with positive weights. In Chapter 4 relations between the different stability concepts such as B-, BN-, A-, AN-stability and algebraic stability are given. To do this the reducibility of a scheme is introduced. The chapter ends with a more detailed investigation on algebraically stable schemes. Chapter 5 deals with the solution of the nonlinear equation which one has to solve at each integration step if one uses implicit schemes. Existence and uniqueness of the solution is investigated. In addition, using BSI- stability, of Frank, Schneid and Ueberhuber the error of the numerical solution is bounded by internal perturbations. Then many known implicit schemes are investigated with respect to their BSI-stability. The chapter ends with some generalizations of B-stability and implementation consideration for implicit Runge-Kutta methods. In the short chapter 6 contractivity of explicit schemes is treated. In chapter 7 B-consistency and B-convergence are introduced to obtain realistic error bounds for the numerical solution in the transient phase of a stiff problem. Then sufficient conditions for these concepts are given. While in the next chapter D-stability is introduced, chapter 10 investigates Runge-Kutta Rosenbrock methods and their stability properties. In the last chapter applications to partial differential equations are considered. In particular a pseudo-linear parabolic problem, the linear advection equation, diffusion-convection problems and the shallow water equations are treated.