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Partial Differential Equations in Classical Mathematical Physics (I. Rubinstein and L. Rubinstein)

codes is discussed. Significant emphasis is given to the choice of error tolerances, types of solution output, user interfaces, and internal (to the code) storage management. Chapter 4 shows the derivation of one-step and multistep methods from first principles. For onestep methods, in addition to deriving formulae by integration and from the equations of condition, there is a straightforward discussion of numerical stability and an introduction to providing interpolants for dense output. In two brief subsections, the Adams and Backward Differentiation Formulae are derived by interpolation. Then, predictor-corrector methods, their accuracy, and their stability are discussed. Finally, the evaluation of implicit methods and the advantages of high order methods are considered. The emphasis of Chapter 5 is on the development of convergence and stability analyses useful in the context of the popular codes. It begins with an analysis of convergence and stability of one-step methods. The techniques developed there are extended to a convergence analysis of explicit multistep methods with constant step size, then to Adams methods (alone or in predictor-corrector mode) with bounded step size sequences, and finally to more general predictor-corrector multistep methods with constant step size. This is followed by a brief discussion of the requirements when starting a method with memory, then of stability of multistep methods all leading to a fixed step size convergence analysis of the backward differentiation formulas. Next, a proof of convergence under a quasi-constant step size assumption is given. This leads into practical considerations of interpolation, of varying order, and of the impact of rounding errors. Mainly with the solution of stiff problems in mind, the next chapter discusses stability when the step size is large. A long section analyzes a variety of methods using the linear system yP Jy + g, then moves on to a discussion of how instability is manifested during numerical integration and its interrelationship with local error control. This leads into a chapter on local error estimation and control. An analysis of the asymptotic behavior of local and global errors for one-step and multistep formulas is followed by a practical discussion of error control and step size selection. There is an extended discussion of error estimation for methods with and without memory and of how its consideration has impacted the choice of Runge-Kutta methods. Finally, there is a discussion of how to choose an "on scale" initial step size using error estimation. A final major chapter discusses the various factors which determine whether a problem is stiff. This is a very practical view which relies more on observation of the behavior and cost of methods used to solve a problem than on a formulistic approach to determining stiffness. Certainly, analyzed by this approach, many problems labeled "stiff" in the literature do not fall into that category. The discussion here is strongly supported by practical examples. There follows a discussion of methods for stiff problems and illustrations of them in use. The whole text is outstanding for its attention to practicality both in the discussion of problems and in the analysis and use of methods and codes. To add to the practical emphasis, the book finishes with an analysis of twelve real problems and with a self-test on the book as a whole. Overall, this book is a resounding success; it fills a void in the numerical ordinary differential equations literature, and it compliments the many excellent, more theoretical texts in this area.