Cambridge Texts in Applied Mathematics. Cambridge: Cambridge Univ. Press. 400 p. £17.95; $ 27.95/pbk; £55.00; $ 74.95/hbk (1995).
There exist few textbooks only which are exclusively dedicated to the numerical solution of ordinary and partial differential equations although this field is of significant importance for courses on numerical analysis. The present book gives a rigorous account of the respective fundamentals. In the exposition it strives to maintain a balance between theoretical, algorithmic and applied aspects of the subject. This is not quite an easy task but the author, a specialist in the field and an experienced teacher excellently realizes the forementioned aims.
The book covers a broad range of material. The first 100 pages are dedicated to the numerical solution of ordinary differential equations including explicit and implicit Runge-Kutta methods, multistep methods including error control devices, solution of stiff equations and iteration for solving nonlinear algebraic systems. The next 160 pages are used to present finite difference and finite element methods for discretizing the Poisson equation and a variety of algorithms for solving the resulting large algebraic systems as Gaussian elimination for banded systems, iterative methods (the alternating directions implicit method and, unfortunately, the conjugate gradient method in the form of a remark only), multigrid techniques, fast Poisson solvers (Hockney method, fast Fourier transform, and, as a remark, odd-even reduction). Evolution type equations (parabolic and hyperbolic) are considered the next 80 pages which are followed by an appendix containing fundamentals in linear algebra, interpolation and quadrature and ordinary differential equations (20 pages). Each chapter is concluded with useful comments and a bibliography as well as a collection of exercises.
This book can be highly recommended as a basis for courses in numerical analysis. Since the emphasis does not lie in presenting deeper mathematical proofs but in providing mainly the unavoidable mathematics for a thorough understanding of the numerical methods it is equally well suited for students in science and engineering. The price for the paperback edition seems to be reasonably calculated also for the normally smaller budget of students.