Numerical Solution of Partial Differential Equations, by K. W. Morton and D. F. Mayers, is one of the best introductory books on the finite difference method available. The book contains reasonably complete coverage of the finite difference approach to solving parabolic, hyperbolic, and elliptic partial differential equations along with a very brief introduction to the finite element method for elliptic problems. I think the authors balance the amount of material covered with what is important to cover in a first course very well. They have added enough recent material (multigrid methods, symplectic integration schemes) to keep the book fresh without abandoning the strengths of the previous edition.
This book only has 278 pages and it doesn't pretend to be a complete treatment of the subject, so there is a good chance that an instructor might find his/her favorite topic gets too little or no coverage at all. For example, I think that every numerical partial differential equations book that is focused on finite difference methods should at least mention the subject of numerical grid generation. Numerical grid generation is a very active area of current research and extends the usefulness of finite difference methods tremendously.
The book has just the briefest possible examination of the finite element method. Instructors that wish to cover this topic will either have to use two books for their course or choose another book. As good as this text is, I think it could be improved by adding more information on the finite element method.
Jeffrey A. Graham teaches at Susquehanna University. His interests include numerical analysis, differential equations, inverse problems, and mathematical biology.
1. Introduction; 2. Parabolic equations in one space variable; 3. 2-D and 3-D parabolic equations; 4. Hyperbolic equations in one space dimension; 5. Consistency, convergence and stability; 6. Linear second order elliptic equations in two dimensions; 7. Iterative solution of linear algebraic equations; Bibliography; Index.