You are here

Finite Difference Schemes and Partial Differential Equations

Number of Pages: 

The book under review is billed as an introduction to the basic theory of finite difference schemes applied to the numerical solution to partial differential equations. It is targeted toward first-year graduate students in scientific and engineering computation. Although most will find the writing style very dry, I think this was a solid book in 1989 and it is still. Strikwerda provides a good introduction to the theory of consistency, stability, and convergence of finite difference schemes. Some books skimp on these topics and they really shouldn’t. Another strength of this book is the two chapters devoted to the basic techniques for solving the large systems of linear equations that occur when approximating the solution to an elliptic partial differential equation.

There are a few things that I think are missing from this text. Characteristics play a huge role in the theory of hyperbolic partial differential equations. Strikwerda’s coverage is adequate, but I think that the numerical solution technique called the method of characteristics deserves a few pages of coverage. Another technique that I think should have been mentioned is called the method of lines. The method of lines allows one to approximate the solution to a partial differential equation by solving a system of ordinary differential equations. It is very intuitive and easy to introduce in the context of parabolic partial differential equations. A more serious flaw in the book is the lack of content relating to irregularly shaped regions in the chapters devoted to elliptic partial differential equations. Not every problem is posed on a square! The author does mention the possibility and suggests a change of coordinates can sometimes help. True enough, but it is also possible to discretize a PDE on an irregular region. At this point in the book, it would have been appropriate to point the reader to the growing body of literature on automatic grid generation and provided a list of references in the bibliography.

In general, there are not nearly enough references in the text and only 5 of the 72 listed postdate the first edition of the book. The latest reference I could find was dated 1996. I expect that there have been a few pertinent papers published in the last nine years. In the 15 years that passed between the first edition of this book and the one under review, I think that there may have been time to update the reference list at least. Maybe even time enough to add a new chapter or some new material here and there. I’m not sure that there is really enough difference between this edition and the previous one to justify calling this a second edition. It seems to me that this book is more of a reprint.

In summary, this is a decent book on numerical partial differential equations even though it is getting a bit long in the tooth. However, before I committed to it for my class, I would look around at some of the recent entries into the field as well as some of the other old standbys.

Jeffrey A. Graham teaches at Susquehanna University. His interests include numerical analysis, differential equations, inverse problems, and mathematical biology.


Date Received: 
Tuesday, March 1, 2005
Include In BLL Rating: 
John C. Strikwerda
Publication Date: 
Jeffrey A. Graham
Preface to the Second Edition; Preface to the First Edition; Chapter 1: Hyperbolic Partial Differential Equations; Chapter 2: Analysis of Finite Difference Schemes; Chapter 3: Order of Accuracy of Finite Difference Schemes; Chapter 4:Stability for Multistep Schemes; Chapter 5: Dissipation and Dispersion; Chapter 6:Parabolic Partial Differential Equations; Chapter 7: Systems of Partial Differential Equations in Higher Dimensions; Chapter 8: Second-Order Equations; Chapter 9: Analysis of Well-Posed and Stable Problems; Chapter 10: Convergence Estimates for Initial Value Problems; Chapter 11: Well-Posed and Stable Initial-Boundary Value Problems; Chapter 12: Elliptic Partial Differential Equations and Difference Schemes; Chapter 13: Linear Iterative Methods; Chapter 14: The Method of Steepest Descent and the Conjugate Gradient Method; Appendix A: Matrix and Vector Analysis; Appendix B: A Survey of Real Analysis; Appendix C: A Survey of Results from Complex Anaylsis; References; Index.
Publish Book: 
Modify Date: 
Saturday, October 15, 2005