Scientific Computing and Differential Equations: An Introduction to Numerical Methods

by Gene H. Golub, James M. Ortega

Scientific Computing and Differential Equations: An Introduction to Numerical Methods, is an excellent complement to Introduction to Numerical Methods by Ortega and Poole. The book emphasizes the importance of solving differential equations on a computer, which comprises a large part of what has come to be called scientific computing. It reviews modern scientific computing, outlines its applications, and places the subject in a larger context.
This book is appropriate for upper undergraduate courses in mathematics, electrical engineering, and computer science; it is also well-suited to serve as a textbook for numerical differential equations courses at the graduate level. * An introductory chapter gives an overview of scientific computing, indicating its important role in solving differential equations, and placing the subject in the larger environment
* Contains an introduction to numerical methods for both ordinary and partial differential equations
* Concentrates on ordinary differential equations, especially boundary-value problems
* Contains most of the main topics for a first course in numerical methods, and can serve as a text for this course
* Uses material for junior/senior level undergraduate courses in math and computer science plus material for numerical differential equations courses for engineering/science students at the graduate level

344 pages, Hardcover

First published September 1, 1991

Reviews

[Tomáš Ševček · Rating 4 out of 5]

A good self-contained introduction to the numerical analysis of differential equations. The book covers standard topics such as initial value problems for ODEs, boundary value problems for ODEs and initial-boundary value problems for PDEs. The book is self-contained in the sense that it covers all the necessary material from numerical mathematics (numerical integration/quadrature, differences, numerical solutions of (non-)linear equations, etc.) and even useful matrix algorithms (QR, Householder and so on). However, while the book provides numerous methods for solving ODE problems, I find the parts on PDEs unsatisfactory since only the most basic approaches are covered (and only for the scalar case). Still, these methods are enough to solve most linear PDEs and non-linear PDEs with non-linearities in the source term (as they can be treated in the similar fashion to the ODE case). If you want to solve (systems of) non-linear PDEs with more difficult non-linearities, you will have to search for methods not covered in the book, such as operator splitting. There is also very little on finite elements and virtually nothing on finite volumes. Finite elements and Galerkin are covered in the part on boundary value problems for ODEs, but not in the PDE case (and their treatment is rather informal as a rigorous treatment would require a solid grasp of functional analysis). All in all, a good intro text, but not suitable if you want to learn more advanced techniques for solving PDEs numerically.