High-Accuracy Finite Difference Methods

by Bengt Fornberg

Overview

Scientific computing plays a critically important role in almost all areas of engineering, modeling, and forecasting. The method of finite differences (FD) is a classical tool that is still rapidly evolving, with several key developments barely yet in the literature. Other key aspects of the method, in particular those to do with computations that require high accuracy, often 'fall through the cracks' in many treatises. Bengt Fornberg addresses that failing in this book, which adopts a practical perspective right across the field and is aimed at graduate students, scientists, and educators seeking a follow-up to more typical curriculum-oriented textbooks. The coverage extends from generating FD formulas and applying them to solving ordinary and partial differential equations, to numerical integration, evaluation of infinite sums, approximation of fractional derivatives, and computations in the complex plane.

Product Details

ISBN-13:	9781009566537
Publisher:	Cambridge University Press
Publication date:	06/05/2025
Series:	Cambridge Monographs on Applied and Computational Mathematics , #42
Pages:	246
Product dimensions:	5.98(w) x 9.02(h) x 0.63(d)

About the Author

Bengt Fornberg, now Professor Emeritus, joined the University of Colorado Boulder in 1995 following positions at CERN, Caltech, and Exxon Corporate Research. He has authored three previous books, including Volume 1 in this present series 'Cambridge Monographs on Applied and Computational Mathematics.' He is a Guggenheim and a SIAM Fellow.

Table of Contents

1. Introduction to finite difference methods; 2. Brief summary of pseudospectral methods; 3. FD approximations for ordinary differential equations; 4. Grid-based FD approximations for partial differential equations; 5. Mesh-free FD approximations; 6. FD in the complex plane; 7. FD-based methods for quadrature and infinite sums; 8. Fractional-order derivatives; A. Polynomial interpolation; B. Splines; C. Fourier transforms, Fourier series, and the FFT algorithm; D. Lagrange multipliers; E. Extrapolation methods; F. Trade-offs between accuracy orders and other approximation features; G. Node sets for FD and RBF-FD-based PDE discretizations; References; Index.

Editorial Reviews

‘In a journey ranging far wider than the title suggests, Fornberg compels the reader to get fingers on keyboard to explore many areas of modern scientific computing (quadrature, differential equations, complex approximation, fractional derivatives). Written in the author's inimitable 'one-picture-worth-a-thousand-proofs'-style and generously annotated with historical tidbits, this is an enjoyable read from beginning to end.’ J. A. C. Weideman, Stellenbosch University

‘Fornberg is the expert in high-order finite differencing - nobody else could have written this remarkable book.’ Nick Trefethen, Harvard University

From the Publisher