Skip to main content Accessibility help
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

2 results

  • 5 - Mesh-Free FD Approximations

  • Bengt Fornberg, University of Colorado Boulder
  • Book:
    High-Accuracy Finite Difference Methods
    Published online:
    16 May 2025
    Print publication:
    05 June 2025, pp 85-111

  • Summary

    Mesh-Free FD Approximations are easy to work with and are in some sense optimal in their representations of functions locally around a single point. However, in more than one dimension, approximations based on polynomial interpolants encounter severe difficulties if the node points are not regularly placed (grid-based). Additional difficulties often arise at boundaries which (above 1-D) may be irregularly shaped, and also from mixtures of scales across a computational domain that may require spatially variable resolution. It transpires that radial basis functions (RBFs) can favorably replace (or supplement) polynomials in such situations. Their use in creating spatially local and highly effective FD-type approximations is quite recent.

High-Accuracy Finite Difference Methods
  • Bengt Fornberg
  • Published online:
    16 May 2025
    Print publication:
    05 June 2025
  • 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.