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An Introductory Guide to Computational Methods for the Solution of Physics Problems

With Emphasis on Spectral Methods

  • George Rawitscher
  • Victo dos Santos Filho
  • Thiago Carvalho Peixoto

Table of contents

  1. Front Matter
    Pages i-xviii
  2. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 1-5
  3. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 7-15
  4. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 17-31
  5. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 33-41
  6. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 43-62
  7. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 63-76
  8. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 77-93
  9. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 95-127
  10. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 129-143
  11. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 145-152
  12. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 153-176
  13. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 177-190
  14. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto
    Pages 191-193
  15. Back Matter
    Pages 195-221

About this book

Introduction

This monograph presents fundamental aspects of modern spectral and other computational methods, which are not generally 
taught in traditional courses. It emphasizes concepts as errors, convergence, stability, order and efficiency applied to 
the solution of physical problems. The spectral methods consist in expanding the function to be calculated into a set of 
appropriate basis functions (generally orthogonal polynomials) and the respective expansion coefficients are obtained via 
collocation equations. The main advantage of these methods is that they simultaneously take into account all available 
information, rather only the information available at a limited number of mesh points. They require more complicated 
matrix equations than those obtained in finite difference methods. However, the elegance, speed, and accuracy of the 
spectral methods more than compensates for any such drawbacks. 

During the course of the monograph, the authors examine the usually rapid convergence of the spectral expansions and 
the improved accuracy that results when nonequispaced support points are used, in contrast to the equispaced points 
used in finite difference methods. In particular, they demonstrate the enhanced accuracy obtained in the solution 
of integral equations.

The monograph includes an informative introduction to old and new computational methods with numerous practical examples, 
while at the same time pointing out the errors that each of the available algorithms introduces into the specific 
solution. It is a valuable resource for undergraduate students as an introduction to the field and for graduate students 
wishing to compare the available computational methods. In addition, the work develops the criteria required for 
students to select the most suitable method to solve the particular scientific problem that they are confronting.

Keywords

Spectral Computational Methods Finite-difference methods Fourier spectral Differentiation Matrices Quadratures Numerical methods

Authors and affiliations

  • George Rawitscher
    • 1
  • Victo dos Santos Filho
    • 2
  • Thiago Carvalho Peixoto
    • 3
  1. 1.University of ConnecticutStorrsUSA
  2. 2.H4D Scientific Research LaboratoryBela VistaBrazil
  3. 3.Federal Institute of Sergipe (IFS)Nossa Senhora da GlóriaBrazil

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