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

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 anysuch 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.