EAN: 9783319427027

AN INTRODUCTORY GUIDE TO COMPUTATIONAL METHODS ...

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 solutionof 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.

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