An Introductory Guide to Computational Methods for the Solution of Physics Problems: With Emphasis on Spectral Methods

ISBN-13: 9783319427027 / Angielski / Miękka / 2018 / 221 str.

George Rawitscher; Victo Dos Santos Filho; Thiago C. Peixoto

Opis

This book disseminates basic aspects of modern spectral computational methods, which are not generally taught in traditional courses. The main advantage of these methods is that they take into account all available information at the same time, rather only the information available at a limited number of meshpoints. That leads to more complicated matrix equations, but the elegance, speed, and accuracy of the method more than compensates for this drawback. The method consists 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.

In the various chapters, 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, and, in particular, they demonstrate the enhanced accuracy obtained in the solution of integral equations.

It includes an informative introduction to the old and new computational methods with numerous practical examples, while at the same time raising awareness of the errors that each of the available algorithms introduces into the specific solution. It is a valuable resource for graduate students wishing to compare the available computational methods (canned or not canned) and judge which is the most suitable to solve the particular scientific problem they are confronting.

Spis treści

Dedication.- Preface.- Numerical Errors.- Methods.- Galerkin and Collocation Methods.- Convergence Theorems.- Chebyshev Polynomials.- The Integral Equation Corresponding to a Differential Equation.- Spectral Finite Element Method.- The Phase-Amplitude Representation of a Wave Function.- The Vibrating String.- Iteratively Calculated Eigenvalues.- Sturmian Functions.- Index.

O autorze

Born in Germany in 1928, George Rawitscher moved with his family to Sao Paulo, Brazil in 1934 in order to escape Nazi persecution. He studied Physics and Mathematics at the University of S. Paulo from 1945 to 1949, (with BA degrees),  received his Ph.D. in Nuclear and Particle Physics in 1956 from Stanford University.  After teaching at the University of Rochester and at Yale University, he went to the University of Connecticut in 1966 and rose to the rank of Full Professor in 1972. In 2009 he retired, and is now a Research Professor in UConn's physics department. His research specializations are in low energy nuclear physics and in computational methods. In addition to his continuing scientific physics involvements, his other interests are on World Order (UN transformation) and Climate Change.

Victo dos Santos Filho, born in 1967 in Lorena, São Paulo State, Brazil, studied Mathematics at the Universidade Salesiana de Lorena from 1985 to 1988, and in sequence he studied Physics at Universidade Estadual Paulista (UNESP) de Guaratinguetá from 1989 to 1992. He received his Master degree in Superconducting Technology in Fundação de Tecnologia Industrial (FTI) in 1994 and his Ph.D. in 2001 in Theoretical Physics at Instituto de Física Teórica (IFT) of UNESP, in the area of Bose-Einstein Condensates. After four years of post-doc in IFT concerning theoretical studies in Bose-Einstein Condensates with dissipative effects, he began to teach in private institutions of São Paulo. From 2015, he has worked in the H4D Scientific Laboratory Research as a scientific researcher in physics and technology. Among his recent scientific research interests are condensed matter physics, mathematical physics, general physics and computational physics.

Thiago Carvalho Peixoto graduated with a Bachelor degree in Physics in 2012 from the Federal University of Sergipe (UFS). He obtained a Master's Degree in Theoretical Physics in 2014 at the Theoretical Physics Institute of São Paulo's State University (UNESP). He worked as a professor of Physics at the Federal Institute of North of Minas Gerais (IFNMG) from 2016 to 2017 and nowadays he is professor of Physics at the Federal Institute of Sergipe (IFS).

Blurb

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.