Spectral Methods: Fundamentals in Single Domains (Scientific Computation)

Book Description

Since the publication of "Spectral Methods in Fluid Dynamics", spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between the theoretical and practical aspects of spectral methods that was the hallmark of the earlier book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since 1988. The initial treatment Fundamentals in Single Domains discusses the fundamentals of the approximation of solutions to ordinary and partial differential equations on single domains by expansions in smooth, global basis functions. The first half of the book provides the algorithmic details of orthogonal expansions, transform methods, spectral discretization of differential equations plus their boundary conditions, and solution of the discretized equations by direct and iterative methods. The second half furnishes a comprehensive discussion of the mathematical theory of spectral methods on single domains, including approximation theory, stability and convergence, and illustrative applications of the theory to model boundary-value problems. Both the algorithmic and theoretical discussions cover spectral methods on tensor-product domains, triangles and tetrahedra. All chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is greatly expanded as are the set of numerical examples that illustrate the key properties of the various types of spectral approximations and the solution algorithms.

A companion book "Evolution to Complex Geometries and Applications to Fluid Dynamics" contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods for complex geometries and provides detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries.

Spectral Methods in MATLAB (Software, Environments, Tools)

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setting a precedent in numerical methods books
Loaded with errors, author defensive and arrogant
A useful introduction to spectral methods

Spectral Methods in MATLAB (Software, Environments, Tools)
Lloyd N. Trefethen
Manufacturer: SIAM: Society for Industrial and Applied Mathematics
ProductGroup: Book
Binding: Paperback

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ASIN: 0898714656

Book Description

This is the only book on spectral methods built around MATLAB programs. Along with finite differences and finite elements, spectral methods are one of the three main technologies for solving partial differential equations on computers. Since spectral methods involve significant linear algebra and graphics they are very suitable for the high level programming of MATLAB. This hands-on introduction is built around forty short and powerful MATLAB programs, which the reader can download from the World Wide Web. This book presents the key ideas along with many figures, examples, and short, elegant MATLAB programs for readers to adapt to their own needs. It covers ODE and PDE boundary value problems, eigenvalues and pseudospectra, linear and nonlinear waves, and numerical quadrature.

Customer Reviews:

setting a precedent in numerical methods books.......2006-05-31

i would have given it a 4-stars, but i feel that i need to counter the negative comment by a reader below who complains that the book is loaded with errors.

first i should say i am not a mathematician although i used mathematics and numerical methods frequently. so there are certainly many details that i overlook or ignore.

in my opinion, most of the errors in the book (inc those not listed on trefethen's website but which i have noticed) are insignificant and inconsequential. they are things like writing 'n' instead of 'N'; the confusions and errors in the indices (this is due to matlab must start from '1', while frequently we are used to start from '0' when writing the mathematics); calling something obvious an inappropriate name, etc. to me, these are trivial errors. (if you are a numerical analysis theorist, i accept that you don't agree with me). only a handful of errors are actually serious in terms of the mathematics, and i believe the number of them are acceptable. every book has errors.

at the level i am reading this book, i just need an introduction to the theory, and then start computing. as far as i am concerned, if the mathematics is seriously wrong, you will notice when you do the computing.

what i like about the book is that it starts you computing right from the first chapter. although mathematical softwares such as matlab have been around for one to two decades, until recently not many books on numerical methods give you an opportunity to do the computations yourself. if the matlab scripts were not provided, i doubt that many would know how to code them (in any language you are familiar with); or even if you do, you may be doing it in an inelegant or inefficient way.

i used to wonder what is the use of a book on advanced numerical methods. for example, i would not know how to write a fluid dynamics code using spectral method to do direct numerical simulation of turbulent flows after reading and even mastering, say, Canuto et al's Spectral Dynamics in Fluid Dynamics. on the other hand, if i am familiar with spectral methods programming, it is also likely that i am familiar with the theory. so what these books are for is just as a store of knowledge and convenient references. you may know that the kind of fluid dynamics code i mentioned above can require a team effort over many years. so most numerical methods books which claim to cover applications are in fact useless, except you are already a practitioner, when come to developing codes for modern advanced simulations.

however, i think introductory books should have examples of how the computer codes to some numerical methods actually look like. before these past few years, i didn't see any such book. only recently, there are at least a dozen books on numerical methods or differential equations with C/C++, Matlab, etc. and i think trefethen's book is setting an example on how to do this without much fuss and long discourse. who knows, you may also learn the basic skills require to work on a much more complicated piece of computational code.

also, trefethen never claims that you can learn matlab with this book. he suggested, for example, higham & higham's matlab guide.

lastly, i should say one should not judge a book by the personality of the author. he may (or may not) be arrogant, but that does not necessarily make his book bad. (it appears to me that the reader below is the arrogant one. just because trefethen probably didn't give him responsive and 'grateful' replies, he wrote this review here).

Loaded with errors, author defensive and arrogant.......2004-06-21

I found glaring errors starting on page 1. I emailed the author for clarification who agreed that his nomenclature was unclear and misleading (come on Professor, we call them mistakes, errata, etc.). The first chapter is available in his web site, go see for yourself. You'll also find a page of errata, which unfortunately is far from complete. My colleagues and I desperately tried to get Professor Trefethen to clarify his nomenclature, which he quite arrogantly dismissed. His matlab code seems to execute fine, but you'll likely spend a good deal of time dissecting it. Any programmer who must use someone elses code despises poorly commented and optimized code. This book just isn't a good place to start for spectral methods or matlab.

A useful introduction to spectral methods.......2001-03-21

The author achieves the hard task of explaining the subject in a very simple and, at the same time, intelligent way. The MATLAB m-files are short, educative and full of useful tricks. The book is well motivated, focused in the essentials and easy to read. I am very fond of most of Prof. Trefethen writings and this book is no disappoitment. It contrasts starkly with the lack of mathematical insight in most of the engineering collocation literature, but at the same time has a hands-on approach to the problems. This book should be useful to anybody interested in solving PDE in regular geometries.

Partial Differential Equations for Computational Science: With Maple and Vector Analysis

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Partial Differential Equations for Computational Science: With Maple and Vector Analysis
David Betounes
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 0387983007

Book Description

This is a classroom-tested and developed textbook designed for use in either one- or two term courses in Partial Differential Equations taught at the advanced undergradute and beginning graduate levels of instruction. In addition to covering the standard PDE topics taught in such courses, there is a heavy focus on vector analysis. Maple examples and exercises for the students are presented strategically within the book, and an Appendix on Maple facilitates the use of this computer algebra language, acting as a quick reference source for readers. In addition, a cross-platform CD-ROM which is packaged with the text contains many analytical examples for students in the use of Maple. However, despite its general computational orientation, this book is designed so that it can also be readily used by instructors not wishing to bring the computational aspects of a computer algebra system into the classroom. This book will have strong appeal to interdisciplinary audiences of engineers and scientists, particulary in regard to its treatments of fluid mechanics, heat equations, and continuum mechanics, inclusion of the latter being a unique feature of the presentation.

Applications of Nonstandard Finite Difference Schemes

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Potentially great

Applications of Nonstandard Finite Difference Schemes

Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Hardcover

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ASIN: 981024133X

Customer Reviews:

Potentially great.......2001-09-21

I have only had a borrowed copy for a few days, but this could be the biggest break through since the PC.
Everyone who models ODEs & PDEs with finite differences
should have this book.
Well written, but the math is dense.

Numerical Approximation of Partial Differential Equations (Springer Series in Computational Mathematics)

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Numerical Approximation of Partial Differential Equations (Springer Series in Computational Mathematics)
Alfio Quarteroni , and Alberto Valli
Manufacturer: Springer
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Binding: Hardcover

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Numerical Partial Differential Equations: Finite Difference Methods (Texts in Applied Mathematics)

ASIN: 0387571116

Book Description

This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method.

Partial Differential Equations: Theory and Technique

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Partial Differential Equations: Theory and Technique
George F. Carrier , and Carl E. Pearson
Manufacturer: Academic Press
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Binding: Hardcover

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Book Description

This is the second edition of the well-established text in partial differential equations, emphasizing modern, practical solution techniques. This updated edition includes a new chapter on transform methods and a new section on integral equations in the numerical methods chapter. The authors have also included additional exercises.

Free Boundary Problems: Theory and Applications (International Series of Numerical Mathematics)

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Free Boundary Problems: Theory and Applications (International Series of Numerical Mathematics)

Manufacturer: Birkhäuser Basel
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Binding: Hardcover

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Many phenomena of interest for applications are represented by differential equations which are defined in a domain whose boundary is a priori unknown, and is accordingly named a "free boundary". A further quantitative condition is then provided in order to exclude indeterminacy. Free boundary problems thus encompass a broad spectrum which is represented in this state-of-the-art volume by a variety of contributions of researchers in mathematics and applied fields like physics, biology and material sciences. Special emphasis has been reserved for mathematical modelling and for the formulation of new problems.

Numerical Solution of Hyperbolic Partial Differential Equations

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Numerical Solution of Hyperbolic Partial Differential Equations
John A. Trangenstein
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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Book Description

Numerical Solution of Hyperbolic Partial Differential Equations is a new type of graduate textbook, with both print and interactive electronic components (on CD). It is a comprehensive presentation of modern shock-capturing methods, including both finite volume and finite element methods, covering the theory of hyperbolic conservation laws and the theory of the numerical methods. The range of applications is broad enough to engage most engineering disciplines and many areas of applied mathematics. Classical techniques for judging the qualitative performance of the schemes are used to motivate the development of classical higher-order methods. The interactive CD gives access to the computer code used to create all of the text's figures, and lets readers run simulations, choosing their own input parameters; the CD displays the results of the experiments as movies. Consequently, students can gain an appreciation for both the dynamics of the problem application, and the growth of numerical errors.

Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (Chapman & Hall/Crc Applied Mathematics and Nonlinear Science)

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Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (Chapman & Hall/Crc Applied Mathematics and Nonlinear Science)
Victor A. Galaktionov , and Sergey R. Svirshchevskii
Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 1584886633

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Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators. Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the book focuses on the existence of new exact solutions on linear invariant subspaces for nonlinear operators and their crucial new properties. This practical reference deals with various partial differential equations (PDEs) and models that exhibit some common nonlinear invariant features. It begins with classical as well as more recent examples of solutions on invariant subspaces. In the remainder of the book, the authors develop several techniques for constructing exact solutions of various nonlinear PDEs, including reaction-diffusion and gas dynamics models, thin-film and Kuramoto-Sivashinsky equations, nonlinear dispersion (compacton) equations, KdV-type and Harry Dym models, quasilinear magma equations, and Green-Naghdi equations. Using exact solutions, they describe the evolution properties of blow-up or extinction phenomena, finite interface propagation, and the oscillatory, changing sign behavior of weak solutions near interfaces for nonlinear PDEs of various types and orders. The techniques surveyed in Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics serve as a preliminary introduction to the general theory of nonlinear evolution PDEs of different orders and types.

Verification of Computer Codes in Computational Science and Engineering

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Verification of Computer Codes in Computational Science and Engineering
Patrick Knupp , and Kambiz Salari
Manufacturer: Chapman & Hall/CRC
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Binding: Hardcover

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Verification and Validation in Computational Science and Engineering

ASIN: 1584882646

Book Description

How can one be assured that computer codes that solve differential equations are correct? Standard practice using benchmark testing no longer provides full coverage because today's production codes solve more complex equations using more powerful algorithms. By verifying the order-of-accuracy of the numerical algorithm implemented in the code, one can detect most any coding mistake that would prevent correct solutions from being computed. Verification of Computer Codes in Computational Science and Engineering sets forth a powerful alternative called OVMSP: Order-Verification via the Manufactured Solution Procedure. This procedure has two primary components: using the Method of Manufactured Exact Solutions to create analytic solutions to the fully-general differential equations solved by the code and using grid convergence studies to confirm the order-of-accuracy. The authors present a step-by-step procedural guide to OVMSP implementation and demonstrate its effectiveness. Properly implemented, OVMSP offers an exciting opportunity to identify virtually all coding 'bugs' that prevent correct solution of the governing partial differential equations. Verification of Computer Codes in Computational Science and Engineering shows you how this can be done. The treatment is clear, concise, and suitable both for developers of production quality simulation software and as a reference for computational science and engineering professionals.

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