Finite Volume Methods for Hyperbolic Problems (Cambridge Texts in Applied Mathematics)

Book Description

This book contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, (including both linear problems and nonlinear conservation laws). These equations describe a wide range of wave propagation and transport phenomena arising in nearly every scientific and engineering discipline. Several applications are described in a self-contained manner, along with much of the mathematical theory of hyperbolic problems. High-resolution versions of Godunov's method are developed, in which Riemann problems are solved to determine the local wave structure and limiters are applied to eliminate numerical oscillations. The methods were orginally designed to capture shock waves accurately, but are also useful tools for studying linear wave-progagation problems, particulary in heterogenous material. The methods studied are in the CLAWPACK software package. Source code for all the examples presented can be found on the web, along with animations of many of the simulations. This provides an excellent learning environment for understanding wave propagation phenomena and finite volume methods.

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Customer Reviews:

an excellent book on hyperbolic equations.......2005-10-18

The author gave almost all the basic knowledge related to hyperbolic equation, at least from the engineering point of view. I read it myself without any help. It's not hard to understand. Moreover, it gives all you need at beginning references.

Good book to start with. Highly recommended........2003-10-24

This book starts from simple things and moves to pretty complicated staff graciously. It is useful even as an introduction to the hyperbolic equations. Finally, this is the only book I use at most every day. This is the book I would strongly recommend to all students who study this field and to researchers. It has a very good and comprehensive reference.

The author develop even the software (unfortunately, this is FORTRAN, not C). The source is available and well discussed in the book (there is a whole chapter). I did not use it but found this is a very good practice. It should be useful for student also.

Many things are really nice. For example, the book gives a very good view of the nature of oscillations in high order schemes, not only formulas. And so on...

However, there are few things I was not satisfied.

1. There are no comprehensive discussion about non-uniform and non-rectangular grids. It is not good, for example, for people who works in spherical coordinates (for example in some brunches of geophysics).

2. There is no information about FCT methods that are still very popular because they give a very straightforward way to use 4th and higher order methods. However, there is a reference to the Oran and Boris book, for instance.

3. It is sometimes really pure mathematical description especially for non-linear equations. It was really inconvenient for me. Fortunately, good reference helped.

There are more things were bothered. However, this is personal. The author works with the advection equation a lot, but does not like to discuss more the conservation form of continuity equation which I would prefer. In spite of author's efforts, I think still that the wave propagation method is not so convenient as flux method even for non-conservative equations. But it depends.

Finally, this book is definitely fine and, I think, it is the best among all books in this field (maybe except the Hirsch book which is "Numerical computation of internal and external flows" 1988). I would highly recommend it to buy.

nice introduction.......2003-07-11

This book provides a nice introduction to the mathematics behind finite-volume methods. After reading through the first half of the book on scalar conservation laws and systems, papers in JCP no longer seem as intimidating. The book is laid out very well, and the notation is consistent throughout. It is the best of the bunch when compared to Toro's Riemann problem book and Laney's Computational Gasdynamics text.

Iterative Methods for Sparse Linear Systems, Second Edition

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Excellent work
Wonderful Book
Great Book

Iterative Methods for Sparse Linear Systems, Second Edition
Yousef Saad
Manufacturer: Society for Industrial and Applied Mathematics
ProductGroup: Book
Binding: Paperback

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Iterative Krylov Methods for Large Linear Systems (Cambridge Monographs on Applied and Computational Mathematics)
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A Multigrid Tutorial
Matrix Computations (Johns Hopkins Studies in Mathematical Sciences)(3rd Edition)
Multigrid

ASIN: 0898715342

Book Description

Tremendous progress has been made in the scientific and engineering disciplines regarding the use of iterative methods for linear systems. The size and complexity of linear and nonlinear systems arising in typical applications has grown, meaning that using direct solvers for the three-dimensional models of these problems is no longer effective. At the same time, parallel computing, becoming less expensive and standardized, has penetrated these application areas. Iterative methods are easier than direct solvers to implement on parallel computers but require approaches and solution algorithms that are different from classical methods. This second edition gives an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations, including a wide range of the best methods available today. A new chapter on multigrid techniques has been added, whilst material throughout has been updated, removed or shortened. Numerous exercises have been added, as well as an updated and expanded bibliography.

Customer Reviews:

Excellent work.......2006-08-12

We used this book to prove a theorem in our studies that is directly related to my PhD thesis on spatial data mining and spatial statistics. This book is a master-piece.
Thanks Dr. Saad.

Wonderful Book.......2005-06-30

This is one of my favorite books in my library on this subject. Also I have used this book for my class as main textbook along with "Iterative Methods for Solving Linear and Nonlinear Equations" by C. T. Kelley , which is another SIAM book.
Highly recommended.

Great Book.......2000-05-24

This is a great book for this subject. The book is easy to follow and Saad does a wonderful job of illustrating with examples. This is a great textbook or a book for reference. This book does a particularly good job with Krylov methods and does a reasonable job with preconditioning.

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An Accessible Multigrid Introduction...No, Really, I Mean It
Good introduction to Multigrid
Excellent basic multigrid theory

A Multigrid Tutorial
William L. Briggs , Van Emden Henson , and Steve F. McCormick
Manufacturer: SIAM: Society for Industrial and Applied Mathematics
ProductGroup: Book
Binding: Paperback

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Multigrid
Iterative Methods for Sparse Linear Systems, Second Edition
An Introduction to Multigrid Methods
Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics
A First Course in the Numerical Analysis of Differential Equations (Cambridge Texts in Applied Mathematics)

ASIN: 0898714621

Book Description

This second edition of the popular A Multigrid Tutorial preserves the introductory spirit of the first edition while roughly doubling the amount of material covered. The topics of the first edition have been enhanced with additional discussion, new numerical experiments, and improved figures. New topics in the second edition include nonlinear equations, Neumann boundary conditions, variable mesh and variable coefficient problems, anisotropic problems, algebraic multigrid (AMG), adaptive methods, and finite elements. This introductory book is ideally suited as a companion textbook for graduate numerical analysis courses. It is written for computational mathematicians, engineers, and other scientists interested in learning about multigrid.

Customer Reviews:

An Accessible Multigrid Introduction...No, Really, I Mean It.......2005-04-28

Mathematicians who write well are a rare commodity. Briggs is one of them. He has the ability to break down complex numerics with understandable prose making concepts accessible to non-specialists. He operates under the rare assumption that his readers weren't born knowing this stuff. This multigrid primer written with Henson and McCormick is no exception. Expository discussions are lucid, compact and informative; examples and plots are helpful and well selected; and pseudo code is well documented with corresponding outputs to verify understanding. If you are embarking on Multigrid from scratch this short (if somewhat pricy) primer is worth the time and money.

Good introduction to Multigrid.......2002-04-05

Anyone with some background in applied mathmatics should find this book accessible. It is well laid-out and focuses on algorithms as opposed to mathematical proofs. It is ideal for someone interested in multigrid methods but who doesn't want to get lost in lots of mathematical theory. It doesn't cover cell centered methods, which is somewhat limiting, but other than that it was useful and informative.

Excellent basic multigrid theory.......1996-12-02

This is a great book. Clearly written, well organized.

Average customer rating:

multigrid explained
for actual programmers/scientists

Multigrid
Ulrich Trottenberg , Cornelius W. Oosterlee , and Anton Schuller
Manufacturer: Academic Press
ProductGroup: Book
Binding: Hardcover

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A Multigrid Tutorial
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Iterative Methods for Sparse Linear Systems, Second Edition
Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics
Iterative Krylov Methods for Large Linear Systems (Cambridge Monographs on Applied and Computational Mathematics)

ASIN: 012701070X

Book Description

Multigrid presents both an elementary introduction to multigrid methods for solving partial differential equations and a contemporary survey of advanced multigrid techniques and real-life applications.

Multigrid methods are invaluable to researchers in scientific disciplines including physics, chemistry, meteorology, fluid and continuum mechanics, geology, biology, and all engineering disciplines. They are also becoming increasingly important in economics and financial mathematics.

Readers are presented with an invaluable summary covering 25 years of practical experience acquired by the multigrid research group at the Germany National Research Center for Information Technology. The book presents both practical and theoretical points of view.

* Covers the whole field of multigrid methods from its elements up to the most advanced applications
* Style is essentially elementary but mathematically rigorous
* No other book is so comprehensive and written for both practitioners and students

Customer Reviews:

multigrid explained.......2006-12-02

as opposed to what another reviewer states, this book has nothing to do with grid computing. instead, it presents a mathematical technique for solving linear systems of equations (with the focus on large, sparse systems arising from the discretization of PDEs). lots of math inside, there's nothing whatsoever for the layman here. the introduction to the basic concepts is quite thorough, and they cover some advanced topics like adaptive multigrid, algebraic multigrid (an extension of MG to general linear systems) and application of MG in computational fluid dynamics. overall, a good book, but I strongly recommend Briggs' and Van Emden Henson's "Multigrid Tutorial" text as a 1st read before tackling this book. the reader is assumed to be familiar with linear algebra (with a focus on matrix calculus and classical iterative methods). to fully understand the later chapters, more advanced exposure to let's say CFD is needed.

for actual programmers/scientists.......2005-01-13

IBM and other companies have been promoting a vision of grid computing. Books have been written, including "Grid Computing" by Joseph and Fellenstein, that describe this approach at a high level. Suitable for system administrators and managers.

By contrast, this book by Trottenberg et al is aimed directly at the scientists and engineers who might be actually writing software for their problems. The mathematical treatment is looser than that in a pure maths book. It's at the typical level for a physic or engineering text. The intent here is not in proving theorems, but in applying them to real world problems.

Specifically, for solving PDEs and linear systems.

Spectral Methods: Fundamentals in Single Domains (Scientific Computation)

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Spectral Methods: Fundamentals in Single Domains (Scientific Computation)
C. Canuto , M.Y. Hussaini , A. Quarteroni , and T.A. Zang
Manufacturer: Springer
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Binding: Hardcover

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Asymptotic Analysis and Boundary Layers (Scientific Computation)
Models of the Atomic Nucleus: With Interactive Software

ASIN: 3540307257

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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Spectral Methods for Time-Dependent Problems (Cambridge Monographs on Applied and Computational Mathematics)

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.

Moving Finite Elements (Numerical Mathematics and Scientific Computation)

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Moving Finite Elements (Numerical Mathematics and Scientific Computation)
M. J. Baines
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover

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ASIN: 0198534671

Book Description

This book is mainly concerned with finite element methods for time-dependent partial differential equations when the grids are allowed to move in time, but also describes grid generation techniques which include grid adjustment. The mechanism for grid movement derives from a generalization of the residual minimization technique which is familiar from the Galerkin finite element method. The book brings together most of the work done over the last decade or so which has been stimulated by Miller's original idea, and discusses the interrelationships between the techniques of the method and the established ideas of the method of characteristics, Hamilton's equations, the Legendre transformation, and grid equidistribution. The book highlights the issues involved and should provide the reader with a clear view of the current state of the subject and prompt further research.

Spectral Methods for Time-Dependent Problems (Cambridge Monographs on Applied and Computational Mathematics)

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Spectral Methods for Time-Dependent Problems (Cambridge Monographs on Applied and Computational Mathematics)
Jan S. Hesthaven , Sigal Gottlieb , and David Gottlieb
Manufacturer: Cambridge University Press
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Binding: Hardcover

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Spectral Methods in MATLAB (Software, Environments, Tools)

ASIN: 0521792118

Book Description

Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

Numerical Methods for Partial Differential Equations (Computer Science and Scientific Computing)

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Very good and helpful book for the engineer with fundamental

Numerical Methods for Partial Differential Equations (Computer Science and Scientific Computing)
William F. Ames
Manufacturer: Academic Press
ProductGroup: Book
Binding: Hardcover

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Book Description

This volume is designed as an introduction to the concepts of modern numerical analysis as they apply to partial differential equations. The book contains many practical problems and their solutions, but at the same time, strives to expose the pitfalls--such as overstability, consistency requirements, and the danger of extrapolation to nonlinear problems methods used on linear problems. Numerical Methods for Partial Differential Equations, Third Edition reflects the great accomplishments that have taken place in scientific computation in the fifteen years since the Second Edition was published. This new edition is a drastic revision of the previous one, with new material on boundary elements, spectral methods, the methods of lines, and invariant methods. At the same time, the new edition retains the self-contained nature of the older version, and shares the clarity of its exposition and the integrity of its presentation.

Key Features
* Material on finite elements and finite differences have been merged, and now constitute equal partners
* Additional material has been added on boundary elements, spectral methods, the method of lines, and invariant methods
* References have been updated, and reflect the additional material
* Self-contained nature of the Second Edition has been maintained
* Very suitable for PDE courses

Customer Reviews:

Very good and helpful book for the engineer with fundamental.......2005-02-16

This book is very detail on how to generate numerical methods for partial differential equations. Staring from basics, the author proceeds with detailed examples and more complicated ideas. This is book will be very helpful for the people having basic computational knowledge and scientific computing experience.

But the book is written in 1977. Some difficult ideas will be easier to be implemented on large scale computers such as Linux clusters or SGI machines. I also found one small shortcoming is that the problem complexity is not detailed discussed. This will be very important now for comparing different algorithms to accomplish the same problem.

Multi-Grid Methods and Applications (Springer Series in Computational Mathematics)

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Multi-Grid Methods and Applications (Springer Series in Computational Mathematics)
W. Hackbusch
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 0387127615

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