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Numerical Solutions of Time-Dependent Advection-Diffusion-Reaction Equations
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From the reviews:
"The numerical solution of time-dependent advection-diffusion-reaction problems draws on different areas of numerical analysis … . We appreciate that the quite thorough, yet not pedantic, analytic part of the presentation is intimately interwoven with numerical tests and examples which will enable the reader to judge on the relative merits of the various approaches and really aid him in developing proper software for the problem at hand." (H. Mutsham, Monatshefte für Mathematik, Vol. 144 (2), 2005)
"Let me say at the outset that I highly recommend this book to practitioners … end-users, and those new to the field. One of its strengths is its in-depth presentation of temporal and spatial discretizations and their interaction … . With each topic, key theoretical results are presented. … I found the present authors’ choice of problems to be one of the highlights of the book." (Peter Moore, SIAM Review, Vol. 46 (3), 2004)
"This excellent research monograph contains a comprehensive discussion of numerical techniques for advection-reaction-diffusion partial differential equations (PDEs). The emphasis is on a method of lines approach, the analysis is careful and complete, and the numerical tests designed to verify the theoretical discussions of stability, convergence, monotonicity, etc. involve solving ‘real life’ equations. … As is to be expected in such a carefully prepared monograph, there is an extensive bibliography and a good index. Highly recommended." (Ian Gladwell, Mathematical Reviews, 2004 g)
"The information, densely packed on roughly 450 pages, is abundant though well-structured, smoothly readable, and with emphasis on explanation of key concepts by means of examples that are stripped from unnecessary complications. … a serious student with a hands-on attitude finds in this book an excellent source for self-studies and investigation. … It is a valuable contribution to the Springer Series in this field of research." (J. Brandts, Nieuw Archief voor Wiskunde, Vol. 7 (1), 2006)
From the Back Cover
This book describes numerical methods for partial differential equations (PDEs) coupling advection, diffusion and reaction terms, encompassing methods for hyperbolic, parabolic and stiff and nonstiff ordinary differential equations (ODEs). The emphasis lies on time-dependent transport-chemistry problems, describing e.g. the evolution of concentrations in environmental and biological applications. Along with the common topics of stability and convergence, much attention is paid on how to prevent spurious, negative concentrations and oscillations, both in space and time. Many of the theoretical aspects are illustrated by numerical experiments on models from biology, chemistry and physics. A unified approach is followed by emphasizing the method of lines or semi-discretization. In this regard this book differs substantially from more specialized textbooks which deal exclusively with either PDEs or ODEs. This book treats integration methods suitable for both classes of problems and thus is of interest to PDE researchers unfamiliar with advanced numerical ODE methods, as well as to ODE researchers unaware of the vast amount of interesting results on numerical PDEs. The first chapter provides a self-contained introduction to the field and can be used for an undergraduate course on the numerical solution of PDEs. The remaining four chapters are more specialized and of interest to researchers, practitioners and graduate students from numerical mathematics, scientific computing, computational physics and other computational sciences.
- Item Weight : 4.14 pounds
- Hardcover : 500 pages
- ISBN-13 : 978-3540034407
- ISBN-10 : 3540034404
- Dimensions : 6.14 x 1.06 x 9.21 inches
- Publisher : Springer (July 21, 2003)
- Language: : English
- Best Sellers Rank: #5,141,858 in Books (See Top 100 in Books)
- Customer Reviews:
Top reviews from the United States
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The MOL is a very practical and very general approach to solving PDE's. Many readers will be familiar with sophisticated ODE solvers, such as ode45 or ode23s from MATLAB. They're probably aware that these solvers are far more efficient than the simple Euler method. In particular, they use a number of tricks to adjust their stepsize, stepping gingerly through bumpy terrain and cruising quickly over flat savannahs. However, it's all too common for engineers and scientists, upon encountering a PDE for the first time, to do their own simple discretization, with constant space and time steps. I know I've done that before. The MOL allows you to get rid of spatial derivatives, reducing your PDE to a set of ODE's - one ODE for every point in space. You then integrate this ODE using a sophisticated ODE solver! Not only does it drastically simplify your algebra (no need to account for the spatial discretization) it allows you to leverage all the power of a fancy ODE solve when solving your PDE.
This book covers MOL's in some detail. After a very useful 100-page overview of the subject, it looks at various ODE algorithms (including most of the standard stiff and non-stiff methods - there's a useful discussion of commercial codes at the end of this section.) They then talk about spatial discretizations, splitting methods, and finally look at ODE solvers specifically designed for solving the large systems of stiff equations that appear in multi-dimensional parabolic PDE's. Throughout, they look at the interplay between choice of spatial discretization and choice of ODE solver.
This book is *not* ideal for someone looking to solve an advection-diffusion-reaction problem today. It doesn't contain simple formulas, and you really have to dig around in the book for some time before picking up hints. For example, it's only mentioned in passing at one point that, for pure diffusion, second-order central discretizations with implicit ODE solvers work well. And this practical knowledge is right after some Fourier Analysis predicting the size of this method's eigenvalues - just the thing to turn off someone scanning the book for a quick solution. While the index and ToC are excellent, and a persistent reader can usually find what they're looking for without too much fuss, I would prefer that the book contained an appendix with 'rules of thumb' for the most common cases.
Nevertheless, where this book shines is as a guide to this interesting subject. While general-purpose pde solvers are becoming and more common (I'll take the chance to mention the new but excellent arb, which will solve almost anything I can throw at in 1-3 dimensions, and syncs with gmsh and vtk) the MOL is worth taking the time to learn. And this book is as good a place as any to start.