Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations

This book deals with numerical methods for solving partial differential equa tions (PDEs) coupling advection, diffusion and reaction terms, with a focus on time-dependency. A combined treatment is presented of methods for hy perbolic problems, thereby emphasizing the one-way wave equation, meth ods for parabolic problems and methods for stiff and non-stiff ordinary dif ferential equations (ODEs). With regard to time-dependency we have at tempted to present the algorithms and the discussion of their properties for the three different types of differential equations in a unified way by using semi-discretizations, i. e. , the method of lines, whereby the PDE is trans formed into an ODE by a suitable spatial discretization. In addition, for hy perbolic problems we also discuss discretizations that use information based on characteristics. Due to this combination of methods, this book differs substantially from more specialized textbooks that deal exclusively with nu merical methods for either PDEs or ODEs. We treat integration methods suitable for both classes of problems. This combined treatment offers a clear advantage. On the one hand, in the field of numerical ODEs highly valuable methods and results exist which are of practical use for solving time-dependent PDEs, something which is often not fully exploited by numerical PDE researchers. Although many problems can be solved by Euler's method or the Crank-Nicolson method, better alter natives are often available which can significantly reduce the computational effort needed to solve practical problems.

List of contents

I Basic Concepts and Discretizations.- II Time Integration Methods.- III Advection-Diffusion Discretizations.- IV Splitting Methods.- V Stabilized Explicit Runge-Kutta Methods.

Summary

This book deals with numerical methods for solving partial differential equa tions (PDEs) coupling advection, diffusion and reaction terms, with a focus on time-dependency. A combined treatment is presented of methods for hy perbolic problems, thereby emphasizing the one-way wave equation, meth ods for parabolic problems and methods for stiff and non-stiff ordinary dif ferential equations (ODEs). With regard to time-dependency we have at tempted to present the algorithms and the discussion of their properties for the three different types of differential equations in a unified way by using semi-discretizations, i. e. , the method of lines, whereby the PDE is trans formed into an ODE by a suitable spatial discretization. In addition, for hy perbolic problems we also discuss discretizations that use information based on characteristics. Due to this combination of methods, this book differs substantially from more specialized textbooks that deal exclusively with nu merical methods for either PDEs or ODEs. We treat integration methods suitable for both classes of problems. This combined treatment offers a clear advantage. On the one hand, in the field of numerical ODEs highly valuable methods and results exist which are of practical use for solving time-dependent PDEs, something which is often not fully exploited by numerical PDE researchers. Although many problems can be solved by Euler's method or the Crank-Nicolson method, better alter natives are often available which can significantly reduce the computational effort needed to solve practical problems.

Additional text

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 theSpringer Series in this field of research." (J. Brandts, Nieuw Archief voor Wiskunde, Vol. 7 (1), 2006)

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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 theSpringer Series in this field of research." (J. Brandts, Nieuw Archief voor Wiskunde, Vol. 7 (1), 2006)

Product details

• Authors: Wille Hundsdorfer, Willem Hundsdorfer, Jan Verwer, Jan G Verwer, Jan G. Verwer
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 13.10.2010
• EAN: 9783642057076
• ISBN: 978-3-642-05707-6
• No. of pages: 472
• Dimensions: 154 mm x 238 mm x 27 mm
• Weight: 722 g
• Illustrations: X, 472 p.
• Series: Springer Series in Computational Mathematics; Springer Series in Computational Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Technology > General, dictionaries; Analysis, B, Numerische Mathematik, Mathematics and Statistics, Mathematical and Computational Engineering, Ordinary Differential Equations, Numerical analysis, Partial Differential Equations, Differential calculus & equations, Differential equations, Engineering mathematics, Applied mathematics, Mathematical and Computational Engineering Applications