Convection-Diffusion Problems: An Introduction to their Analysis and Numerical Solution
update: 2019-01-22 09:29:55

Many physical problems involve diffusive and convective (transport) processes. When diffusion dominates convection, standard numerical methods work satisfactorily. But when convection dominates diffusion, the standard methods become unstable, and special techniques are needed to compute accurate numerical approximations of the unknown solution. This convection-dominated regime is the focus of the book [1]. After discussing at length the nature of solutions to convection-dominated convection-diffusion problems, the authors motivate and design numerical methods that are particularly suited to this class of problems.

At first they examine finite-difference methods for two-point boundary value problems, as their analysis requires little theoretical background. Upwinding, artificial diffusion, uniformly convergent methods, and Shishkin meshes are some of the topics presented. Throughout, the authors are concerned with the accuracy of solutions when the diffusion coefficient is close to zero. Later in the book they concentrate on finite element methods for problems posed in one and two dimensions.

For numerical analysts working in this area, the standard reference book is Roos, Stynes and Tobiska, which has already reached its 2nd edition [2,3]. This book contains a lot of useful information, but it is pitched at a fairly high level and so is daunting for those beginners who have some familiarity with numerical methods and their analysis but who have not previously worked with convection-diffusion and other singularly perturbed differential equations. An easier, more introductory book is needed to encourage new people to enter this fascinating research area. This gap in the published literature is demonstrated by the popularity of an introductory survey article written by M.Stynes for Acta Numerica in 2005 [4].

The new book [1] is an extended and updated version of [4], where exercises and other material are added to make the book more attractive and more useful for the novice reader. It is published by the American Mathematical Society in their “Graduate Studies in Mathematics” series, as the main target audience for the book is graduate students in numerical analysis. In addition, researchers from other areas who wish to learn quickly the basic techniques needed to deal with singularly perturbed differential equations and convection-dominated problems will find the book useful.

The book is based on a course of lectures presented by M.Stynes at the AARMS (Atlantic Association for Research in the Mathematical Sciences) Summer School at Dalhousie University in Halifax, Canada during July 2015.

References:

1.          M. Stynes and D. Stynes, Convection-Diffusion Problems: An Introduction to Their Analysis and Numerical Solution, Graduate Studies in Mathematics Vol. 196, American Math. Society, Providence, 2018.  Print ISBN: 978-1-4704-4868-4. Electronic ISBN: 978-1-4704-5021-2.

2.          H.-G.Roos, M.Stynes and L.Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics Vol. 24, Springer-Verlag, Berlin, 1996. ISBN: 3-540-60718-8

3.          H.-G.Roos, M.Stynes and L.Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics Vol. 24, Springer-Verlag, Berlin, 2008. ISBN: 978-3-540-34466-7

4.       M.Stynes, Steady-state convection-diffusion problems, in Acta Numerica 2005 (A.Iserles, ed.), Cambridge University Press, Cambridge, 2005, pp.445--508.