High-order methods have gained increasing attention in recent years. Their theoretical development has reached a high level of sophistication, and at the same time the range of applications has been broadening, including such diverse topics as global atmospheric modeling, aerodynamics, oceanography, thermal convection, and theoretical chemistry. Specialized conferences on the subject like the International Conference on Spectral Applications and High-Order Methods (ICOSAHOM) have been launched to bring mathematicians, engineers, and computer scientists together in order to stimulate further work in the field and to prospect new areas: high-order time schemes, treatment of singularities, complex geometries, mixed discretization techniques, domain decomposition, and parallelism. These topics were once considered as the stumbling block of spectral methods. As time goes on, this is no longer true, and high-order methods apply more and more to real-life engineering problems.

The monograph by Gottlieb and Orszag [163] and the book by Canuto et al. [64] remain milestones in the subject. They are cited in almost every paper written on the topic. Gottlieb and Orszag's monograph was the first on the subject and contains very little about applications. Moreover, it is silent on the topics mentioned above. Most of the developments covered by Canuto et al. are devoted to simple geometries, but a last chapter entitled “Domain Decomposition Methods” introduces extensions to more complex geometries. Recent achievements in the field of high-order methods have far-reaching consequences for geometrically complex configurations.