Higher-Order Numerical Methods for Transient Wave Equations , Gary C. Cohen Springer–Verlag, New York, 2002. $69.95 (348 pp.). ISBN 3-540-41598-X
Problems involving wave propagation occur in many disciplines, including acoustics, electromagnetics, geophysics, elasticity, and fluid dynamics. Long ago, in Methods for the Approximation of Time Dependent Problems (GARP Publications Series, No. 10, 1973), Heinz-Otto Kreiss and Joseph Oliger advocated the use of high-order discretization for such problems, especially for long-time integration. Kreiss and Oliger argued that the number of grid points needed in a unit interval of wavelength depends on the time interval of integration: The higher the formal order of the discretization scheme, the weaker is its dependence on the time interval. To my knowledge, Higher-Order Numerical Methods for Transient Wave Equations , by Gary C. Cohen, is the first book to address specifically the use of high-order discretizations in the time domain to solve wave equations. Traditionally, the engineering community solved such problems in the frequency domain, but recently have tended to abandon that harmonic approach amid an increase of interest in complex pulselike sources containing a large range of frequencies. Cohen, a researcher at the French National Institute for Research in Computer Science and Control (INRIA), has been working on this topic for more than 15 years.
The book is divided into three parts. Part 1 (chapters 1–3) presents the governing equations for acoustics, electromagnetism, and material elasticity. This part provides a useful overview of functional spaces and introduces the appropriate Sobolev spaces. Chapter 3 presents plane-wave solutions to Maxwell’s equations and to the elastics system. This chapter is useful, but touches only briefly on the major issue of boundary conditions.
Part 2 of the book (chapters 4–10) deals with construction of second- and fourth-order standard finite-difference schemes for both space and time. The material is fairly standard except for the modified-equation approach introduced in the mid-1980s, which evolved to uniform fourth-order spatiotemporal schemes. The notation in chapter 4 is perhaps unnecessarily heavy: It renders cryptic and hard-to-recognize even simple Taylor expansions that should be familiar to undergraduate students! I also expected to find more on compact (implicit) schemes and treatment of stability issues associated with boundary conditions—including reduction of the order of accuracy at the boundaries. However, the book does not contain such material. A one-line reference to spectral methods dismisses them as “… very difficult to implement …”(p. 57). Other topics in part 2 include stability of the discrete schemes (including energy techniques), numerical dispersion and anisotropy, reflection-transmission analysis, and construction of schemes in heterogeneous media. The description is quite rigorous but easy to follow, and the remarks at the end of each section are very useful.
Part 3 (chapters 11–14) presents various finite-element formulations and contains by far the most interesting material in the book. The key development in the use of finite elements in this context is the mass-lumping concept. In mass lumping, one constructs a diagonal mass matrix in a way that does not impair the formal accuracy of the finite-element discretization. The author presents mass lumping in detail, although his account of its history is somewhat erroneous. The use of Gauss–Lobatto points was introduced at MIT in the early 1980s in the context of spectral elements. In talking about the MIT contribution, the author refers to spectral elements as any finite elements of order higher than three and thus seems to contradict the vast majority of literature on the subject. The quadrilateral/hexahedral elements favored by the author have good approximation properties. However, the author seems unaware that tensorproduct bases developed in the past 10 years make triangular/tetrahedral elements competitive with—perhaps even better than—quadrilateral/hexahedral ones. Therefore, the text’s discussion entitled “Tetrahedra or Hexahedra?” is not justified.
The section in Part 3 on mixed formulations, which is quite good, extends the classical finite-difference Yee scheme to finite elements. For transient wave equations, Cohen’s extension of the Yee scheme is more efficient and accurate than the classical Lagrange finite elements. The improvements result from Cohen’s use of a new class of elements, called edge elements, to ensure continuity at the elemental interfaces of the tangential components of the finite-element basis functions.
The book’s last chapter (chapter 14), devoted to outflow boundary conditions, analyzes in detail the method of perfectly matched layers (PML). The chapter mentions other methods, but not in enough detail for the unfamiliar reader to compare their merits with those of PML.
Cohen’s book should be useful, especially to new researchers, and could even be a reference in a course. I would not recommend it as a textbook, because it has no problem sets, is not self-contained, and does not treat other useful topics such as compact schemes, explicit and implicit filtering, and discontinuous Galerkin methods. However, I recommend the book for its clear and cogent coverage of the material selected by its author.
