9R9. Higher-Order Numerical Methods for Transient Wave Equations. Scientific Computation. - GC Cohen (Domaine de Voluceau, INRIA, Rocquencourt, BP 105, Le Chesnay, 78153 Cedex, France). Springer-Verlag, Berlin. 2002. 348 pp. ISBN 3-540-41598-X. $69.95.
Reviewed by GC Gaunaurd (Code AMSRL-SE-RU, Army Res Lab, 2800 Powder Mill Rd, Adelphi MD 20783-1197).
This is a book reviewing techniques for the numerical solution of the various types of wave equations present in various fields of classical physics, such as acoustics, electromagnetism, and elastodynamics. Portions could be used in undergraduate courses, but the bulk of the book relates to graduate work in numerical analysis.
The book, like Gaul, is divided in three parts. The first part is devoted to the presentation and derivation of the basic wave equations that appear in the three classical fields of physics mentioned above. There are chapters dealing with the equations themselves, on the boundary conditions for each case, and on functional issues such as variational formulations and energy relations. The plane-wave solutions are given special treatment in several space dimensions in the final chapter of Part I.
Part II deals with Finite Difference Methods (FDM) and Part III with Finite Element Methods (FEM), and that is the extent of the book.
The finite difference approximation is the easiest, oldest, and most popular method used in many fields of engineering. Key methods presented here start with the centered second-order approximations to the Laplacian operator and to the second time-derivative operator. Advantages and disadvantages of the method are quickly revealed and several examples are analyzed. These examples dealt with elastodynamic cases (for geophysics) and with electromagnetic cases (for radar applications). The reader is referred to the work of Dr Taflove for further applications of the FDM in the time-domain for the electromagnetic case of Maxwell equations. In general, all the FDM examples discussed here are centered schemes since the uncentered ones generate numerical dissipation. Henceforth, it will only be possible to name some of the chapter titles contained in this part. The author uses construction schemes in homogeneous media in one chapter. The next chapter discusses the dispersion relations generated by the schemes. The stability and accuracy of the solutions are discussed in later chapters on the basis of the dispersion relations. This part concludes with three chapters providing pointers for the construction of schemes for heterogeneous media. Many such schemes are presented and discussed in 1D and 2D.
Complex geometries cannot be treated appropriately using FDM. There are various examples treating this deficiency. The use of FEM is the key to the solution of these problems and deficiencies. The FEM is the subject of Part III.
These FEM approaches introduce a mass matrix. This matrix has to be inverted at each time-step with the consequent cost-increase of the FEM schemes. That was the reason why the finite element methods were initially considered impractical.
This difficulty was solved in the 1980s by using the so-called Gauss-Lobatto quadrature formula to evaluate the mass matrix. The technique was applied to wave equations by a number of researchers, and their works are summarized in various subsequent chapters. Some of the title chapters of this portion deal with “Spectral Elements,” in 2D and 3D. Mixed finite elements eventually overcame those previous difficulties. Such results are extended to quadrilateral (and higher) elements as spectral elements. The mass-lumping technique originally proposed was then extended to triangles and tetrahedrons. This is the key for the use of mixed finite elements for all three types of wave equations considered here. There are some additional problems related to the modeling of unbounded domains. It is often necessary to model very large (but finite) domains as infinite ones. This is particularly true at low frequencies. In general, the sub-domain of propagation providing interesting results is much smaller than the whole domain. But the whole domain may be too large to be incorporated into a computerized numerical model. So, one must search for ways to replace the whole (infinite) domain by a smaller one without distorting or altering the solution. This is the subject of the last chapter on unbounded domains. “Absorbing boundary conditions” (ABC) are useful to handle such situations, but they become problematic above one-space dimension. The use of damping layers has also provided some valid solutions, which unfortunately we cannot describe in the last paragraph of this brief review. This has led to the perfectly matched layer (PML) which is a most active and convenient area of current research for problems in unbounded media.
This ends Part III and the book. There is a final Appendix on Conforming Isomorphisms, a long Bibliography with 124 entries, and an Index. The Schroedinger wave equation of Quantum Mechanics is not discussed.
Higher-Order Numerical Methods for Transient Wave Equations is well written and moderately priced, fills a most needed gap, and contains many jewels that will be more appreciated by the serious student rather than by the casual reader. It will be a valuable addition for libraries, graduate students, and professional researchers dealing with wave propagation problems of classical physics.