Finite-difference time-domain methods (Q2914219)

This chapter of the handbook represents a survey article on finite difference time domain (FDTD) methods for solving initial-boundary value problems of the Maxwell equations. The authors do not intend to provide an extensive summary on all efficient or promising FDTD techniques. Alternatively, the crucial ideas or concepts are explained and some important classes of FDTD methods are discussed. The focus of the article is on perfectly matched layer (PML) absorbing boundary conditions for the simulation of unbounded electromagnetic problems in a time domain. Furthermore, the construction of grids is outlined in the case of two and three space dimensions, where also non-uniform grids and staggered meshes are considered. In the chapter, the contents of the book [\textit{A. Taflove} and \textit{S. C. Hagness}, Computational electrodynamics: the finite-difference time-domain method. 2nd ed. London: Artech House (2000; Zbl 0963.78001)] is partly reused.NEWLINENEWLINEThe authors start with a description of the historical background and a general classification of FDTD methods in Section 1. Section 2 introduces the Maxwell equations both in differential form and integral form for three space dimensions. Reduced systems including two space dimensions are also presented for a special case. In Section 3, the authors review the algorithm of \textit{K. S. Yee} [IEEE Trans. Antennas Propag. 14, No. 3, 302--307 (1966; Zbl 1155.78304)], i.e., an explicit FDTD method on a uniform grid. The application of more sophisticated spatial meshes is discussed briefly in Sections 4 and 5. The phenomenon of numerical dispersion is analysed in Section 6 and results of numerical simulations are presented to illustrate this effect. Section 7 includes a summary of some strategies to reduce the errors caused by numerical dispersion. The authors explain the concept of numerical stability for the FDTD methods in Section 8, where numerical simulations are presented again to demonstrate an instable behaviour. It follows that explicit FDTD schemes are stable only if the time-step size is below a bound determined by the Courant number. To achieve an unconditionally stable method, an alternating-direction implicit time-stepping algorithm, which was introduced in [\textit{F. Zheng}, \textit{Z. Zhen} and \textit{J. Zhang}, ``Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method'', IEEE Trans. Microw. Theory Techn. 48, No. 9, 1550--1558 (2000; \url{doi:10.1109/22.869007})], is analysed in Section 9. Section 10 represents the largest part of the article, where PMT absorbing boundary conditions are considered. The authors review several techniques starting from [\textit{J.-P. Berenger}, J. Comput. Phys. 114, No. 2, 185--200 (1994; Zbl 0814.65129)] and continuing with stretched-coordinate PML as well as uniaxial PML, which includes also material by S. D. Gedney and A. Taflove [loc. cit.]. Numerical simulations of illustrative examples in both two and three space-dimensions are presented and the errors of the methods are quantified.NEWLINENEWLINEThe article is written carefully and the contents are presented comprehensibly. The authors provide a survey on important and promising FDTD methods in computational electrodynamics for the simulation of real-life applications. This survey article is appropriate for both engineers and mathematicians.NEWLINENEWLINEFor the entire collection see [Zbl 1217.78003].