Introduction

The finite difference time domain method, usually referred to as the FDTD, is a particular implementation of a general class of methods known as finite difference techniques. The FDTD is so widely used in the CEM community that although finite difference methods cover a wide spectrum of complexity and accuracy, it is the FDTD which is almost always implied in CEM when finite differences are mentioned.

Finite difference methods are numerical methods in which derivatives are directly approximated by finite difference quotients. The general class of such methods is the most intuitive numerical approach, and was the first to be extensively developed by the scientific computing community. To this day, it probably remains the most universally applicable numerical technique and the one most widely used for scientific computation. As just discussed, for dynamic problems in CEM, the most popular is the FDTD method. The opening discussion in this chapter will discuss finite differences in general, before moving on to the specifics of the FDTD.

At this point, a general comment about the philosophy underlying the mathematical treatment of the computational algorithms in this book would be in order. Although we endeavor not to be “sloppy” mathematically, the emphasis in this book is in presenting well-known methods for well-known problems in CEM, rather than on the basic mathematical requirements of the methods, as one would expect to find in an applied mathematics text, for instance. An example of the type of issue which we will gloss over, at least initially, is the differentiation of discontinuous functions, which requires the generalized (weak) derivative, properly the field of functional analysis.