There is a fundamental difference between space and time. Space extends from negative infinity to positive infinity, whereas time only increases in one direction. This inherent difference requires finite difference approximation for time derivatives to be different from spatial derivatives.
In general, there are two ways to form a discretized approximation of a time derivative. They are the single time step method and the multilevel time discretization method. In this chapter, the single time step method will be discussed briefly. The multilevel discretization method will be discussed in greater detail. One major difference between the single time step method and multilevel methods is that, for wave propagation problems, the latter, when properly implemented, would lead to dispersion-relation-preserving (DRP) schemes (see Tam and Webb, 1993). DRP property is very desirable for computing wave propagation problems. DRP schemes will be discussed in detail in the next two chapters.