6 DISCRETIZATION IN TIME
In the previous chapter the spatial discretization of operators was considered. We now turn our attention to temporal discretizations. We could operate as before, and treat the temporal dimension as just another spatial dimension. This is possible, and has been considered in the past (Zienkiewicz and Taylor (1988)). There are three main reasons why this approach has not found widespread acceptance.
(a) For higher-order schemes, the tight coupling of several timesteps tends to produce exceedingly large matrix systems.
(b) For lower-order schemes, the resulting algorithms are the same as finite difference schemes. As these are easier to derive, and more man-hours have been devoted to their study, it seems advantageous to employ them in this context.
(c) Time, unlike space, has a definite direction. Therefore, schemes that reflect this hyperbolic character will be the most appropriate. Finite difference or low-order finite elements in time do reflect this character correctly.
In what follows, we will assume that we have already accomplished the spatial discretization of the operator. Therefore, the problem to be solved may be stated as a system of nonlinear ordinary differential equations (ODEs) of the form
Timestepping schemes may be divided into explicit and implicit schemes.
6.1. Explicit schemes
Explicit schemes take the RHS vector r at a known time (or ...