Unformatted text preview: f all points at t = 0 that determine the solution at (x t). Remark 1. De nition 2.2.1 is particular to (2.2.1). It will have to be amended to
account for inhomogeneous equations, vector systems, and initial-boundary value problems.
As noted, the domain of dependence of the point (x t) for the initial value problem
(2.2.1) is the single point (x ; at 0). In a similar manner, let us use De nition 2.2.1
to identify domains of dependence of the nite di erence schemes (2.2.2b) and (2.2.4).
The solution of the forward time-backward space scheme (2.2.4) at a point (j x n t)
is determined by the initial data on the interval (j ; n) x j x] at t = 0 (Figure 2.2.5).
Thus, following De nition 2.2.1, we'll call this interval the domain of dependence of the
point (j x n t) for the di erence scheme (2.2.4). The domain of dependence of the
forward time-forward space scheme (2.2.2b), however, is the interval j x (j + n) x],
which cannot possibly be correct (Figure 2.2.5). This scheme does not use the correct
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