The nite di erence equation 222b involves three

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Unformatted text preview: erences for approximating the spatial derivatives in (2.2.1a). Thus, using (2.1.6, 2.1.8) in (2.2.1a) we obtain the forward time-backward space di erence scheme U j x j U j U j U n Uj +1 = (1 ; ) n n j Uj + (2.2.4) nn j Uj ;1 : Using (2.1.7, 2.1.8) in (2.2.1a) yields the forward time-centered space di erence scheme +1 = U n ; n j (n; n) (2.2.5) 2 j+1 j;1 These two schemes have the computational stencils shown in Figure 2.2.2. They are used in exactly the same way as the forward time-forward space scheme (2.2.2b). Each scheme has about the same computational complexity. The question to ask at this juncture is whether or not there are any signi cant di erences between the three schemes (2.2.2b), (2.2.4), and (2.2.5). We have not yet studied n Uj j U U : 10 Finite Di erence Methods n+1 n j-1 j j+1 j-1 j j+1 Figure 2.2.2: Computational stencils of the forward time-backward space scheme (2.2.4) (left) and the forward time-centered space scheme (2.2.5) (right). their discretization errors however, based on...
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