Example 222 consider 221 with 1 and a sin x x hence

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Unformatted text preview: longer times. (The x = x = t = 12 Finite Di erence Methods 0 −0.1 −0.2 U −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 0.25 0.2 0.5 0.15 0.1 0 0.05 0 t −0.5 x 0 −0.1 −0.2 U −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 0.25 0.2 0.5 0.15 0.1 0 0.05 0 t −0.5 x Figure 2.2.3: Solutions of Example 2.2.1 obtained by the forward time-forward space scheme (2.2.2b) (top) and forward time-backward space scheme (2.2.4) (bottom). Each solution has a Courant number of 1 2. = 2.2. A Kinematic Wave Equation 13 oscillations take longer to develop with smoother initial data.) Once again, the backward space scheme is producing a reasonable approximation of the exact solution. Let us postpone treatment of the forward time-centered space scheme (2.2.5) and seek to understand the di culty with scheme (2.2.2b). Thus, consider (2.2.1) when is a positive constant. The solution of (2.2.1) at a point ( ) is determined by the initial data at the point ( ; 0) (Figure 2.2.5). a xt x at De nition 2.2.1. The domain of dependence of a point ( ) for the initial value probxt lem (2.2.1) is the set o...
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