# 24 the solution of the forward time backward space

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Unformatted text preview: nitial data to determine the solution at (j x n t). These simple arguments lead to the famous Courant, Friedrichs, Lewy Theorem 2, 1] which we'll state in the context of (2.2.1). Theorem 2.2.1. (Courant, Friedrichs, Lewy). A necessary condition for the conver- gence of the solution of a nite-di erence approximation to the solution of (2.2.1) for arbitrary initial data is that the domain of dependence of the nite-di erence approximation contain the domain of dependence of the partial di erential equation (2.2.1). Remark 2. We'll give a formal de nition of convergence in the next section however, informally, convergence implies that the solution of the di erence scheme approaches the 14 Finite Di erence Methods 3 2 U 1 0 −1 −2 −3 4 8 3 6 2 4 1 2 0 t 0 x 1 U 0.5 0 −0.5 −1 4 8 3 6 2 4 1 2 0 t 0 x Figure 2.2.4: Solutions of Example 2.2.2 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...
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