31 computational grid used for the nite di erence

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Unformatted text preview: re r = (2.3.3b) t 2: x The initial and boundary conditions (2.3.1b, 2.3.1c) are 0 Uj =( j ) x j =0 1 ::: J (2.3.3c) 18 Finite Di erence Methods n 0 U = n UJ =0 n> 0 (2.3.3d) : The computational stencil for (2.3.3a) is shown in Figure 2.3.2. The parameter analogous to the Courant number (2.2.3) for the kinematic wave equation (2.2.1). 11 00 11 00 is n+1 11 00 j-1 r 11 00 j n j+1 Figure 2.3.2: Computational stencil of the forward time-centered space scheme (2.3.3a) for the heat conduction equation (2.3.1a). The solution of (2.3.3a) is obtained in the same manner as the nite di erence solutions of (2.2.2b) and (2.2.4) for the kinematic wave equation (2.2.1). Thus, using the initial data (2.3.3c), we calculate a solution j1 at the interior mesh points, =12 ; 1, of time level 1 using (2.3.3a) with ranging from 1 to ; 1. The 1 boundary conditions (2.3.3d) with = 1 determine 01 and J . Knowing the discrete solution at time level 1, we proceed to determine it at time level 2, etc. in the sam...
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