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# Iterative solution techniques can reduce the

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Unformatted text preview: method were introduced by Douglas 1] and Peaceman and Rachford 5]. The ADI method is a predictor-corrector scheme where part of the di erence operator is implicit in the initial (prediction) step and another part is implicit in the nal (correction) step. In the Peaceman-Rachford 5] variant of ADI, the predictor step consists of solving (5.1.2) for a time step t=2 using the backward Euler method for the x derivative terms and the forward Euler method for the y derivative terms, i.e., x y U +1 2 n jk = = U + r 2 U +1 2 + r 2 U : 2 2 n jk x n x jk = y n y jk (5.1.5a) 6 Multi-Dimensional Parabolic Problemss The corrector step completes the solution process for a time step by using the forward Euler method for x derivative terms and the backward Euler method for y derivative terms thus, U +1 n jk = U +1 2 + r2 2 U +1 2 + r2 2 U +1 : n jk = x n x jk = y n y jk (5.1.5b) The computation stencils for both the predictor and corrector steps are shown in Figure 5.1.4. The predictor (5.1.5a) is implicit in the x direction and the corrector (5.1.5b) is implicit in the y direction. 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 Figure 5.1.4: Computational stencil for the predictor (top) and corrector (bottom) steps of the ADI method (5.1.5a, 5.1.5b). Predicted solutions are shown in red and corrected solutions are black. On a rectangular region, the predictor equations (5.1.5a) are solved by the tridiagonal algorithm with the unknowns ordered by rows. Thus, assuming that Dirichlet boundary 5.1. ADI Methods 7 data is prescribed, we write (5.1.5a) at all interior points j = 1 2 : : : J ; 1, in a given row k to obtain (I + C )U +1 2 = g n x where 2 U +1 2 6 1 +1 2 6 U2 6 n = k U +1 2 = n k = n 6 4 = ... k U +1 2 ;1 n 3 2 7 7 7 7 5 (5.1.6a) n yk C = r2 x x 6 6 6 4 2 ;1 ;1 2 = J = k k 2 g =6 4 n yk U ;1 n J k +r n ... y y y y k k 2U ;1 n J k 1 ... ;1 2 ; 7 7 7 5 (5.1.6b) 3 U1 + r 2 U1 =2 n 3 =2 7 5: (5.1.6c) Thus, U +1 2 is determined by solving (5.1.6) using the tridiagonal algorithm for all interior rows k = 1 2 : : : K ; 1. The corrector system (5.1.5b) is solved by ordering the unknowns by columns. In particular, writing (5.1.5b) for all interior points in column j gives n = k (I + C )U +1 = g +1 2 n y n j xj = (5.1.7) where U , C , and g +1 2 follow from (5.1.6b, 5.1.6c) upon replacement of x by y and k by j and interchange of the spatial subscripts. Equation (5.1.7) is then solved by columns for j = 1 2 : : : J ; 1, using the tridiagonal algorithm. The horizon...
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