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# 13b y the computational stencil for 513 is shown in

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Unformatted text preview: ed exactly as in one dimension. Thus, beginning with the initial conditions 111 000 111 000 j n 111 000 111 000 111 000 k 111 000 Figure 5.1.2: Computational stencil for the explicit scheme (5.1.3). U 0 = (j x k y) j = 0 1 ::: J jk k = 0 1 ::: K (5.1.3c) and assuming that the solution U , j = 0 1 : : : J , k = 0 1 : : : K , has been calculated, (5.1.3a) is used to compute U +1 at all interior points j = 1 2 : : : J ;1, k = 1 2 : : : K ; 1. The boundary conditions n jk n jk U +1 n jk = (j x k y (n + 1) t) furnish the solution on @ . j=0 J k=0 K (5.1.3d) 4 Multi-Dimensional Parabolic Problemss Absolute stability of (5.1.3a) is assured in the maximum norm when r +r x y 1=2: If x = y the stability requirement is t= x2 1=4, which is more restrictive than in one dimension. Thus, there is an even greater motivation to study implicit methods in two dimensions. When the Crank-Nicolson method is applied to (5.1.2), we nd U +1 ; U n n jk jk t " = # 2 (U +1 + U ) (U +1 + U ) + y2 x2 2 2 2 n n jk x jk n n jk y jk or 1; r x 2+r 2 x 2 y y rU +1 = 1 + x n jk 2+r 2 x y 2 y U : (5.1.4a) n jk The computational stencil of (5.1.4a) is shown in Figure 5.1.3. 11 00 11 00 11 00 11 00 11 00 11 00 j 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 n 11 00 k 11 00 11 00 11 00 Figure 5.1.3: Computational stencil for the Crank-Nicolson scheme (5.1.4). For simplicity, assume that trivial boundary data is prescribed, i.e., = 0 in (5.1.2c) order the equations (5.1.4a) and unknowns by rows and write (5.1.4a) in the matrix form (I + 1 C)U +1 = (I ; 1 C)U 2 2 n n (5.1.4b) 5.1. ADI Methods 5 where U = U1 1 : : : U ;1 1 U1 2 : : : U ;1 2 : : : U1 n n n n n J n J K 2 DD 6D D D C=6 x 6 4 x ... y DD y 2 D =6 4 7 7 7 5 x ... ba ;1 ;1] n J K T (5.1.4c) 7 7 7 5 (5.1.4d) x 3 ab 6b a b 6 ::: U 3 y y ;1 2 3 c 6 D = 6 c ... 6 4 y c 7 7 7 5 (5.1.4e) and a = 2(r + r ) x y b= r ; x c= r : ; y (5.1.4f) The matrices D and D are (J ; 1) (J ; 1) tridiagonal and diagonal matrices, respectively, so C is a (J ;1)(K ;1) (J ;1)(K ;1) block tridiagonal matrix. This system may be solved by an extension of the tridiagonal algorithm (Figure 4.1.4) to block systems ( 3], Chapter 2) however, this method requires approximately (5=3)KJ 3 multiplications per time step. This would normally be too expensive for practical computation. Iterative solution techniques can reduce the computational cost and we will reconsider these in Chapter 9. For the present, let us discuss a solution scheme called the alternating direction implicit (ADI) method. Variations of this...
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