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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 MultiDimensional 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 CrankNicolson 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 CrankNicolson 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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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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