Unformatted text preview: method were introduced by Douglas 1] and
Peaceman and Rachford 5].
The ADI method is a predictorcorrector 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 PeacemanRachford 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 MultiDimensional 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
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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.
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis

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