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Implicit approximation can be solved using:
• Point iteration (G/S, SOR)
• Direct (matrix) solution
• Combination of matrix soln and iteration
(used in MODFLOW)
1D
 2h
S h

2
T t
x
Implicit Approximation
hi 1n 1  2hi n 1  hi 1n 1
(x) 2
hi
n 1
n 1
n
S hi  hi

T
t
1
  [Ghin  hin11  hin11 ]
E
Solve by iteration
or
2
2
S
(

x
)
S
(

x
)
hi 1n 1  (2 
)hi n 1  hi 1n 1  
hi n
Tt
Tt
In this form, the equation can be solved directly
using matrix methods. See W&A, p. 95.
2
2
S
(

x
)
S
(

x
)
hi 1n 1  (2 
)hi n 1  hi 1n 1  
hi n
Tt
Tt
All known terms are on the RHS;
all unknown terms are on the LHS.
Ahi 1
n 1
 Bhi
n 1
 Chi 1
n 1
F
A  C 1
S (x) 2
B  2 
Tt
S (x) 2 n
F 
hi
Tt
2D
 2h
x
2
hi 1, j n 1  2hi , j n 1  hi 1, j n1
(x)
2


 2h
S h

2
T t
y
hi, j 1n 1  2hi , j n 1  hi , j 1n1
(y) 2
n 1
n
h

h
S i, j
i, j

T
t
Let x =  y = a
2
2
Sa
Sa
hi 1, j n 1  hi 1, j n 1  hi, j 1n 1  hi , j 1n 1  (4 
)hi , j n 1  
hi, j n
Tt
Tt
The motivation behind the Alternating Direction Implicit Procedure
is to keep the coefficient matrix tridiagonal so that we can use the
Thomas algorithm to solve the matrix equation.
Not tridiagonal
2
2
Sa
Sa
hi 1, j n 1  hi 1, j n 1  hi, j 1n 1  hi , j 1n 1  (4 
)hi , j n 1  
hi, j n
Tt
Tt
Tridiagonal
2
2
Sa
Sa
n 1
n 1
n 1
n
n
n
hi , j 1  hi , j 1 (4 
)hi , j

hi, j  hi 1, j  hi 1, j
Tt
Tt
solution oriented along columns
In the next time step, the solution is oriented along rows.
2
2
Sa
Sa
hi 1, j n 1  hi 1, j n 1  hi, j 1n 1  hi , j 1n 1  (4 
)hi , j n 1  
hi, j n
Tt
Tt
Tridiagonal
2
2
Sa
Sa
n 1
n 1
n 1
n
n
n
hi 1, j  hi 1, j (4 
)hi , j

hi, j  hi, j 1  hi , j 1
Tt
Tt
solution oriented along rows
In point iteration, the 5-point operator moves over
each node in the grid….
In the ADI matrix solution, the 5-point equations are
assembled into one matrix equation for each column
(or row).
Examples of solution techniques that combine
matrix solution with iteration:
IADI (see chapter 5 of W&A)
SSOR*
SIP*
PCG2*
*Used in MODFLOW