Computational Fluid Dynamics
Lecture 14
Iterative Methods:
Consider the 1D analog:
()
2
2
11
2
2
2
22
ii
i
i
i
p
Rx
x
pp
p
R
x
i
p
px
p
+−
∂
=
∂
−+
=
∆
+∆
=−
R
Jacobi Method:
12
1
2
nn
n
p
pp x
R
+
=+
−
∆
Here
n
means the iteration level, not the time step.
1
+
GaussSiedel Method:
1
2
n
i
2
i
p
x
R
++
−
∆
Uses updated
(
values as soon as they become available.
)
2
1
n
+
GaussSiedel is typically twice as fast as Jacobi iteration, but can’t be accelerated with over
relaxation.
S.O.R. – Successive over relaxation
Relaxation parameter
0
λ
<<
necessary for convergence.
2
1
1
2
n
i
i
n
i
p
x
R
p
λλ
−
∆
+
−
Finding the right value of
is important to the rate of convergence.
There is an optimal value, but finding it is as expensive as solving the original problem.
Typical values near
0.5 or
1.5
==
1
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01
λ
<<
then the current value becomes a weighted average of the GaussSiedel value and the
value from the previous iteration. This is termed
underrelaxation
.
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 Spring '07
 Slinn
 Articles with example pseudocode, Gauss–Seidel method, Jacobi method, Iterative method, Successive overrelaxation

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