Spring 2003 · Title: Spring 2003 Created Date: 20100615110158-05'00'
Capstone Final Presentation Spring 2003
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Numerical Analysis
Miao LiTom Schreck
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21( ) sin cos
ug g
t
u u u i k
3 2 3 31[ cos sin ]
3
hh h gh h gh
t
i
33 2 3[ ] ( ) [ ]
h hh h D h h
t x
Simplification of N.S. Equation
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Simplification of N.S. Equation cont.
From:
We get the 1D equation:
33 2 3[ ] ( ) [ ]
h hh h D h h
t x
3 3 3[ ] ( )[ ] ( )xxx x x x xh
h h D h h ht
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Two Methods Used
An explicit scheme using forward Euler.
An implicit scheme using Newton’s Method
( ) ( )( ) (1 ) ( )
i ii i
y t t y tf t f t t
t
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Program RunsH(X) vs. X
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20 25
X
T=0
T=1
T=2
T=3
T=4
T=5
T=6
T=7
T=8
T=9
T=10
T=11
T=12
T=13
T=14
T=15
T=16
T=17
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Program Runs Cont.Compare H(x) for D=0,1,2
x
0 5 10 15 20 25
H(x
)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
D=0 D=1 D=2
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Program Runs Cont.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25
Series1
Series2
Series3
Series4
Series5
Series6
Series7
Series8
Series9
Series10
Series11
Series12
Series13
Series14
Series15
Series16
Series17
Series18
Series19
Series20
Series21
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Convergence TestsMaxH vs. Time
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 5 10 15 20 25
Time (s)
dx = .05
dx = .1
dx = .2
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Implicit Method1 *n
i i iy y q 1n
if
=*if +
*|i jj
fq
q
*
, (1 )i
ii j j
j
ft q Rq
1,..., 1xi N
* *(1 )i
n ni i i iR y y tf tf 1,..., 1xi N
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