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Interpolations. Achi Brandt. Two Dimensional Interpolation. First direction Second direction Bileanear. Bicubic … Bipolynomial. Linear Interpolation 2h →h. ”second order” interpolation. P-order interpolation. Linear interpolation: p = 2 Cubic interpolation: p = 4. u (x j ). u (x). - PowerPoint PPT Presentation
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Interpolations
Achi Brandt
Two Dimensional Interpolation
First direction Second direction
Bileanear. Bicubic … Bipolynomial
Linear Interpolation 2h →h
”second order” interpolation
1121 (Error iii UUU
UU y oscillatorfor 0
)("
"2
2
2211
2
2
22
ih
hiiih
xU
Uh
UUU
UUh smooth for 0 2
P-order interpolation
• Linear interpolation: p = 2• Cubic interpolation: p = 4
u (xj)
u (x)
Xi+1 Xi+2 Xi+pX
1110)(
pp xaxaaxw
)()(1 xwxw p )()( 1 pii xuwu
ji
piij
jij
jj xx
xxxxx
x
xxw
1)(,
)(
)()(
1
1
1)(
)(1
,,2
,
if5.
if1
if )(Error
)(!
)()(Error
qiqi
pii
jjpp
ppii
xxxqp
xxx
hxxuh
up
xxxxxwxu
Fine-to Coarse Transfer (Restriction)
Injection
Full Weighting
hf 2 hhI2 hf
hi
hi ff 22
hif2 h
ihi
hi fff 122124
1 2
h
h
h
h
f
f
f
f2
3
22
21
2
~
hhI2
25.5.25.
25.5.25.
25.5.25.
h
h
h
h
h
h
h
f
f
f
f
f
f
f
7
6
5
4
3
2
1
hf~
Linear
interpolation:
Second order transfers.
General order, general dimension d:
121
121
121
412h
hI
1
2
11
211
2
1
21
2hhI
Thh
hh II 22
12
Thh
hh II d 22
12
1-D linear interpolation 2h h
Adjoint h 2h transfer (anterpolation)
h
k
k
k
k
k
22
12
2
12
22
hI2
1
5.
00
0
0
5.
15.
0
0
0
05.
1
h
k
k
k
V
V
V
V
2
)1(2
2
)1(2
=
=
h
k
k
k
V
V
V
V
2
)1(2
2
)1(2
=
=
ThI221
5.0
0
25.25.
0
05.
0
025.
25.
00
5.
h
k
k
k
k
k
22
12
2
12
22
Coarse-Grid Aliasing
Coarse-Grid Aliasing
