1
Atms 502 / CS 505 / CSE 566
Numerical Fluid Dynamics
Tue., Nov. 7, 2006
12/6/06
Atms 502  Fall 2006  Jewett
2
van Leer: Concepts
Concepts
:
•
Grid zones
Not just point values
Local functions describe behavior in zone
If look only at zone mean:
piecewise constant
Next step:
piecewise linear
This is the mean
+ a constant slope
12/6/06
Atms 502  Fall 2006  Jewett
3
•
van Leer
grid zone averages
piecewise constant
van Leer: Upstream scheme
•
Familiar…
grid point values
q
n
+
1
=
q
n
"
c
#
t
#
x
q
i
"
q
i
"
1
( )
=
q
n
"
cq
i
"
cq
i
"
1
[ ]
#
t
#
x
q
1/ 2
=
q
1/ 2
"
u
q
1/ 2
"
u
q
"
1/ 2
[ ]
#
t
#
x
12/6/06
Atms 502  Fall 2006  Jewett
4
van Leer (1977)
•
Key concept:
integrate
under
function at time t
that will be
in
zone
[0,
∆
x]
at
t+
∆
t
.
q
1/ 2
=
q
1/ 2
dx
0
1
"
#
$
+
q
"
1/ 2
dx
"
0
$
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12/6/06
Atms 502  Fall 2006  Jewett
5
van Leer (1977)
•
Fluxes
Grid zone
boundary
velocities (C
grid) shown
q
1/ 2
=
q
1/ 2
dx
0
1
"
#
$
+
q
"
1/ 2
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 Fall '09
 JEWETT
 Derivative, Piecewise linear function, Linear function, Jewett, Van Leer, piecewise linear

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