1 lecture 24: flux limiters 2 last time… l developed a set of limiter functions l second order...
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Lecture 24: Flux Limiters
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Last Time…
Developed a set of limiter functions
Second order accurate
3
This Time…
Examine physical rationale for limiter functions
Application to unstructured meshes
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Recall Higher-Order Scheme for e
Consider finding face value using a second-order
scheme with the gradient found at the upwind cell:
Recall:
What is the limiter function trying to do?
( )
2P W
e P e
xr
x
E Pe
P W
r
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Limiter Functions
=2r
0 for r 0r
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Physical Interpretation
The value of r can be thought of as the ratio of two
gradients:
Limiter chooses gradient adaptively to avoid creating
extrema
E Pe
P W
r
Downwind cell gradient
Upwind cell gradient
ww
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Case (a): Linear Variation
Since:
If variation is a straight line, on a
uniform mesh, r=1
From our limiter function range,=1
for r=1
Can use either gradient and get the
right value at e
E P
P W
r
r=1
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Case (b): 2>r>1
r>1 means
If we used =1, we would not
create overshoot
In fact we can use up to r and
not create
E P P W
e E
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Case (b): 2>r>1 (Cont’d)
Consider case when re >1, i.e.,
Say we choose the =re line
When =re :
( )2
1
2
21 1
2 2
P We P e
E PP P W
P W
E PP
P E
E
xr
x
E P P W
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Case (b’): r>2
Consider case when re >2, i.e.,
For re>2, say we choose the =2 line
When =2:
( )2
1 11
P We P e
P P W
E PP
e
P Ee e
E
xr
x
r
r r
E P P W
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Case (c): 0< r<1
If r<1:
E P P W
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Case (c): 0<r<1 (Cont’d)
Consider case when 0<re <1, i.e.,
Say we choose =re
When =re :
( )2
1
2
21 1
2 2
P We P e
E PP P W
P W
E PP
P E
E
xr
x
E P P W
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Case (d): r<0
When r<0, this implies local
extremum
Our limiter has =0 for r<0
This implies e P
Defaults to first order upwind scheme
0 for r 0r
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Unstructured Meshes
Find face value using:
No easy way to define rf
0 0 0( )f f fr r
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Unstructured Meshes
1 0
00
ff
rr
• Create fictitious point U
•Find value at U by using cell gradient
•Hence define rf
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Closure
In this lecture, we
Considered the physical meaning of the limiter
function
Saw that it was an adaptive way to choose either an
upwind or a downwind gradient to find face value
Looked at difficulties in implementing for unstructured
meshes