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

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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