1
Atms 502
Numerical Fluid Dynamics
Thu., Nov. 2, 2006
Monotonicity;
van Leer (1977) methods
Conservative, monotonic schemes
References:
Wilhelmson, Wicker Notes
Droegemeier notes
Durran chapters. 3, 5
11/6/06
Atms 502  Fall 2006  Jewett
3
Monotonicity
•
A monotonic scheme
is one which:
1
Given an initial distribution that is
monotonic
before
advection …
Produces a distribution that is
monotonic
after
advection.
•
Durran
makes an important distinction:
A monotone
scheme guarantees that
2
:
In a
monotonicitypreserving
scheme:
1
Droegemeier
if
a
j
n
"
b
j
n
for all j, at time n, then
a
j
n
+
1
"
b
j
n
+
1
will hold for all j and n.
2
wilhelmson
if
a
j
0
"
a
j
+
1
0
for all j, at time 0, then
a
j
n
"
a
j
+
1
n
will hold for all j and n.
11/6/06
Atms 502  Fall 2006  Jewett
4
•
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
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2
11/6/06
Atms 502  Fall 2006  Jewett
5
Monotonicity
•
When we refer to
monotonicity
we will
mean
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 Fall '09
 JEWETT
 Types of functions, Step function, Van Leer

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