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1
Atms 502
Numerical Fluid Dynamics
Tue., Sep. 19, 2006
9/29/06
Atms 502  Fall 2006  Jewett
2
Program 2  in C
•
The following approach seems to work:
•
…
#deﬁne NX 25
#deﬁne BC_WIDTH 1
#deﬁne I1 BC_WIDTH
#deﬁne I2 I1+NX1
#deﬁne NXDIM NX+2*BC_WIDTH
update(u1,i1,i2,nx)
ﬂoat u1[ ];
int i1,i2,nx;
{
int i;
for (i=i1; i<=i2; i++)
u1[i]=u2[i];
}
update(u1,nx)
ﬂoat u1[ ];
int nx;
{
int i;
for (i=1; i<=nx; i++)
u1[i]=u2[i];
}
OLD
NEW
9/29/06
Atms 502  Fall 2006  Jewett
3
DiscreteDispersion Relation
•
Physical root behavior:
•
2
∆
x speed is …
•
Lagging phase error (decelerating)
how do we know this?
c
phys
=
"
phys
k
=
1
k
#
t
arcsin
μ
sin
k
#
x
( )
c
phys
$
c
1
%
k
2
#
x
2
6
1
%
2
( )
’
(
)
*
+
For small k
∆
x 
Durran calls “in limit
of good spatial resolution”
9/29/06
Atms 502  Fall 2006  Jewett
4
Takacs (1985)
•
Considerations
:
Oddorder
schemes generally dissipative
(upstream)
Evenorder
schemes generally dispersive
(leapfrog)
Constructs scheme with “some of improved phase
characteristics associated w/thirdorder scheme”
•
Secondorder schemes
:
4thorder amplification errors
3rdorder phase errors
Extra point => 3rdorder accuracy, reducing phase error
Takacs pp. 10511052
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9/29/06
Atms 502  Fall 2006  Jewett
5
Takacs (1985)
•
Amplitude error
(Fig. 1)
θ
is
k
∆
x
;
μ
is Courant number
Oddorder schemes most dissipative
Evenorder schemes least dissipative
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This note was uploaded on 10/06/2009 for the course ATMOSPHERI 502 taught by Professor Jewett during the Spring '09 term at University of Illinois at Urbana–Champaign.
 Spring '09
 JEWETT

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