Computational Fluid Dynamics
Lecture 6
Space differencing errors.
0
C
tx
ψ
∂∂
+=
Seek traveling wave solutions.
(
)
ik
x
t
e
ω
−
k
is wave number and
is frequency.
=kC
is dispersion relation.
where
C
is phase speed.
C
k
=
, true solution is non dipersive for constant
C
.
note that the group velocity,
g
C
, the speed of energy propagation is defined as
g
C
k
∂
=
∂
, in this case
g
CC
=
2
nd
order spatial differences.
()
11
2
2
2
2
2
0
2
seek discrete solutions.
2
2
sin
sin
sin
Note that this is the sin
jj
j
j x
t
j
ik
x
ik
x
t
t
ik
x
ik
x
C
C
C
C
C
C
et
ee
ie
C
e
x
Ce
e
xi
C
kx
x
C
kx C
kk
x
c
ωω
φ
φφ
+−
∆−
∆
∆
∂−
∂∆
=
−
+
∆
−
=
∆
=∆
∆
=
∆
∆
=
∆
sin
sin
function,
, which has the nice property
1 as
0.
xx
→→
1
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View Full DocumentWhich is a function of
k
, dispensive unlike solutions to the true advection equation.
()
3
2
2
For good resolution
1 and Taylor series sin
6
1
1
6
C
x
kx
x x
CC
k
x
∆≈
≈−∆
±
−
Phase speed error is time lagging and second order.
Phase error larger for larger
, worst case is:
∆
2
2
,
2
sin
0
C
k
x
x
x
x
x
π
λ
=−
∆
=
∆
∆
∆
==
∆
∆
The phase speed of the 2
x
∆
wave is zero. Noise doesn’t propagate at all.
Since
2
C
ω
is real, there is no change in wave amplitude with time, and no amplitude error.
The group velocity of waves
2
cos
C
Ck
k
x
∂
=
∂
∆
is approximate correct for small
∆
.
But for poorly resolved waves, larger
2
, 0
2
x
x
∆
<∆<
∆=
∆
cos
1
and the energy propagation for poorly resolved waves is
2
C
C
k
∂
= −
∂
The energy propagates backwards!
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 Spring '07
 Slinn
 Wavelength, group velocity, phase velocity, ∆x, phase speed

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