Computational Fluid Dynamics
Lecture 7
Artificial Dissipation
Lack of dissipation in centered spatial differencing can be a disadvantage.
Dispersed, small scale waves propagate in arbitrary directions without loss of amplitude.
Adding scale selective dissipation is often useful.
Nonlinear problems can cause a cascade of
energy to small scales and lead to numerical instabilities.
Diffusive/Dissapative effects realized by even order derivatives:
()
21
1
e.g.
2
j
jj
j
t
φ
γφ
+−
∂
=−
+
∂
Where
2
γ
is a constant that sets the magnitude of the filter.
( )
2
2
solution
2
1
m
ikx
m
m
m
DA
tx
A
DA t
k
t
ψψ
ψ
∂∂
==
∂
∂
±²³ ²´
e
Damping is largest for largest
k
(smallest wavelength) for the 2
nd
order diffusion operator above,
this leads to.
2
c
o
s
2
so 2
wave
is damped the strongest.
2
A
kxA
t
x
x
π
λ
∂
−
∆
∂
∆=
∆
Assume, for example, second ordered central differences:
N
23
24
11
1
2
truncation error
1
2
26
jj j
j
x
x
xx
x
φφ
αφ
α
σ
∂−
∂
≈+
−
+
=
+
∆ − +
∂∆
∂
∆
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 Spring '07
 Slinn

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