Computational Fluid Dynamics
Lecture 8
Combining Time and Space Differencing
0
uu
C
tx
∂∂
+=
1. Suppose forward time and centered space
unstable.
→
2. Leap Frog and centered space
→
neutral, conditionally stable
3. Forward time and one sided space
could be stable, conditionally stable
→
4. Leap Frog and onesided space
→
unstable, because LF has waves going both ways.
It is possible that damping from one sided spatial differences with amplification from a time
discretization to produce a stable scheme.
Consider LF and Central differences, LF has positive phase speed error, and Central differences
have negative (decelerating) phase speed errors.
Sometimes it is useful to examine time and space differencing properties together.
We use a traveling wave solution of the form:
()
with
r
i
r
ik
j x
n t
n
j
ri
nt
n
j
n
n
j
e
i
ee
Ae
ω
φ
ωω
∆−
∆
∆
∆
∆
=
=+
=
=
Determining the imaginary part of
is equivalent to a Von Neumann stability analysis.
Phase speed
properties are contained in
r
.
0
nd
11
Examine LF and 2
Central differences
0
22
nn
jj
A
C
φφ
+−
=
−−
∆∆
Substitute in wave solution
1
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i
22
or
sin
sin
Discrete
sin
sin
Dispersion
Relation
where
If
1
will be real and the scheme is neutral.
0
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 Spring '07
 Slinn
 Frequency, John von Neumann, ∆x, Cphys, e− ik ∆x

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