MPO 662 – Problem Set 4
1. The LaxWendroff method replaces timederivatives with spatial derivatives and the lat
ter are approximated with centered differences.
An alternate algorithm known as the
Warming and Beam (Tannehill
et al.
, 1997; Durran, 1999) replaces the 1st and 2nd
spatial derivatives with upstream differences:
u
n
+1
j
=
u
n
j

μ
u
n
j

u
n
j

1

μ
(1

μ
)
2
u
n
j

2
u
n
j

1
+
u
u
j

2
(1)
where
μ
is the Courant number.
Show that the truncation error of this scheme is
O
(Δ
t
2
,
Δ
x
2
,
Δ
x
Δ
t
) and that it is stable for 0
≤
μ
≤
2.
Hint:
Derive the modified
equation up to second order to show that second order nature of the scheme.
2. Determine the order of accuracy and the stability properties of the slant derivative ap
proximation to the advection equation:
u
n
+1
j

u
n
j
Δ
t
+
c
2
u
n
+1
j

u
n
+1
j

1
Δ
x
+
u
n
j
+1

u
n
j
Δ
x
!
= 0
(2)
Hint: The expansions must be in space and time, be careful in your derivations
3. The advection operator does not produce dissipation at all, and for this reason it is often
discretized with a centered difference scheme that is also free of numerical dissipation.
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 Spring '08
 Iskandarani,M
 Numerical Analysis, Computational fluid dynamics, Tannehill, amplification factor, Durran

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