MAE 290B, Winter 2010
HOMEWORK 3
Due Wed 02242010 in class
Provide source codes used to solve all problems
PROBLEM 1
Consider the advection  diffusion equation
∂
t
u
+
c∂
x
u
=
ν∂
xx
u,
1. Discretize this equation spatially using 2ndorder centered finite difference formulae
for the convective terms and the diffusive terms.
2. Discretize the resulting set of ODEs temporally using a lowstorage, mixed RK3
θ
method. Express this discretization as a function of the two nondimensional parate
mers
CN
=
c
Δ
t/
Δ
x
(Courant Number) and
V N
=
ν
Δ
t/
(Δ
x
)
2
.
3. Apply VonNeumann stability analysis to determine the stability of the resulting spatio
temporal scheme. For this purpose,
(a) Find the amplification factor
σ
=
u
n
+1
j
/u
n
j
as a function of
CN
,
V N
and
k
Δ
x
.
(b) Plot the contour

σ

= 1 as a function of
k
Δ
x
and
CN
for
V N
= 0
,
5
,
10
,
20.
Use this plot to discuss the stability of the scheme
(c) Consider now the case
V N
= 0. Find the maximum value of
CN
for which the
scheme is stable. Relate this result to the stability region of the RK3 scheme.
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 Spring '10
 MAE290B
 Numerical Analysis, Expression, convective terms, courant number

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