MPO 662 – Problem Set 4
1
Consistency and stability
You have reverseengineered a computer program and deciphered that it updates the variable
u
according to the following rule:
u
n
+1
j

u
n
j
Δ
t
+
c
3
u
n
j

4
u
n
j

1
+
u
n
j

2
2Δ
x
= 0
(1)
•
Determine the continuous PDE this finite difference equation is trying to approximate.
•
Determine the truncation error for this scheme and derive the leading terms of the
modified equations.
•
Determine the stability characteristics of this scheme.
•
With the timederivative kept in continuous form, study the dispersion characteristics of
the spatial discretization.
2
Heat equation
The concepts of stability, consistency and convergence extend readily to parabolic equations.
Here we will study several instances of such equations. The 1D heat equation,
u
t
=
νu
xx
, is
discretized using a
θ
scheme in time according to:
u
n
+1
j

u
n
j
Δ
t
=
νθ
u
n
+1
j
+1

2
u
n
+1
j
+
u
n
+1
j

1
Δ
x
2
+
ν
(1

θ
)
u
n
j
+1

2
u
n
j
+
u
n
j

1
Δ
x
2
(2)
1. Derive the Taylor series of this scheme around the spacetime point (
x
n
, t
n
+
1
2
) and show
that the scheme is secondorder when
θ
= 1
/
2, and firstorder otherwise.
Hint: Start
with the spatial expansion and then the temporal one.
2. Derive the amplification factor as a function of
θ
and deduce the stability conditions for
the cases
θ
= 0
,
1
/
2 and 1.
3. Discuss the algorithmic and computational advantages or disadvantages of each choice
of
θ
.
4. Plot the decay factor as a function of
k
Δ
x/π
for the three different values of
θ
and
compare it to that of the analytical factor
A
a
=
e

αk
2
Δ
x
2
where
α
=
ν
Δ
t
Δ
x
2
. Try different
values of
α
: 0.1 0.25, 0.5, 0.75, 1 and 10 (please use only those that are stable in the
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 Spring '08
 Iskandarani,M
 Numerical Analysis, Boundary value problem, initial condition, Boundary conditions, amplification factor, computational grid

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