pb4_10 - MPO 662 Problem Set 4 1 Consistency and stability...

MPO 662 – Problem Set 4
1 Consistency and stability
You have reverse-engineered 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 time-derivative 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 space-time point ( x
n
, t
n +
1
2
) and show
that the scheme is second-order when θ = 1 / 2, and first-order 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