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Homework # 3
Due: Friday, 9/27
1. (20 pts) Consider the third order Adams-Bashforth method:
Y
i+1
= Y
i
+
∆t
12
23f(t
i
, Y
i
) − 16f(t
i−1
, Y
i−1
) + 5f(t
i−2
, Y
i−2
)
a. Write a computer code to implement this multistep method where you use the third
order RK method from Lab # 2 to get your starting values at Y
1
and Y
2
. Test your code
on the IVP
y
0
(t) = t
2
e
t
3
, 0 < t < 1 y(0) =
1
3
whose exact solution is y(t) =
1
3
e
t
3
and verify that it is indeed third order by computing
the numerical rate of convergence for ∆t =
1
10
,
1
20
,
1
40
,
1
80
.
b. Modify the code from (a) to implement a predictor/corrector pair where we predict
with the explicit method in (a) to get Y
p
i+1
and correct with
Y
i+1
= Y
i
+
∆t
12
5f(t
i+1
, Y
p
i+1
) + 8f(t
i
, Y
i
) − f(t
i−1
, Y
i−1
)
.
Repeat your calculations in (a) using this pair. What do you conclude about the numerical
rate and the magnitude of the error?
2. (8 pts) Write the following fourth order problem as a system of first order IVPs using
the unknowns w
i
(t), i = 1, 2, 3, 4.
y
0000
(t)+4y
00
(t)−y
0
(t) = 4t+y
2
(t) 0 < t ≤ 4 y(0) = 1, y
0
(0) = −2, y
00
(0) = 5, y
000
(0) = 0
3. (12pts) Consider the so-called “theta method”
Y
i+1
= Y
i
+ ∆t
θf(t
i
, Y
i
) + (1 − θ)f(t
i+1
, Y
i+1
)
Note that when θ = 1 we have the forward Euler scheme, when θ = 0 we have the backward
Euler scheme and when θ = 1/2 we have the trapezoidal method. We want to apply this
scheme to solve the IVP y
0
(t) = λy(t), y(0) = 1.
a. Write the method in the form Y
i+1
= ζ(λ∆t)Y
i
. Explicitly give the amplification
factor ζ(λ∆t). Show that your result reduces to the amplification factor we derived for the
forward Euler when θ = 1 and the backward Euler when θ = 0.
b. (UG) (i) Assuming that λ < 0 is real, determine the amplification factor ζ for the
trapezoidal rule where θ = 1/2. (ii) Is the trapezoidal rule explicit or implicit? For what
value(s) of θ is the general scheme implicit?
1
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