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nde_homework2.pdf
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Homework # 2
Due: Wednesday, 9/18
1. (20 pts) In Section 2.1 we derived a second order explicit Taylor series method.
a. Modify the derivation in Section 2.1 to derive the second order implicit Taylor series
method
Y
0
= y
0
Y
i
= Y
i−1
+ ∆tf(t
i
, Y
i
) −
∆t
2
2
[f
t
(t
i
, Y
i
) + f(t
i
, Y
i
)f
y
(t
i
, Y
i
)] , i = 1, 2, . . . N
for the IVP (1.2)
b. (G) Derive a third order accurate explicit Taylor series method for the IVP (1.2).
c. Do a hand calculation to perform one step of the implicit method from (a) for the
IVP
y
0
(t) = ty(t) 0 < t ≤ 1 y(0) = 2
using ∆t = 0.1. Compute to the actual error. Give your answers to six digits of accuracy.
2. (15pts) The Heun method is given by
Y
i+1
= Y
i
+
1
4
∆tf(t
i
, Y
i
) +
3
4
∆tf
t
i
+
2
3
∆t, Y
i
+
2
3
∆tf(t
i
, Y
i
)
.
a. (UG) Verify that the local truncation error is at least order (∆t)
3
; this means that
you have to verify that terms through (∆t)
2
disappear but you don’t have to verify that
the terms of order (∆t)
3
don’t disappear (although they don’t so the local truncation error
is exactly third order).
b. (G) Verify that the local truncation error is exactly (∆t)
3
.
3. (15 pts) Consider the two-step method
Y
i+1
= Y
i−1
+ 2∆tf(t
i
, Y
i
) .
This differs from the Adams-Bashforth methods that we derived because it uses the ap-
proximation Y
i−1
and not just Y
i
. These types of multistep methods can still be derived
by using an interpolating polynomial for f but in this case we use a polynomial over the
interval [t
i−1
, t
i+1
] and thus we must integrate the differential equation y
0
(t) = f(t, y) over
this interval.
a. Use an appropriate constant interpolating polynomial for f(t, y) to derive this method.
b. Determine the exact local truncation error for the method; i.e., don’t just say for
example order ∆t but rather give the constant in front of ∆t.
1
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