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4.6 Exercises
Exercise 4.16 Motivated by the result of Exercise 4.15, we study, in this
exercise, the difference scheme 4.49 with θ =1/2. This difference scheme
is usually referred to as the Crank-Nicholson scheme. In component form
the scheme is given by

m
m
v
− v
1
m+1
j
j
j
j
=
j+1
j+1
j−1
∆t
(∆x)
2
(∆x)
for j =1, 2,... ,n and m ≥ 0.
(a) Show that this implicit scheme is unconditionally stable in the sense
of von Neumann. v m+1 − 2v m+1 2 + v m+1 + v j−1 − 2v m 2 + v m 153
n
m
(b) Discuss how the vectors v m+1 ∈ R can be computed from v .
(c) Show that the solution of the Crank-Nicholson scheme for the initial-
boundary value problem (4.1) admits the representation
n
m
m
v j = γ k a(µ k ) sin(kπx j ),
k=1
∆t ∆t −1 !
n
0
where a(µ)= 1 − µ 1+ µ and γ k =2∆x v sin(kπx j ).
j
2 2 j=1
(d) Show that the amplification factor of the difference scheme, a(µ),
satisfies

|a(µ) − e −µ∆t | = O (∆t) .
3
How does this result relate to the corresponding result for the explicit
scheme (4.4)? Compare your result with the conclusions you derived
in Exercise 4.15.
(e) Implement the Crank-Nicholson scheme. Choose the initial function
f(x) as in Example 4.2 and try to verify that the scheme is uncondi-
∆t
tionally stable by varying the parameter r = (∆x) 2 .
Exercise 4.17 Consider the general scheme (4.49). Use the von Neumann
method to discuss the stability for any θ ∈ [0, 1].
Exercise 4.18 Consider the equation
u t + cu x = u xx ,
with Dirichlet-type boundary conditions. Here c ≥ 0 is given constant.
(a) Show that this problem has a family of particular solutions of the
form
2
−(ikπc+(kπ) )t ikπx
e e .