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4.6 Exercises
151
Exercise 4.11 Consider the problem
u t =(α(x, t)u x ) x + c(x, t)u x + q(x, t)u
u x ( 2 ,t)=b(t),
u x ( 1 ,t)= a(t),
u(x, 0) = f(x),
where 2 > 1 are given constants, and a(t),b(t),α(x, t),c(x, t), and q(x, t)
are given functions.
(a) Derive an explicit scheme.
(b) Derive an implicit scheme. for x ∈ ( 1 , 2 ), t > 0,
Exercise 4.12 Consider the equation
u t = αu xx ,
with Dirichlet boundary conditions. Here α> 0 is a given constant. We
define an explicit scheme
v m+1 − v m v m − 2v m + v m
j j j−1 j j+1
= α for j =1,... ,n, m ≥ 0,
∆t ∆x 2
and an implicit scheme

v m+1 − v m v m+1 − 2v m+1 + v m+1
j j j
= α j−1 j+1 for j =1,... ,n, m ≥ 0.
∆t ∆x 2
(a) Derive a stability condition for the explicit scheme using the von
Neumann method.

(b) Show that the implicit scheme is unconditionally stable in the sense
of von Neumann.

Exercise 4.13 Consider the equation

u t = u xx ,

with Neumann-type boundary conditions and the following explicit scheme

v m+1 − v m−1 v m − 2v m + v m
j j j
= j−1 j+1 for j =1,... ,n, m ≥ 0.
2∆t ∆x 2
(a) Use the Taylor series to explain the derivation of this scheme.