Page 154 -

4.4 An Implicit Scheme
lot of methods available. In this section, we will present the simplest and
probably most popular method: the standard implicit scheme. We do not
want to go too deep into the discussion of different methods here; the topic
is far too large and the interested reader may consult e.g. Thomee [27] and
references given there for further discussions.
Before we start presenting the implicit scheme, let us remind ourselves
of the basic difference between explicit and implicit schemes. We stated
above, on page 120, that a scheme is called explicit if the solution at one
time step can be computed directly from the solution at the previous time
step. On the other hand, we call the scheme implicit if the solution on the
next time level is obtained by solving a system of equations. 141
We want to derive an implicit scheme for the following equation:
u t = u xx for x ∈ (0, 1), t > 0,
u(0,t)=0, u(1,t)=0, (4.38)
u(x, 0) = f(x).
Borrowing the notation from the explicit scheme, we apply the following
approximations:
u(x, t +∆t) − u(x, t)
u t (x, t +∆t) ≈ ,
∆t
and
u(x − ∆x, t +∆t) − 2u(x, t +∆t)+ u(x +∆x, t +∆t)
u xx (x, t +∆t) ≈ .
∆x 2
This leads to the following scheme:
v m+1 − v m v m+1 − 2v m+1 + v m+1
j j j
= j−1 j+1 for j =1,... ,n, m ≥ 0.
∆t ∆x 2
The computational molecule of this scheme is depicted in Fig. 4.7.
The boundary conditions of (4.38) imply that
v m = 0 and v m =0
0 n+1
for all m ≥ 0, and the initial condition gives

v = f(x j ) for j =1,... ,n.
0
j
In order to write this scheme in a more convenient form, we introduce the
m
n
m T
vector v m ∈ R with components v m =(v ,... ,v ) . Then we observe
n
1
that the scheme can be written as
m
(I +∆tA)v m+1 = v , m ≥ 0. (4.39)