Lecture 37
Implicit Methods
The Implicit Difference Equations
By approximating
u
xx
and
u
t
at
t
j
+1
rather than
t
j
, and using a backwards difference for
u
t
, the
equation
u
t
=
cu
xx
is approximated by
u
i,j
+1

u
i,j
k
=
c
h
2
(
u
i

1
,j
+1

2
u
i,j
+1
+
u
i
+1
,j
+1
)
.
(37.1)
Note that all the terms have index
j
+ 1 except one and isolating this term leads to
u
i,j
=

ru
i

1
,j
+1
+ (1 + 2
r
)
u
i,j
+1

ru
i
+1
,j
+1
for
1
≤
i
≤
m

1
,
(37.2)
where
r
=
ck/h
2
as before.
Now we have
u
j
given in terms of
u
j
+1
. This would seem like a problem, until you consider that
the relationship is linear. Using matrix notation, we have
u
j
=
B
u
j
+1

r
b
j
+1
,
where
b
j
+1
represents the boundary condition.
Thus to find
u
j
+1
, we need only solve the linear
system
B
u
j
+1
=
u
j
+
r
b
j
+1
,
(37.3)
where
u
j
and
b
j
+1
are given and
B
=
1 + 2
r

r

r
1 + 2
r

r
.
.
.
.
.
.
.
.
.

r
1 + 2
r

r

r
1 + 2
r
.
(37.4)
Using this scheme is called the
implicit method
since
u
j
+1
is defined implicitly.
Since we are solving (37.1), the most important quantity is the maximum absolute eigenvalue of
B

1
, which is 1 divided by the smallest
ew
of
B
. Figure 37.1 shows the maximum absolute
ew
’s of
B

1
as a function of
r
for various size matrices. Notice that this absolute maximum is always less
than 1. Thus errors are always diminished over time and so this method is always stable. For the
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 Fall '08
 Young,T
 Equations, Numerical differential equations, Implicit Method, Explicit and implicit methods

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