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point
(t
m
,x
i
)
. We collect the discretization errors in the grid points at time
t
m
in the
error vector
ε
m
,i.e.
ε
i
u
i
:=
u(t
m
,x
i
)
−
,
0
≤
i
≤
N
+
1
,
0
≤
m
≤
M.
(2.13)
ε
m
M
m
The error vectors
{
}
satisfy the difference equation
=
0
k
−
1
I
θ
G
ε
m
+
1
+
−
θ)
G
ε
m
k
−
1
I
E
m
+
+
(
1
−
=
(2.14)
or, in explicit form,
ε
m
+
1
A
θ
ε
m
η
m
,
=
+
(2.15)
where
η
m
:=
(k
−
1
I
+
θ
G
)
−
1
E
m
, and where
A
θ
:=
(k
−
1
I
+
θ
G
)
−
1
(
−
k
−
1
I
+
(
1
θ)
G
)
, is called an
amplification matrix
.
The recursion (
2.15
) shows that the discretization error
ε
i
is related to the con-
sistency error
E
i
. Estimates on
ε
i
can be obtained by taking norms in the recursion
(
2.15
). Using induction on
m
,wehave
−
Proposition 2.3.3
Fo r a l l M
∈ N
,1
≤
m
≤
M
,
one has
m
−
1
ε
m
m
ε
0
m
−
1
−
n
E
n
2
≤
A
θ
2
,
2
+
k
0
A
θ
2
,
(2.16)
2
n
=
2
=
N
+
1
ε
m
2
ε
i
2
.
where
|
|
i
=
0
We see from (
2.16
) that the discretization errors
ε
i
can be controlled in terms of
the consistency errors
E
i
A
θ
2
is bounded by 1. The condition
that the norm of the amplification matrix
A
θ
is bounded by 1 is a
stability condition
for the FDM. We obtain immediately
provided
the norm
Theorem 2.3.4
If the stability condition
A
θ
2
≤
1
(2.17)
,
the FDM
(
2.10
)
converges and
,
if
ε
0
→∞
→∞
=
holds
,
then
,
as M
,
N
0
,
ε
m
E
m
m
2
≤
T
sup
m
2
.
(2.18)
sup
We want to discuss the validity of the stability condition (
2.17
)for
G
as in (
2.9
).
Therefore, we need the following lemma to obtain the eigenvalues for a tridiagonal
matrix. It follows immediately by elementary calculations.
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