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and the tridiagonal matrices
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
2
−
1
1
.
.
.
.
.
.
.
.
.
1
h
2
−
1
N
N
sym
×
N
×
N
G
=
∈ R
,
I
=
∈ R
.
.
.
.
.
.
.
.
.
.
sym
−
1
−
12
1
(2.9)
Then, after multiplication by
k
, the finite difference scheme (
2.8
) becomes:
Find
u
m
+
1
N
∈ R
such that for
m
=
0
,...,M
−
1
,
I
θk
G
u
m
+
1
=
I
θ)k
G
u
m
k(θ
f
m
+
1
θ)
f
m
),
+
−
(
1
−
+
+
(
1
−
(2.10)
u
0
=
u
0
.
We now show that the vectors
u
m
converge towards the exact solution as
k
→
0 and
h
→
0.
2.3.2 Convergence of the Finite Difference Method
Naturally, by the transition from the PDE (
2.4
) to the finite difference equa-
tions (
2.8
), which is called the
discretization
of the PDE, an error is introduced, the
so-called
discretization error
, which we analyze next. We begin with the definition
of a related
consistency error
.
Definition 2.3.1
The consistency error
E
i
at
(t
m
,x
i
)
is the difference scheme (
2.7
)
with
u
i
m
i
in
E
replaced by
u(t
m
,x
i
)
.
Using Taylor expansions of the exact solution at the grid point
(t
m
,x
i
)
, we can
readily estimate the consistency errors
E
i
in terms of powers of the mesh width
h
and the time step size
k
.
Proposition 2.3.2
If the exact solution u(t, x) of
(
2.4
)
is sufficiently smooth
,
then
,
as h
→
0,
k
→
0,
the following estimates hold for m
=
1
,...,M
−
1
and
i
=
1
,...,N
:
|
E
i
|≤
C(u)(h
2
+
k),
0
≤
θ
≤
1
,
(2.11)
1
2
,
E
i
C(u)(h
2
k
2
),
|
|≤
+
=
θ
(2.12)
where the constant C(u) >
0
depends on the exact solution u and its derivatives
.
For the convergence of the FDM, we are interested in estimating the error be-
tween the finite difference solution
u
i
and the exact solution
u(t, x)
at the grid
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