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θ<
2
,
≤
and in the case of
0
1
+
(
4
−
C
1
)σ
0
<C
1
<
2
−
σ,
C
2
≥
.
(3.33)
−
−
2
σ
C
1
Proof
Define
X
m
u
N
2
L
2
u
m
+
1
N
2
L
2
f
m
+
θ
2
u
m
+
θ
N
2
:=
−
+
C
2
k
∗
−
C
1
k
a
.
The assertion follows if we show that
X
m
≥
0. Then, adding these inequalities for
=
−
m
0
,...,M
1 will obviously give (
3.31
).
u
m
+
1
N
u
N
, then
u
m
+
θ
u
m
+
1
N
(u
N
+
1
:=
−
=
+
−
Let
w
)/
2
(θ
2
)w
and
N
u
m
+
1
N
2
L
2
u
N
2
L
2
(u
m
+
1
N
u
N
,u
m
+
1
u
N
)
(w,
2
u
m
+
θ
N
−
=
−
+
=
−
(
2
θ
−
1
)w).
N
By the definition of the
θ
-scheme, we have
k
−
A
f
m
+
θ
,u
m
+
N
=
k
−
)
(w, u
m
+
θ
N
u
m
+
θ
N
u
m
+
θ
N
2
(f
m
+
θ
,u
m
+
θ
N
)
=
+
a
+
k
−
a
.
u
m
+
θ
N
u
m
+
θ
N
2
f
m
+
θ
≤
a
+
∗
This gives
k
(
2
.
X
m
2
L
2
u
m
+
θ
N
2
f
m
+
θ
u
m
+
θ
N
f
m
+
θ
2
∗
≥
(
2
θ
−
1
)
w
+
−
C
1
)
a
−
2
∗
a
+
C
2
1
1, we now obtain
X
m
In the case of
2
≤
θ
≤
≥
0 if condition (
3.32
) is satisfied.
θ<
2
, we have by the definition of the
θ
-scheme that
(w, v
N
)
=
In the case 0
≤
k(
−
A
u
m
+
θ
N
+
f
m
+
θ
,v
N
)
, yielding
k
A
∗
λ
1
/
2
λ
1
/
2
A
u
m
+
θ
N
f
m
+
θ
w
L
2
≤
A
w
∗
≤
∗
+
k
u
m
+
θ
N
∗
,
=
λ
1
/
2
A
a
+
f
m
+
θ
u
m
+
θ
N
u
m
+
θ
N
u
m
+
θ
N
u
m
+
θ
N
since
(
A
,v
N
)
≤
a
v
N
a
gives
A
∗
≤
a
and choosing
u
m
+
θ
N
u
m
+
θ
N
u
m
+
θ
N
v
N
:=
gives
A
∗
≥
a
. Hence,
k
−
1
X
m
u
m
+
θ
N
2
f
m
+
θ
u
m
+
θ
N
f
m
+
θ
2
∗
≥
(
2
−
C
1
−
σ)
a
−
2
(
1
+
σ)
∗
a
+
(C
2
−
σ)
.
Therefore, we have
X
m
≥
0 if conditions (
3.30
), (
3.33
) hold.
Remark 3.5.2
The conditions (
3.30
), (
3.33
) are time-step restrictions of CFL
1
-type.
Here, time-step restrictions are formulated in terms of the matrix property
λ
.If
A
1
CFL is an acronym for Courant, Friedrich and Lewy who identified an analogous condition as
being necessary for the stability of explicit timestepping schemes for first order, hyperbolic equa-
tions.
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