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we increase the temporal approximation accuracy for the linearised problem in the time
interval
(
t
n
,t
n
+1
)
up to the second order. Note that here
T
n
+
4
kl
+
T
n
+
p−
1
4
kl
T
kl
:=
,
p
= 1
,
4
.
(24)
2
Theorem 1.
Finite difference schemes (12)-(13) are balanced.
Proof.
Consider, e.g., (16). Multiplying the left-hand side by
τRΔλ
,wehave
τ
+
m
k
τRΔλ
=
T
n
+
2
k
−T
n
+
2
k
+1
m
k
+1
+
T
n
+
2
− T
n
+
2
k−
1
m
k−
1
RΔλ
k
T
n
+
2
τ
2
R
cos
2
ϕ
l
Δλ
.
k
+1
D
k
+1
/
2
+
T
n
+
2
k−
1
D
k−
1
/
2
+
T
n
+
2
−
(
D
k
+1
/
2
+
D
k−
1
/
2
)
(25)
k
Summing all over the
k
's, due to the periodicity in
λ
the terms with
D
k±
1
/
2
cancel.
Doing in the same manner with the right-hand side of (16), we find
RΔλ
k
T
n
+
2
k
− T
k
RΔλ
k
τ
2
f
n
+
2
k
=
,
(26)
that is scheme (12) is balanced. In particular, under
f
n
+
2
k
=0
we obtain the mass
conservation law at a fixed latitude
ϕ
l
=
k
T
n
+
2
k
T
k
.
(27)
k
Theorem 2.
The finite difference operators
A
Δλ
and
A
Δϕ
in (12)-(13) are negative
definite.
Calculations for (13) can be done in a similar way.
Proof.
Consider (13) on grid (11). Multiply the right-hand side by
T
l
|
cos
ϕ
l
|
and sum
all over the
l
's. It holds
D
l
+1
/
2
T
l
|
cos
ϕ
l
|
1
R
2
|
cos
ϕ
l
|
Δϕ
T
l
+1
− T
l
Δϕ
T
l
− T
l−
1
Δϕ
− D
l−
1
/
2
l
1
R
2
Δϕ
2
=
D
l
+1
/
2
(
T
l
+1
− T
l
)
T
l
−
D
l−
1
/
2
(
T
l
− T
l−
1
)
T
l
l
l
1
R
2
Δϕ
2
=
D
l
+1
/
2
(
T
l
+1
− T
l
)
T
l
−
D
l
+1
/
2
(
T
l
+1
− T
l
)
T
l
+1
l
l
1
R
2
Δϕ
2
=
D
l
+1
/
2
(
T
l
+1
− T
l
)(
T
l
− T
l
+1
)
l
1
R
2
Δϕ
2
D
l
+1
/
2
(
T
l
+1
− T
l
)
2
≤
0
.
(28)
=
−
l
Here
l
=
l −
1
, and we used the periodicity of the solution in
ϕ
.
Calculations for (12) are similar.
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