Biomedical Engineering Reference
In-Depth Information
Stability analysis
In order to render the analysis easier we make the following simplifying assumption
on the structure of the ionic models (see [23, Sect. 3.2.2] and Remark 5 below):
C
I
|
|
,
I
ion
(
V
m
,
w
)
≤
V
m
|
+
|
w
C
g
|
|
(4.19)
g
(
V
m
,
w
)
≤
V
m
|
+
|
w
def
=
def
=
for all
V
m
,
w
, and we set
α
1
+
3
C
I
+
C
g
and
β
C
I
+
3
C
g
. In what follows,
the symbol
indicates an inequality up to a multiplicative constant proportional to
e
T
/
(
1
−
τ max
{
α
,
β
}
)
.
Remark 5.
Assumption (4.19) is not satisfied by the usual ionic models. Neverthe-
less, this simplification allows to motivate, from a theoretical point of view, the
numerically-observed stability properties of the electro-diffusive splittings reported
in Algorithm 1. Note that, for the ionic models (4.10)-(4.13) the assumption holds if
one assumes
a priori
that
V
m
remains uniformly bounded, which allows to estimate
empirically the constants
C
I
and
C
g
(see [23, Remark 3.1]).
The next result, proved in [25], establishes the energy-based stability of Algo-
rithm 1, in terms of
u
e
and
V
m
.
Theorem 2.
Assume that
(4.19)
holds and let
w
n
V
m
,
u
e
)
(
,
be given by Algorithm 1.
Then, under the condition
1
τ
<
{
α
,
β
}
,
(4.20)
max
we have:
•
u
e
,
V
m
)=(
u
n
+
1
e
V
n
+
1
m
For
(
,
)
:
w
n
,
Ω
H
+
χ
m
V
m
m
=
0
τ
σ
n
−
1
u
m
+
e
e
2
0
2
0
2
0
,
Ω
H
+
2
∇
,
Ω
H
n
m
=
0
τ
σ
−
1
)
,
Ω
H
w
0
2
2
0
2
0
V
m
+
1
m
u
m
+
1
e
+
2
i
∇
(
+
,
Ω
H
+
χ
m
V
m
m
=
0
τ
I
m
+
1
n
−
1
app
2
0
2
0
,
Ω
H
+
,
Ω
H
,
with
1
≤
n
≤
N.
u
e
,
V
m
)=(
u
e
,
V
n
+
1
m
•
For
(
)
:
w
n
m
V
m
σ
u
e
1
2
2
0
,
Ω
H
+
χ
2
0
,
Ω
H
+
τ
2
0
,
Ω
H
∇
i
m
=
0
τ
σ
n
−
1
u
m
+
e
m
=
0
τ
σ
n
−
1
)
1
e
1
2
2
0
,
Ω
H
+
2
0
,
Ω
H
V
m
+
1
m
u
m
+
1
e
+
2
∇
i
∇
(
+
w
0
,
Ω
H
+
χ
m
V
m
,
Ω
H
+
τ
σ
u
e
m
=
0
τ
I
m
+
1
n
−
1
app
2
2
0
2
0
2
0
2
0
i
∇
,
Ω
H
+
,
Ω
H
,
with
1
≤
n
≤
N.
