Biomedical Engineering Reference
In-Depth Information
Algorithm 2
Decoupled time-marching for the heart-torso system.
1. Ionic state: find
w
n
+
1
∈
X
H
,
h
such that
∂
τ
w
n
+
1
)
ξ
=
+
g
(
V
m
,
w
n
+
1
0
Ω
H
X
H
,
h
;
2. Transmembrane potential: find
V
n
+
1
m
for all
ξ
∈
∈
X
H
,
h
such that
m
Ω
H
∂
τ
V
n
+
1
V
n
+
1
m
u
e
·
∇φ
χ
φ
+
Ω
H
σ
∇
·
∇φ
+
Ω
H
σ
∇
i
i
m
I
n
+
1
app
)
φ
V
m
,
w
n
+
1
=
−
I
ion
(
Ω
H
for all φ
∈
X
H
,
h
;
3. Extracellular potential: find
u
n
+
1
e
∈
X
H
,
h
such that
t
T
V
m
·
∇ψ
+
γσ
u
n
+
1
e
u
n
+
1
e
Ω
H
(
σ
i
+
σ
)
∇
·
∇ψ
+
Ω
H
σ
i
∇
ψ
e
h
Σ
T
Σ
σ
T
∇
u
T
·
n
T
ψ
+
γσ
u
T
ψ
=
−
h
Σ
for all ψ
∈
X
H
,
h
;
4. Torso potential: find
u
n
+
1
T
∈
X
T
,
h
T
T
·
∇ζ
+
γσ
Σ
σ
T
∇
u
T
·
n
T
ζ
+
γσ
Ω
T
σ
T
∇
u
n
+
1
u
n
+
1
T
u
n
+
1
e
ζ
=
ζ
T
h
h
Σ
Σ
X
T
,
h
;
5. Go to next time-step.
for all
ζ
∈
Note that, since the (quasi-static) time discretization of (4.8)
3
,
4
does not generate
numerical dissipation in time, a naive Dirichlet-Neumann explicit coupling, obtained
by enforcing
u
n
+
1
T
u
e
=
Σ
,
on
u
n
+
1
e
u
n
+
T
·
σ
e
∇
·
n
=
−
σ
T
∇
n
T
on
Σ
,
might lead to numerical instability.
The next result, proved in [25], establishes the energy-based stability of Algo-
rithm 2. There,
E
H
u
e
,
V
m
) denotes the discrete bidomain energy
arising in the stability estimates provided by Theorem 2. For instance, in the case
(
V
m
(resp.
E
H
u
e
,
u
e
,
V
m
)=(
u
e
,
V
m
)
,wehave
E
H
u
e
,
V
m
def
=
w
0
,
Ω
H
+
χ
m
V
m
,
Ω
H
+
τ
σ
V
m
,
Ω
H
+
τ
σ
u
e
2
2
2
0
2
0
2
0
2
0
i
∇
i
∇
,
Ω
H
,
