Biomedical Engineering Reference
In-Depth Information
u
e
,
V
m
)=(
u
e
,
V
m
)
•
For
(
:
w
n
,
Ω
H
+
χ
m
V
m
,
Ω
H
+
τ
σ
u
e
,
Ω
H
+
τ
σ
V
m
1
2
1
2
2
0
2
0
2
0
2
0
i
∇
i
∇
,
Ω
H
m
=
0
τ
σ
n
−
1
u
m
+
e
,
Ω
H
w
0
,
Ω
H
+
χ
m
V
m
e
2
0
2
0
2
0
+
2
∇
,
Ω
H
+
τ
σ
V
m
,
Ω
H
+
τ
σ
u
e
m
=
0
τ
I
m
+
1
n
−
1
app
2
2
2
0
2
0
2
0
i
∇
i
∇
,
Ω
H
+
,
Ω
H
,
with
1
≤
n
≤
N.
Theorem 2 shows that electro-diffusive Gauss-Seidel and Jacobi splittings are
energy-stable under condition (4.20), as for the unsplit case
u
e
,
V
m
)=(
u
n
+
1
e
V
n
+
1
m
)
(analyzed in [23]), but with slightly altered energy norms. As a result, stability is not
compromised.
(
,
Remark 6.
The proof of Theorem 2 does not depend on the time discretization con-
sidered in steps 1 and 2 of Algorithm 1. Indeed, we do not make use of any numer-
ical dissipation produced by the scheme, apart from the one directly provided by
the splitting. Therefore, the backward Euler quotients
∂
τ
V
n
+
m
, can be
safely replaced by a second-order backward difference formula, and then corrected
(see, e.g., [59, 60]) to recover overall second-order accuracy.
∂
τ
w
n
+
1
and
Remark 7.
The above stability result can be adapted, with minor modifications, to
the case
u
e
,
V
m
)=(
u
n
+
1
e
V
m
)
(
,
. The full Jacobi splitting, obtained after replacing
V
m
,
w
n
+
1
V
m
,
w
n
I
ion
(
)
by
I
ion
(
)
in step 2, could also be considered.
We conclude this subsection with a few numerical illustrations. The results re-
ported in Fig. 4.7 (top) confirm that the electro-diffusive Gauss-Seidel and Jacobi
splittings do not introduce additional constraints on the time-step size
, as stated
in Theorem 2. Fig. 4.7 (bottom) shows that the Coupled, the Gauss-Seidel and the
Jacobi time-marching schemes all provide the expected first-order accuracy
τ
O
(
τ
)
in time. Note that, at a given time-step size, Gauss-Seidel is slightly more accurate
than Jacobi and that the Coupled scheme is slightly more accurate than Gauss-Seidel.
This accuracy shifting could be related to the energy-norm weakening observed in
the stability analysis.
4.5.2 Coupled heart-torso system
In this subsection, we propose a series of time-marching procedures for the heart-
torso system (4.8) that allow a decoupled computation of the transmembrane, ex-
tracellular and torso potentials. The main idea consists in combining the bidomain
splittings of the previous section with a specific explicit Robin-Robin treatment of
the heart-torso coupling conditions. The proposed schemes are presented in Algo-
rithm 2, with
γ
>
0 a dimensionless free parameter (fixed below) and where we have
t
assumed that
σ
T
|
Σ
=
σ
T
I
(without loss of generality).