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).
Search WWH ::




Custom Search