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Multiscale in Science and Engineering manuscript No.(will be inserted by the editor)
Asymptotic convergence rates of Schwarz
Waveform Relaxation algorithms for
Schrodinger equations with an arbitrary
number of subdomains
Xavier ANTOINE1, Emmanuel LORIN2,3?
1 Institut Elie Cartan de Lorraine, UMR CNRS 7502, Universite de Lorraine,
Inria Nancy-Grand Est, SPHINX Team, F-54506 Vandoeuvre-les-Nancy Cedex,
France.
2 School of Mathematics and Statistics, Carleton University, Ottawa, Canada,
K1S 5B6.
3 Centre de Recherches Mathematiques, Universite de Montreal, Montreal, Canada,
H3T 1J4.
The date of receipt and acceptance will be inserted by the editor
Abstract We derive some asymptotic estimates of the rate of convergence
of Schwarz Waveform Relaxation (SWR) domain decomposition methods for
the Schrodinger equation when using an arbitrary number of subdomains.
Hence, we justify that under certain conditions, the rates of convergence
? Present address: Insert the address here if needed
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2 Xavier ANTOINE, Emmanuel LORIN
mathematically obtained for two subdomains [8–10] are still asymptotically
valid for a larger number of subdomains, as it is usually numerically observed
[25].
1 Introduction
We are interested in this paper in the analysis of the rate of convergence of
some SWR Domain Decomposition Methods (DDMs) by using an arbitrary
number of subdomains. This study is an extension of existing results about
the convergence of SWR algorithms on 2 subdomains [8–10]. We show that
the convergence rates established for 2 subdomains are actually still accu-
rate estimates for an arbitrary number of sufficiently large subdomains and
bounded potentials. In this paper, we will mainly focus on the computa-
tion of contraction factors from Lipschitz continuous mappings involved in
the proof of convergence of SWR methods. As a consequence, we will not
introduce technical details about the full proof of convergence. Instead, we
refer to several papers where the reader could find them, depending on the
equation under consideration. For linear advection and diffusion reaction
equations, we refer to [19]. The analysis and derivation for the Schrodinger
equation in the time-dependent case is presented in [10,11,17,18], while the
stationary equation is studied in [8,9].
The SWR algorithms are well-established methods for the parallel so-
lution of evolution partial differential equations by allowing computations
of the underlying PDE on small subdomains with very good speed-up. The
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Title Suppressed Due to Excessive Length 3
convergence of the overall SWR DDM is strongly dependent on the choice
of the transmission conditions between the subdomain interfaces. Typically,
classical Dirichlet transmission conditions will usually provide very slow
convergence, while transparent transmission conditions provide a very fast
converging solution. In this paper, we do not discuss the space discretization
of the algorithm, but focus on the convergence of the continuous in-space
algorithms. We refer to [1,16,18–26] for details about the SWR methods. In
this paper, we analyze the convergence rate for the Classical SWR (CSWR)
algorithm which is based on Dirichlet transmission conditions, and Optimal
SWR algorithm (OSWR) based on transparent transmission conditions. In
the latter case, convergence of the 2-subdomain algorithm is optimal in the
sense that it occurs in only 2 iterations [9,25].
This paper is organized as follows. In Section 2, we analyze the rate of
convergence of the CSWR and OSWR DDMs for the stationary Schrodinger
equation solved by the imaginary-time method. The analysis is extended
in Section 3 to real-time dynamics, and some numerical experiments are
proposed in Section 4.
2 Stationary states problems
We study the convergence of the SWR methods by using an arbitrary
number m > 2 of subdomains, for computing the point spectrum of the
Schrodinger Hamiltonian by using the imaginary-time method [3,5–7,12–
15]. We refer to [8,19] regarding the well-posedness of the equation and of
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4 Xavier ANTOINE, Emmanuel LORIN
the convergence of the algorithms. We intend to determine the ground state
to the following one-dimensional Schrodinger Hamitonian −4+V (x) (used
in quantum mechanics, acoustics wave propagation, optics), where the po-
tential V is supposed to be smooth and bounded with bounded derivative.
The imaginary-time method, which is also called Normalized Gradient Flow
(NGF) method, reads: for t0 = 0 < t1 < · · · < tn < tn+1 < · · · , solve
∂tφ(t, x) = 4φ(t, x)− V (x)φ(t, x)φ(t, x), x ∈ Ω, tn < t < tn+1,
φ(t, x) = 0, x ∈ ∂Ω, tn < t < tn+1,
φ(tn+1, x) := φ(, t+n+1, x) =φ(t−n+1, x)
||φ(·, t−n+1)||L2(R),
φ(0, x) = ϕ0(x), x ∈ R,with ||ϕ0||2L2(R) = 1,
(1)
where ϕ0 is a given initial guess, usually built from an ansatz. The procedure
is repeated until the convergence is reached, that is when the following
stopping criterion is satisfied
||φ(tn+1, ·)− φ(tn, ·)||L∞(R) 6 δ, (2)
for δ > 0 small enough. We propose the following decomposition (with
possible overlap): Ω = ∪mi=1Ωi, such that Ωi = (ξ−i , ξ+i ), for 2 6 i 6 m− 1,
Ω1 = (−a, ξ+1 ) and Ωm = (ξ−m,+a) (see Fig. 1). Moreover, the overlapping
size is: ξ+i − ξ−i+1 = ε > 0, and |Ωi| = L+ ε, where L is assumed to be much
larger than ε. We also have ξ±i+1 − ξ±i = L. In the following, we will denote
by φ(k)i the local solution in the subdomain Ωi at Schwarz iteration k.
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Title Suppressed Due to Excessive Length 5
Fig. 1 Domain decomposition with possible overlapping.
We study the convergence rate of the CSWR and OSWR algorithms. The
convergence rate is here defined as the slope of the logarithm residual history
with respect to the Schwarz iteration number, i.e. (k, log(E(k))) : k ∈ N,
where
E(k) :=
m−1∑i=1
∥∥ ‖φcvg,(k)i∣∣(ξ−i+1,ξ
+i )− φcvg,(k)
i+1∣∣(ξ−i+1,ξ
+i )‖∞∥∥L2(0,T (kcvg))
6 δSc, (3)
and φcvg,(k)i (resp. T (kcvg)) denotes the NGF converged solution (resp. time)
in Ωi at Schwarz iteration k, δSc being a small parameter. More specifically,
we intend to prove that the convergence rate is, in first approximation,
independent of the number of subdomains as already numerically mentioned
in [25] and Section 4. Finally, φ(k)i is the local solution in Ωi, for any i ∈
1, · · · ,m, at Schwarz iteration k ∈ N.
2.1 Potential-free equation
We first consider the potential-free Schrodinger equation in imaginary-time,
with P (∂t, ∂x) = ∂t − ∂2x. The NGF algorithm consists in solving for any
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6 Xavier ANTOINE, Emmanuel LORIN
n ∈ N, from tn to t−n+1:
(∂t − ∂2x
)φ(t, x) = 0, (t, x) ∈ (tn, t
−n+1)×Ω,
φ(t,−a) = 0, t ∈ (tn, t−n+1),
φ(t, a) = 0, t ∈ (tn, t−n+1).
We then L2(R)-normalize the solution
φ(tn+1, ·) =φ(tn+1− , ·)
‖φ(tn+1−)‖L2(Ω).
The procedure is repeated until convergence following the stopping criterion
(2).
Let us start by studying the rate of convergence of the CSWR, then
considering the OSWR algorithm. We will first generalize the ideas and
results presented in [19] to m subdomains, as well as [9].
2.1.1 CSWR algorithm The CSWR algorithm reads as follows: for k > 1
and i ∈ 2, · · · ,m− 1, from time tn to t−n+1, solve
P (∂t, ∂x)φ(k)i = 0, (t, x) ∈ (tn, t
−n+1)×Ωi,
φ(k)i (0, ·) = ϕ0, x ∈ Ωi,
φ(k)i (t, ξ+i ) = φ
(k−1)i+1 (t, ξ+i ), t ∈ (tn, t
−n+1),
φ(k)i (t, ξ−i ) = φ
(k−1)i−1 (t, ξ−i ), t ∈ (tn, t
−n+1).
In Ω1, we get
P (∂t, ∂x)φ(k)1 = 0, (t, x) ∈ (tn, t
−n+1)×Ω1,
φ(k)1 (0, ·) = ϕ0, x ∈ Ω1,
φ(k)1 (t,−a) = 0, t ∈ (tn, t
−n+1),
φ(k)1 (t, ξ+1 ) = φ
(k−1)2 (t, ξ+2 ), t ∈ (tn, t
−n+1),
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Title Suppressed Due to Excessive Length 7
and in Ωm
P (∂t, ∂x)φ(k)m = 0, (t, x) ∈ (tn, t
−n+1)×Ωm,
φ(k)m (0, ·) = ϕ0, x ∈ Ωm,
φ(k)i (t, ξ−m) = φ
(k−1)m−1 (t, ξ−m), t ∈ (tn, t
−n+1),
φ(k)m (t,+a) = 0, t ∈ (tn, t
−n+1).
To analyze the convergence of this DDM, we set ei := φexact|Ωi − φi and
we introduce h±i ∈ H3/40 (0, T ) =
φ ∈ H3/4(0, T ) : φ(0) = 0
, for i ∈
1, · · · ,m. We consider the following system(iτ − ∂2x
)ei(τ, x) = 0, (τ, x) ∈ R×Ωi,
ei(τ, ξ±) = h±i (τ), τ ∈ R ,
where we denote by τ the dual variable to t in Fourier space, and by ·
the Fourier transform F with respect to t. We set α(τ) := eiπ/4√τ , and
we get ei(τ, x) = Ai(τ)eα(τ)x + Bi(τ)e−α(τ)x, for any τ ∈ R, and for i ∈
2, · · · ,m− 1
ei(τ, ξ±i ) = Ai(τ)eα(τ)ξ
±i +Bi(τ)eα(τ)ξ
±i = h±i (τ) .
By using the boundary/transmission conditions, we find, for i ∈ 2, · · · ,m−
1, Ai(τ) =
(h+i (τ)e−α(τ)ξ
−i − h−i (τ)e−α(τ)ξ
+i
)eα(τ)(L+ε) − e−α(τ)(L+ε)
,
Bi(τ) =
(h−i (τ)eα(τ)ξ
−i − h+i (τ)eα(τ)ξ
+i
)eα(τ)(L+ε) − e−α(τ)(L+ε)
.
Similarly, we haveA1(τ) =
h+1 (τ)
eα(τ)(L+ε/2−a) − e−α(τ)(L+ε/2+a),
B1(τ) =h+1 (τ)
eα(τ)(L+ε/2+a) − e−α(τ)(L+ε/2−a),
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8 Xavier ANTOINE, Emmanuel LORIN
and Am(τ) =
h−m(τ)
eα(τ)(a−L−ε/2) − eα(τ)(L+ε/2+a),
Bm(τ) = − h−m(τ)
e−α(τ)(L+ε/2+a) − eα(τ)(L+ε/2−a).
We introduce the mapping G(C) from(H3/4(R)
)2(m−1)to itself, defined as
follows: G(C) : 〈h+i , h
−i+1
16i6m−1〉 = 〈
ei(·, ξ−i+1), ei+1(·, ξ+i )
16i6m−1〉 .
Thus, we deduce that
F(G(C)〈
h+i , h
−i+1
16i6m−1〉
)(τ) = 〈
ei(τ, ξ
−i+1), ei+1(τ, ξ+i )
16i6m−1〉,
where
ei(τ, ξ−i+1) =
(h−i (τ)(eεα(τ) − e−εα(τ)) + h+i (τ)(eLα(τ) − e−Lα(τ))
)eα(τ)(L+ε) − e−α(τ)(L+ε)
and
ei+1(τ, ξ+i ) =
(h+i+1(τ)(eεα(τ) − e−εα(τ)) + h−i+1(τ)(eLα(τ) − e−Lα(τ))
)eα(τ)(L+ε) − e−α(τ)(L+ε)
.
We then introduce
G(C) : 〈h+i , h
−i+1
16i6m−1〉 = 〈
ei(·, ξ−i+1), ei+1(·, ξ+i )
16i6m−1〉,
Following the same strategy as for the two subdomains case [9,25], we com-
pute G(C),2(〈h+i , h
−i+1i〉
). Let us set h
+,(2)i (τ) := ei(τ, ξ
−i+1) and h
−,(2)i+1 (τ) :=
ei+1(τ, ξ+i ). We get
e(2)i (τ, ξ−i+1) =
(ei(τ, ξ
−i+1)
(eα(τ)L − e−α(τ)L
)+ ei(τ, ξ
+i−1)
(eα(τ)ε − e−α(τ)ε
))eα(τ)(L+ε) − e−α(τ)(L+ε)
and
e(2)i+1(τ, ξ+i ) =
(ei+1(τ, ξ+i )
(eα(τ)L − e−α(τ)L
)+ ei+1(τ, ξ−i+2)
(eα(τ)ε − e−α(τ)ε
))eα(τ)(L+ε) − e−α(τ)(L+ε)
.
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Title Suppressed Due to Excessive Length 9
We then obtain
e(2)i+1(τ, ξ+i ) =
1(eα(τ)(L+ε) − e−α(τ)(L+ε)
)2h−i+1(τ)
(eα(τ)L − e−α(τ)L
)2+ h−i+1(τ)
(eα(τ)ε − e−α(τ)ε
)2+h−i+1(τ)
(eα(τ)L − e−α(τ)L
)2+ 2h+i+1(τ)
(eα(τ)ε − e−α(τ)ε
)(eα(τ)L − e−α(τ)L
)=
e−2εα(τ)
1− e−2α(τ)(L+ε))2h−i+1(τ)
(1− e−2α(τ)L
)2+ h−i+1(τ)
(eα(τ)(ε−2L) − e−α(τ)(ε−2L)
)2+2h+i+1(τ)
(eα(τ)ε − e−α(τ)ε
)(e−α(τ)L − e−3α(τ)L
)and
e(2)i (τ, ξ−i+1) =
1(eα(τ)(L+ε) − e−α(τ)(L+ε)
)2h+i (τ)
(eα(τ)L − e−α(τ)L
)2+ h+i (τ)
(eα(τ)ε − e−α(τ)ε
)2+2h−i (τ)
(eα(τ)ε − e−α(τ)ε
)(eα(τ)L − e−α(τ)L
)=
e−2α(τ)ε(1− e−2α(τ)(L+ε)
)2h+i (τ)(1− e−2α(τ)L
)2+ h+i (τ)
(eα(τ)(ε−2L) − e−α(τ)(ε−2L)
)2+2h−i (τ)
(eα(τ)ε − e−α(τ)ε
)(e−α(τ)L − e−3α(τ)L
).
For m > 2 subdomains, we have
G(C),2(〈h+i , h
−i+1i〉
)(τ) = e−2α(τ)ε〈
h+i (τ), h−i+1(τ)
i〉+O
(e−α(τ)L
).
In particular, for τ < 0, we obtain
∣∣G(C),2(〈h+i , h
−i+1i〉
)(τ)∣∣ 6 e−ε
√2|τ |∣∣〈h+i (τ), h−i+1(τ)
i〉∣∣+ Ce−L
√2|τ |.
Following exactly the same procedure as [19] or [9], we claim for any n ∈ N∗
∥∥〈e2k+1i , e2k+1
i+1 16i6m−1〉∥∥Πm−1i=1 H2,1(Ωi×(0,T ))×H2,1(Ωi+1×(0,T ))
6 (C(C)ε )k
×‖〈e(0)i (ξ+i ), e
(0)i+1(ξ−i+1
16i6m−1〉‖(H3/4(0,T )
)2(m−1)
for some T , with C(C)ε a positive constant lower that 1. This justifies the
fact that, as numerically observed in [25] and in Section 4, the convergence
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10 Xavier ANTOINE, Emmanuel LORIN
rate is independent of the number of subdomains of length L. However,
the overall convergence is linearly dependent of the number of subdomains
through the summation over the m subdomains in (3). As a consequence,
we expect that the logscale slope of the residual history, i.e. the convergence
rate, to be independent of m. Finally, the overall error (k,E(k)) is still
shifted in logscale, by a positive constant linearly dependent on log(m).
We conclude by the following
Proposition 1 The convergence rate C(C)ε of the CSWR method with sub-
domains of length L is of the form
C(C)ε = sup
τe−ε√
2|τ | +O(e−L√
2|τ |) .
2.2 OSWR algorithm
We now study the OSWR algorithm for m > 2 subdomains. To this end, we
define the transparent boundary operator ∂x± ∂1/2t . The OSWR algorithm
reads as follows: for k > 1 and for i ∈ 1, · · · ,m− 1, solve
P (∂t, ∂x)φ(k)i = 0, (t, x) ∈ (tn, t
−n+1)×Ωi,
φ(k)i (0, ·) = ϕ0, x ∈ Ωi,
(∂x + ∂1/2t )φ
(k)i (t, ξ+i ) = (∂x + ∂
1/2t )φ
(k−1)i+1 (t, ξ+i ), t ∈ (tn, t
−n+1),
(∂x − ∂1/2t )φ(k)i (t, ξ−i ) = (∂x − ∂1/2t )φ
(k−1)i−1 (t, ξ−i ), t ∈ (tn, t
−n+1).
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Title Suppressed Due to Excessive Length 11
In Ω1, we get
P (∂t, ∂x)φ(k)1 = 0, (t, x) ∈ (tn, t
−n+1)×Ω1,
φ(k)1 (0, ·) = ϕ0, x ∈ Ω1,
φ(k)1 (t,−a) = 0, t ∈ (tn, t
−n+1),
(∂x + ∂1/2t )φ
(k)1 (t, ξ+1 ) = (∂x + ∂
1/2t )φ
(k−1)2 (t, ξ+1 ), t ∈ (tn, t
−n+1),
and in Ωm
P (∂t, ∂x)φ(k)m = 0, (t, x) ∈ (tn, t
−n+1)×Ωm,
φ(k)m (0, ·) = ϕ0, x ∈ Ωm,
(∂x − ∂1/2t )φ(k)i (t, ξ−m) = (∂x − ∂1/2t )φ
(k−1)m−1 (t, ξ−m), t ∈ (tn, t
−n+1),
φ(k)m (t,+a) = 0, t ∈ (tn, t
−n+1).
We then consider(iτ − ∂2x
)ei(τ, x) = 0, (τ, x) ∈ R×Ωi,
(∂x ± α(τ))ei(τ, ξ±) = h±i (τ), τ ∈ R .
By defining α(τ) := eiπ/4√τ , we obtain
ei(τ, x) = Ai(τ)eα(τ)x +Bi(τ)e−α(τ)x .
By again using the boundary conditions, we find, for i ∈ 2, · · · ,m− 1,
Ai(τ) =h+i (τ)e−α(τ)ξ
+i
2α(τ), Bi(τ) = − h
−i (τ)eα(τ)ξ
−i
2α(τ).
We introduce a mapping G(O) defined as follows
G(O) : 〈h+i , h
−i+1
16i6m−1〉
= 〈
(∂x + ∂1/2t )ei+1(·, ξ+i ), (∂x − ∂1/2t )ei(·, ξ−i+1)
16i6m−1〉 .
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12 Xavier ANTOINE, Emmanuel LORIN
Thus, we can write that
G(O)(〈h+i , hi+116i6m−1〉
)(τ)
= 〈
(∂x + α(τ))ei+1(τ, ξ+i ), (∂x − α(τ))ei(τ, ξ−i+1)
16i6m−1〉
= 2α(τ)〈Ai+1(τ)eα(τ)ξ
+i ,−Bi(τ)e−α(τ)ξ
−i+116i6m−1〉.
Let us introduce
h+,(2)i (τ) = 2α(τ)Ai+1(τ)eα(τ)ξ
+i , h
−,(2)i+1 (τ) = −2α(τ)Bi(τ)e−α(τ)ξ
−i+1 ,
so that we get: e(2)i (τ, x) = A
(2)i (τ)eα(τ)x+B
(2)i (τ)e−α(τ)x. From the bound-
ary conditions, we find, for i ∈ 2, · · · ,m− 1,A
(2)i (τ) =
h+,(2)i (τ)e−α(τ)ξ
+i
2α(τ)= Ai+1(τ),
B(2)i (τ) = − h
−,(2)i (τ)eα(τ)ξ
−i
2α(τ)= Bi−1(τ).
In addition, we have
G(O),2(〈h+i , h
−i+1
16i6m−1〉
)(τ)
= 〈
(∂x + α(τ))e(2)i+1(τ, ξ+i ), (∂x − α(τ))e
(2)i (τ, ξ−i+1)
16i6m−1〉
= 2α(τ)〈A
(2)i+1(τ)eα(τ)ξ
+i ,−B(2)
i (τ)e−α(τ)ξ−i+116i6m−1〉
= 〈h+i+2(τ)e−α(τ)(ξ
+i+2−ξ
+i ), h−i−1(τ)e−α(τ)(ξ
−i+1−ξ
−i−1)16i6m−1〉
= 〈h+i+2(τ)e−2α(τ)(L+ε), h−i−1(τ)e−2α(τ)(L+ε)
16i6m−1〉.
We can finally state the following result.
Proposition 2 The convergence rate C(O)ε of the OSWR method with sub-
domains of length L is O(e−2α(τ)L
).
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Title Suppressed Due to Excessive Length 13
Remark 1 By using a simple unitary transformation, it is trivial to extend
the above results to the case of a linear equation with time-dependent po-
tential V (t) in L1loc(R). Indeed, from
(i∂t + ∂2x + V (t)
)φ(t, x) = 0, it is
sufficient to define the new unknown φ(t, x) = e−i∫ t0V (s)φ(t, x) by gauge
change, φ satisfying then the potential-free Schrodinger equation. This idea
was previously used in several papers (see e.g. [2,4]).
2.3 The space variable potential case
We now assume that the potential is space-dependent. Then, the argument
used in Remark 1 is no longer valid. As it whas studied in [9], we can no
more get a simple expression of the exact solution on each subdomain. We
then have to use approximations. Let us set : P (t, x,D) = ∂t − ∂2x − V (x),
where V is smooth, bounded with bounded derivative.
2.3.1 CSWR algorithm Based on the same notations as above, we directly
consider the error equation for the CSWR algorithm:(iτ − ∂2x + V (x)
)ei(τ, x) = 0, (τ, x) ∈ R×Ωi,
ei(τ, ξ±) = h±i (τ), τ ∈ R.
(4)
We also assume that V is positive, and that ‖V ‖∞ exists. From Nirenberg’s
factorization theorem [9,27], we have
P (t, x, ∂t, ∂x) = (∂x + iΛ−)(∂x + iΛ+) +R, (5)
where R ∈ OPS−∞ is a smoothing pseudodifferential operator. The oper-
ators Λ± are pseudodifferential operators of order 1/2 (in time) and order
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14 Xavier ANTOINE, Emmanuel LORIN
zero in x. Furthermore, their total symbols λ± := σ(Λ±) can be expanded
in the symbol class S1/2S as
λ± ∼+∞∑j=0
λ±1/2−j/2, (6)
where λ±1/2−j/2 are symbols corresponding to operators of order 1/2− j/2,
see [9]. We denote by ei an approximate solution to (4) in Ωi of the following
form (neglecting the scattering effects)
ei(τ, x) = Ai(τ) exp(− i
∫ x
ξ+i
λ+(y, τ)dy)
+ Bi(τ) exp(− i
∫ x
ξ−i
λ−(y, τ)dy).
By construction, λ− = −λ+ if we select λ+1/2 = −λ−1/2 (see [9]). Now, by
using the boundary conditions, we find that, for i ∈ 2, · · · ,m− 1,
Ai(τ) =h+i (τ)− h−i (τ) exp
(− i
∫ ξ+iξ−i
λ−(y, τ)dy)
1− exp(− 2i
∫ ξ+iξ−i
λ−(y, τ)dy) ,
Bi(τ) =h−i (τ)− h+i (τ) exp
(− i
∫ ξ+iξ−i
λ−(y, τ)dy)
1− exp(− 2i
∫ ξ+iξ−i
λ−(y, τ)dy) .
Let us introduce
G(C) : 〈h+i , hi+1
16i6m−1〉 = 〈
ei(·, ξ−i+1), ei+1(·, ξ+i )
16i6m−1〉.
Thus we have
G(C) : 〈h+i , hi+116i6m−1〉 = 〈ei(·, ξ−i+1), ei+1(·, ξ+i )
16i6m−1〉 .
Next, some computations yield
ei(τ, x) =
(h+i (τ)− h−i (τ) exp
(− i
∫ ξ+iξ−i
λ−(y, τ)dy))
exp(i∫ xξ+iλ−(y, τ)dy
)1− exp
(− 2i
∫ ξ+iξ−i
λ−(y, τ)dy)
+
(h−i (τ)− h+i (τ) exp
(− i
∫ ξ+iξ−i
λ−(y, τ)dy))
exp(− i
∫ xξ−iλ−(y, τ)dy
)1− exp
(− 2i
∫ ξ+iξ−i
λ−(y, τ)dy)
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Title Suppressed Due to Excessive Length 15
and
ei+1(τ, ξ+i ) =
(h+i+1(τ)− h−i+1(τ) exp
(− i
∫ ξ+i+1
ξ−i+1
λ−(y, τ)dy))
exp(− i
∫ ξ+i+1
ξ+iλ−(y, τ)dy
)1− exp
(− 2i
∫ ξ+i+1
ξ−i+1
λ−(y, τ)dy)
+
(h−i+1(τ)− h+i+1(τ) exp
(− i
∫ ξ+i+1
ξ−i+1
λ−(y, τ)dy))
exp(− i
∫ ξ+iξ−i+1
λ−(y, τ)dy)
1− exp(− 2i
∫ ξ+i+1
ξ−i+1
λ−(y, τ)dy)
=
(h+i+1(τ)− h−i+1(τ) exp
(− i
∫ L+ε0
λ−(y + ξ−i+1, τ)dy))
exp(− i
∫ L0λ−(y + ξ+i , τ)dy
)1− exp
(− 2i
∫ L+ε0
λ−(y + ξ−i+1, τ)dy)
+
(h−i+1(τ)− h+i+1(τ) exp
(− i
∫ L+ε0
λ−(y + ξ−i+1, τ)dy))
exp(− i
∫ ε0λ−(y + ξ−i+1, τ)dy
)1− exp
(− 2i
∫ L+ε0
λ−(y + ξ−i+1, τ)dy) .
Then, we get
ei(τ, ξ−i+1) =
(h+i (τ)− h−i (τ) exp
(− i
∫ ξ+iξ−i
λ−(y, τ)dy))
exp(i∫ ξ−i+1
ξ+iλ−(y, τ)dy
)1− exp
(− 2i
∫ ξ+iξ−i
λ−(y, τ)dy)
+
(h−i (τ)− h+i (τ) exp
(− i
∫ ξ+iξ−i
λ−(y, τ)dy))
exp(− i
∫ ξ−i+1
ξ−iλ−(y, τ)dy
)1− exp
(− 2i
∫ ξ+iξ−i
λ−(y, τ)dy)
=
(h+i (τ)− h−i (τ) exp
(− i
∫ L+ε0
λ−(y + ξ−i , τ)dy))
exp(− i
∫ ε0λ−(y + ξ−i+1, τ)dy
)1− exp
(− 2i
∫ L+ε0
λ−(y + ξ−i , τ)dy)
+
(h−i (τ)− h+i (τ) exp
(− i
∫ L+ε0
λ−(y + ξ−i , τ)dy))
exp(− i
∫ L0λ−(y + ξ−i , τ)dy
)1− exp
(− 2i
∫ L+ε0
λ−(y + ξ−i , τ)dy) .
Let us set h+,(2)i (τ) := ei(τ, ξ
−i+1) and h
−,(2)i+1 (τ) := ei+1(τ, ξ+i ). Then we are
led to
e(2)i+1(τ, ξ+i ) =
ei+1(τ, ξ−i+2)− ei+1(τ, ξ+i ) exp(− i
∫ L+ε0
λ−(y + ξ−i+1, τ)dy)
1− exp(− 2i
∫ L+ε0
λ−(y + ξ−i+1, τ)dy)
× exp(− i
∫ L
0
λ−(y + ξ+i , τ)dy)
+ei+1(τ, ξ+i )− ei+1(τ, ξ−i+2) exp
(− i
∫ L+ε0
λ−(y + ξ−i+1, τ)dy)
1− exp(− 2i
∫ L+ε0
λ−(y + ξ−i+1, τ)dy)
× exp(− i
∫ ε
0
λ−(y + ξ−i+1, τ)dy)
= h−i+1(τ) exp(− 2iελ−(τ, ξ−i+1)
)+ R1
(τ, ε, L, max
16i6m‖h±i ‖
)
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16 Xavier ANTOINE, Emmanuel LORIN
and
e(2)i (τ, ξ−i+1) =
ei(τ, ξ−i+1)− ei(τ, ξ
+i−1) exp
(− i
∫ L+ε0
λ−(y + ξ−i , τ)dy)
1− exp(− 2i
∫ L+ε0
λ−(y + ξ−i , τ)dy)
× exp(− i
∫ ε
0
λ−(y + ξ−i+1, τ)dy)
+ei(τ, ξ
+i−1)− ei(τ, ξ
−i+1) exp
(− i
∫ L+ε0
λ−(y + ξ−i , τ)dy)
1− exp(− 2i
∫ L+ε0
λ−(y + ξ−i , τ)dy)
× exp(− i
∫ L
0
λ−(y + ξ−i , τ)dy)
= h+i (τ) exp(− 2iελ−(τ, ξ−i+1)
)+ R2
(τ, ε, L, max
16i6m‖hi‖
),
where R` = O(
max16i6m ‖hi‖e−α(τ)L), ` = 1, 2. We then have
G(C),2i
(〈h+i (τ), h−i+1(τ)i〉
)= Ci(ε, τ, V )〈
h+i (τ), h−i+1(τ)
i〉+O
(max
16i6m‖hi‖e−α(τ)L
).
We yet refer [9], where for large τ , we can rigorously expand the symbol
λ±. We do not proceed to these laborious expansions in this paper. In the
first approximation [9], the local convergence rate between Ωi, and Ωi+1 is
given by
Ci(ε, τ) ≈ exp(− ε(√
2|τ |+ V (ξ−i+1)/√
2|τ |).
In general, the above procedure does not allow to directly construct a con-
traction factor, i.e. subdomain-independent coefficients Ci. This is a con-
sequence to the fact that the convergence rate is dependent on the values
of the potential at the subdomain interfaces. However we argue that the
contraction factor (related to the convergence rate of the CSWR method)
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Title Suppressed Due to Excessive Length 17
C(C)ε is such that
supτ exp(− ε(
√2|τ |+ maxi∈1,··· ,m−1 V (ξ−i+1)/
√2|τ |)
). C
(C)ε
. supτ exp(− ε(
√2|τ |+ mini∈1,··· ,m−1 V (ξ−i+1)/
√2|τ |)
)6 supτ exp
(− ε√
2|τ |).
The following Lemma actually justifies that the convergence rate ob-
tained in the case of m subdomains is close to the one for two subdomains
and numerically seen in [9] and in Section 4.
Lemma 1 Let us assume that f : x ∈ RN 7→ f(x) ∈ RN , is such that
f(x) = γx+ ε, with γ < 1 and ‖ε‖ = o(γ). We define the sequence xn+1 =
f(xn) + ε, with x0 ∈ RN given. Then, we have: ‖xn‖ 6 γn‖x0‖+ o(γ).
For ‖V ‖∞ small enough, we directly get
C(C)ε ≈ sup
τexp
(− 2iεα(τ)
).
For the sake of simplicity, we do not detail more the computations.
We conclude that, as in the potential-free case the contraction factor for
m sufficiently large subdomains (L 1) is close to the one for the two-
subdomains case.
Proposition 3 The convergence rate C(C)ε of the CSWR method with m
subdomains of length L is the same as for two-subdomains (see [9]) up to a
term of the order of O(e−L√
2|τ |).Although, the convergence rate C
(C)ε is determined, the overall conver-
gence is also proportional to the number of subdomains. In particular, the
transmission from one subdomain to the next one naturally slows down the
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18 Xavier ANTOINE, Emmanuel LORIN
overall convergence of the SWR method, but without changing the slope of
the residual history in logscale [9].
2.3.2 OSWR algorithm We now consider the OSWR algorithm. More specif-
ically, from time tn to tn+1− , the OSWR algorithm with potential reads as
follows:
P (∂t, ∂x)φ(k)i = 0, (t, x) ∈ (tn, t
−n+1)×Ωi,
φ(k)i (0, ·) = ϕ0, x ∈ Ωi,(∂x + iΛ+(τ, x)
)φ(k)i (t, ξ+i ) =
(∂x + iΛ+(τ, x)
)φ(k−1)i+1 (t, ξ+i ), t ∈ (tn, t
−n+1),(
∂x + iΛ−(τ, x))φ(k)i (t, ξ−i ) =
(∂x + iΛ−(τ, x)
)φ(k−1)i−1 (t, ξ−i ), t ∈ (tn, t
−n+1).
(7)
The operators Λ± are coming from (5) and the corresponding symbols are
denoted by λ±. The latter can be constructed as an asymptotic series of the
form (6). In practice, the series∑+∞j=0 λ
±1/2−j/2 is truncated and, for p ∈ N,
we can define λ±p :=∑pj=0 λ
±1/2−j/2 and the corresponding operators Λ±p . In
[9], the SWR convergence rate is established for the transmission operator
∂t ± iΛ±p . In this paper, we will only consider Λ± but the results could be
extended to Λ±p in a similar way. The error equation in Fourier (resp. real)
space in time (resp. space) is
(iτ − V (x)− ∂2x
)ei(τ, x) = 0, (τ, x) ∈ R×Ωi,(
∂x + iΛ±(τ, x))ei(τ, ξ
±) = h±i (τ), τ ∈ R.
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Title Suppressed Due to Excessive Length 19
The convergence analysis is identical to the one that we presented in Sub-
section 2.2. Basically, we define
G(O) : 〈h+i , h
−i+1
16i6m−1〉
= 〈(∂x + iΛ+(·, x)
)ei+1(·, ξ+i ),
(∂x + iΛ−(·, x)
)ei(t, ξ
−i+1)
16i6m−1〉.
Thus, we can define
G(O) : 〈h+i , h
−i+1
16i6m−1〉
= 〈(∂x + iλ+(·, x)
)ei+1(·, ξ+i ),
(∂x + iλ−(·, x)
)ei(·, ξ−i+1)
16i6m−1〉.
We define approximate solutions to (7) on each Ωi, neglecting again the
scattering effets
ei(τ, x) = Ai(τ) exp(− i
∫ x0λ+(y, τ)dy
)+ Bi(τ) exp
(i∫ x0λ−(y, τ)dy
).
Then by construction λ+ = −λ−, we get
Ai(τ) =h+i (τ)e
∫ ξ+i
0 λ+(τ,y)dy
2iλ+(τ, x), Bi(τ) = − h
−i (τ)e
∫ ξ−i
0 λ−(τ,y)dy
2iλ+(τ, x),
leading to
G(O),2(〈h+i , h
−i+1
16i6m−1〉
)(τ)
= 〈
(∂x + iλ+(τ, x))e(2)i+1(τ, ξ+i ), (∂x − iλ+(τ, x))e
(2)i (τ, ξ−i+1)
16i6m−1〉
= 〈h+i+2(τ)e
−i∫ ξ+i+2
ξ+i
λ+(τ,y), h−i−1(τ)e
−i∫ ξ−i+1
ξ−i−1
λ+(τ,y)dy16i6m−1〉
= 〈h+i+2(τ)e−2i
∫ L+ε0
λ+(τ,y+ξ+i ), h−i−1(τ)e−2i∫ L+ε0
λ+(τ,y+ξ−i−1)dy16i6m−1〉 .
Similarly to the CSWR method, the OSWR DDM asymptotically has a
convergence rate mainly independent, up to a multiplicative constant, of
the number of subdomains of length L. Details of the analysis for the con-
vergence over 2 subdomains for the OSWR and quasi-OSWR can be found
in Section 2.3 of [9].
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20 Xavier ANTOINE, Emmanuel LORIN
2.4 Scalability
We notice that using a large number of subdomains does not allow for
an acceleration of the convergence rate, which is the same as for the two
subdomains case up to a multiplication coefficient. We however benefit from
i) an embarrassingly parallel algorithm and ii) local computations on each
subdomain.
3 Extension to time-dependent problems
The principle for analyzing the rate of convergence in the time-dependent
equation is closely related to the stationary case (see also [10]). Basically, we
have to replace t (resp. τ) by it (resp. iτ) in the equations. Let Q(∂t, ∂x) =
i∂t+∂2x be the Schrodinger operator in real time. Let us consider the IBVP
on Ω = (−a, a)
(i∂t + ∂2x
)φ(t, x) = 0, (t, x) ∈ (0, T )×Ω,
φ(0, ·) = ϕ0, x ∈ Ω,
φ(t,−a) = 0, t ∈ (0, T ),
φ(t, a) = 0, t ∈ (0, T ),
where T > 0 and ϕ0 ∈ L2(R) are given. The convergence rate is defined as
the slope of the logarithm residual history according to the Schwarz iteration
number, that is (k, log(E(k))) : k ∈ N, where now
E(k) :=
m−1∑i=1
∥∥ ‖φ(k)i∣∣(ξ−i+1,ξ
+i )− φ(k)
i+1∣∣(ξ−i+1,ξ
+i )‖∞∥∥L2(0,T )
6 δSc, (8)
δSc being a small parameter.
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Title Suppressed Due to Excessive Length 21
First, the CSWR algorithm reads as follows, for k > 1 and for i ∈
2, · · · ,m− 1,
Q(∂t, ∂x)φ(k)i = 0, (t, x) ∈ (0, T )×Ωi,
φ(k)i (0, ·) = ϕ0, x ∈ Ωi,
φ(k)i (t, ξ+i ) = φ
(k−1)i+1 (t, ξ+i ), t ∈ (0, T ),
φ(k)i (t, ξ−i ) = φ
(k−1)i−1 (t, ξ−i ), t ∈ (0, T ).
In Ω1, we get
Q(∂t, ∂x)φ(k)1 = 0, (t, x) ∈ (0, T )×Ω1,
φ(k)1 (0, ·) = ϕ0, x ∈ Ω1,
φ(k)1 (t,−a) = 0, t ∈ (0, T ),
φ(k)1 (t, ξ+1 ) = φ
(k−1)2 (t, ξ+2 ), t ∈ (0, T ),
and in Ωm
Q(∂t, ∂x)φ(k)m = 0, (t, x) ∈ (0, T )×Ωm,
φ(k)m (0, ·) = ϕ0, x ∈ Ωm,
φ(k)i (t, ξ−m) = φ
(k−1)m−1 (t, ξ−m), t ∈ (0, T ),
φ(k)m (t,+a) = 0, t ∈ (0, T ).
In order to analyze the convergence of the SWR methods, it is sufficient
to replace τ by iτ in all the equations from Section 2. In particular, for m
subdomains, we can write that
G(C),2(〈h+i , h
−i+1i〉
)(τ) = e−2β(τ)ε〈
h+i (τ), h−i+1(τ)
i〉+O
(e−α(τ)L
),
where β(τ) :=√τ .
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22 Xavier ANTOINE, Emmanuel LORIN
A similar study is possible with OSWR algorithm for m subdomains.
First, let us define the transparent operator ∂x±e−iπ/4∂1/2t at the interfaces.
The OSWR algorithm reads as follows: for k > 1 and for i ∈ 2, · · · ,m−1
Q(∂t, ∂x)φ(k)i = 0, (t, x) ∈ (tn, t
−n+1)×Ωi,
φ(k)i (0, ·) = ϕ0, x ∈ Ωi,
(∂x + e−iπ/4∂1/2t )φ
(k)i (t, ξ+i ) = (∂x + e−iπ/4∂
1/2t )φ
(k−1)i+1 (t, ξ+i ), t ∈ (tn, t
−n+1),
(∂x − e−iπ/4∂1/2t )φ(k)i (t, ξ−i ) = (∂x − e−iπ/4∂1/2t )φ
(k−1)i−1 (t, ξ−i ), t ∈ (tn, t
−n+1).
In Ω1, we get
Q(∂t, ∂x)φ(k)1 = 0, (t, x) ∈ (tn, t
−n+1)×Ω1,
φ(k)1 (0, ·) = ϕ0, x ∈ Ω1,
φ(k)1 (t,−a) = 0, t ∈ (tn, t
−n+1),
(∂x + e−iπ/4∂1/2t )φ
(k)1 (t, ξ+1 ) = (∂x + e−iπ/4∂
1/2t )φ
(k−1)2 (t, ξ+1 ), t ∈ (tn, t
−n+1),
and in Ωm
Q(∂t, ∂x)φ(k)m = 0, (t, x) ∈ (tn, t
−n+1)×Ωm,
φ(k)m (0, ·) = ϕ0, x ∈ Ωm,
(∂x − e−iπ/4∂1/2t )φ(k)i (t, ξ−m) = (∂x − e−iπ/4∂1/2t )φ
(k−1)m−1 (t, ξ−m), t ∈ (tn, t
−n+1),
φ(k)m (t,+a) = 0, t ∈ (tn, t
−n+1).
We then consider(iτ − ∂2x
)ei(τ, x) = 0, (τ, x) ∈ R×Ωi,
(∂x ± β(τ))ei(τ, ξ±) = h±i (τ), τ ∈ R,
where we have set β(τ) := eiπ/4√τ . We get:
ei(τ, x) = Ai(τ)eβ(τ)x +Bi(τ)e−β(τ)x.
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Title Suppressed Due to Excessive Length 23
From the boundary conditions, we find, for i ∈ 2, · · · ,m− 1,
Ai(τ) =h+i (τ)e−β(τ)ξ
+i
2β(τ), Bi(τ) = − h
−i (τ)eβ(τ)ξ
−i
2β(τ).
We introduce a mapping G(O) defined as follows
G(O) : 〈h+i , h
−i+1
16i6m−1〉
= 〈
(∂x + e−iπ/4∂1/2t )ei+1(·, ξ+i ), (∂x − e−iπ/4∂1/2t )ei(·, ξ−i+1)
16i6m−1〉.
We again find the same convergence rate as in [10] for two subdomains, but
the overall error is still shifted in logscale, by a positive constant linearly
dependent on log(m).
4 Numerical experiments
4.1 Eigenvalue problem
We present a simple test case illustrating the theoretical results presented
in Section 2. We take V (x) = 5x2/2, and Ω = (−2, 2) and solve the
Schrodinger equation in imaginary-time by using a three-point finite differ-
ence scheme, with meshsize∆x = 1/64, and for a time step∆t = 0.2. Details
about the solver implementation can be found in [9]. We apply the CSWR
method on m = 2, 4, 8 subdomains and compare the convergence rate as a
function of the Schwarz iterations. Initially, we take φ0(x) = φ0(x)/‖φ0‖2,
with φ0(x) = e−4x2
. The m − 1 overlapping zones have a length equal to
ε = ∆x, corresponding to 2 nodes. We observe on Fig. 2 that the asymptotic
residual history is numerically independent of the number of subdomains.
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24 Xavier ANTOINE, Emmanuel LORIN
0 200 400 600 800 1000 1200 1400 160010
-15
10-10
10-5
100
2 subdomains
4 subdomains
8 subdomains
600 800 1000 1200 1400 1600
10-12
10-11
10-10
10-9
2 subdomains
4 subdomains
8 subdomains
Fig. 2 Comparison of the asymptotic convergence rates for 2, 4 and 8 subdo-
mains, including a zoom in the asymptotic regime.
We also report on Fig. 3 the converged solution
(x, φcvg,(k(cvg))(x) =
φg(x)), x ∈ (−2, 2)
, the initial guess
(x, φ0(x)), x ∈ (−2, 2)
, as well as
the NGF converged solution after 20 Schwarz iterations with 8 subdomains,
(x, φcvg,(20)(x)), x ∈ (−2, 2). We however numerically observe that for
8 subdomains the asymptotic rates of convergence seems to be relatively
different than 2 and 4. This can be explained by the fact that, for a large
number of subdomains, the size of each of these subdomains is relatively
small, which induces an inaccuracy in the theoretical slope of convergence.
In addition, the overall convergence rate depends on the value of the poten-
tial at the subdomain interfaces. In the next example, the overall domain
is larger, leading to larger subdomains, then better accuracy of the conver-
gence rate.
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Title Suppressed Due to Excessive Length 25
-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
Initial guess
Ground state: |g
(x)|
Fig. 3 Initial guess, potential and converged ground state.
4.2 Time-dependent equation
In this example, the potential is chosen as V (x) = −5 exp(−2x2), with
Ω = (−8, 8) and final time T = 10. We solve the Schrodinger equation
in real-time with the three-point finite difference scheme, for ∆x = 1/32
and ∆t = 10−2. We refer to [10] for the details concerning the solver. The
CSWR method is applied for m = 2, 4, 8 and 16 subdomains. We compare
the convergence rate as a function of the Schwarz iterations. The initial data
is defined by φ0(x) = φ0(x)/‖φ0‖2, with φ0(x) = e−5(x+2)2+ik0x, for k0 = 5.
The m− 1 overlapping zones have a length equal to ε = ∆x, corresponding
to 2 nodes. We observe on Fig. 4 that, as proven above, the asymptotic
residual history is essentially independent of the number of subdomains.
We also report in Fig. 5 the amplitude of the converged solution(x, |φN,(k(cvg))(x)|, x ∈ (−8, 8)
, the initial guess
(x, |φ0(x)|), x ∈ (−8, 8)
,
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26 Xavier ANTOINE, Emmanuel LORIN
0 100 200 300 400 500 600 70010
-15
10-10
10-5
100
105
2 subdomains
8 subdomains
16 subdomains
32 subdomains
250 300 350 400 450 500 550 600 650
10-12
10-10
10-8
10-6
2 subdomains
8 subdomains
16 subdomains
32 subdomains
Fig. 4 Comparison of the asymptotic convergence rates for 2, 8, 16 and 32 sub-
domains, including a zoom in the asymptotic regime.
-8 -6 -4 -2 0 2 4 6 80
0.5
1
1.5
2
2.5
|0
(x)|: initial condition
|N,(7)
|: solution after 10 Schwarz iterations
|N,(cvg)
|: modulus of converged solution
Fig. 5 Amplitudes of the initial guess, converged solution and solution after 7
Schwarz iterations on 16 subdomains.
as well as the TDSE converged solution after 10 Schwarz iterations with 16
subdomains, and finally
(x, |φN,(7)(x)|), x ∈ (−8, 8)
, where N is such that
TN = T = 10.
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Title Suppressed Due to Excessive Length 27
5 Conclusion
In this paper, we presented an asymptotic analysis of the convergence rate
of the multi-domains CSWR and OSWR DDMs for the one-dimensional
linear Schrodinger equation in imaginary- and real-time. Asymptotic esti-
mates show that the already existing estimates stated for the two-domains
configuration extend to m > 2 domains. This is illustrated through some
numerical examples.
Acknowledgments. X. Antoine was supported by the French National
Research Agency project NABUCO, grant ANR-17-CE40-0025. E. Lorin
thanks NSERC for the financial support via the Discovery Grant program.
References
1. M. Al-Khaleel, A.E. Ruehli, and M.J. Gander. Optimized waveform relaxation
methods for longitudinal partitioning of transmission lines. IEEE Transac-
tions on Circuits and Systems, 56:1732–1743, 2009.
2. X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schadle. A review
of transparent and artificial boundary conditions techniques for linear and
nonlinear Schrodinger equations. Commun. Comput. Phys., 4(4):729–796,
2008.
3. X. Antoine, W. Bao, and C. Besse. Computational methods for the dynam-
ics of the nonlinear Schrodinger/Gross-Pitaevskii equations. Comput. Phys.
Commun., 184(12):2621–2633, 2013.
![Page 28: Asymptotic convergence rates of Schwarz Waveform ...elorin/MultiSub.pdfMultiscale in Science and Engineering manuscript No. (will be inserted by the editor) Asymptotic convergence](https://reader030.fdocuments.in/reader030/viewer/2022040901/5e721658e210e848dc1f09db/html5/thumbnails/28.jpg)
28 Xavier ANTOINE, Emmanuel LORIN
4. X. Antoine, C. Besse, and S. Descombes. Artificial boundary conditions for
one-dimensional cubic nonlinear Schrodinger equations. SIAM J. Numer.
Anal., 43(6):2272–2293 (electronic), 2006.
5. X. Antoine and R. Duboscq. GPELab, a Matlab toolbox to solve Gross-
Pitaevskii equations I: Computation of stationary solutions. Comput. Phys.
Commun., 185(11):2969–2991, 2014.
6. X. Antoine and R. Duboscq. Robust and efficient preconditioned Krylov spec-
tral solvers for computing the ground states of fast rotating and strongly in-
teracting Bose-Einstein condensates. J. Comput. Phys., 258C:509–523, 2014.
7. X. Antoine and R. Duboscq. Modeling and Computation of Bose-Einstein
Condensates: Stationary States, Nucleation, Dynamics, Stochasticity. In
Besse, C. and Garreau, J.C., editor, Nonlinear Optical and Atomic Systems:
at the Interface of Physics and Mathematics, volume 2146 of Lecture Notes in
Mathematics, pages 49–145. Springer, 2015.
8. X. Antoine, F. Hou, and E. Lorin. Asymptotic estimates of the conver-
gence of classical Schwarz waveform relaxation domain decomposition meth-
ods for two-dimensional stationary quantum waves. ESAIM: M2AN, DOI:
https://doi.org/10.1051/m2an/2017048, 2018.
9. X. Antoine and E. Lorin. An analysis of Schwarz waveform relaxation domain
decomposition methods for the imaginary-time linear Schrodinger and Gross-
Pitaevskii equations. Numer. Math., 137(4):923–958, 2017.
10. X. Antoine and E. Lorin. On the rate of convergence of Schwarz waveform
relaxation methods for the time-dependent Schrodinger equation. Submitted.
https://hal.archives-ouvertes.fr/hal-01649736, 2018.
11. X. Antoine, E. Lorin, and A. Bandrauk. Domain decomposition method and
high-order absorbing boundary conditions for the numerical simulation of the
![Page 29: Asymptotic convergence rates of Schwarz Waveform ...elorin/MultiSub.pdfMultiscale in Science and Engineering manuscript No. (will be inserted by the editor) Asymptotic convergence](https://reader030.fdocuments.in/reader030/viewer/2022040901/5e721658e210e848dc1f09db/html5/thumbnails/29.jpg)
Title Suppressed Due to Excessive Length 29
time dependent Schrodinger equation with ionization and recombination by
intense electric field. J. Sci. Comput., 64(3):620–646, 2015.
12. W. Bao and Y. Cai. Mathematical theory and numerical methods for Bose-
Einstein condensation. Kinetic and Related Models, 6(1):1–135, 2013.
13. W. Bao, I-L. Chern, and F.Y. Lim. Efficient and spectrally accurate numer-
ical methods for computing ground and first excited states in Bose-Einstein
condensates. J. Comput. Phys., 219(2):836–854, 2006.
14. W. Bao and Q. Du. Computing the ground state solution of Bose-Einstein con-
densates by a normalized gradient flow. SIAM J. Sci. Comput., 25(5):1674–
1697, 2004.
15. W. Bao and W. Tang. Ground-state solution of Bose-Einstein condensate by
directly minimizing the energy functional. J. Comput. Phys., 187(1):230–254,
2003.
16. C. Besse and F. Xing. Domain decomposition algorithms for two-dimensional
linear Schrodinger equation. J. Sci. Comput., 72(2):735–760, 2017.
17. C. Besse and F. Xing. Schwarz waveform relaxation method for one-
dimensional Schrodinger equation with general potential. Numer. Algor.,
74(2):393–426, 2017.
18. V. Dolean, P. Jolivet, and F. Nataf. An introduction to domain decomposition
methods. Society for Industrial and Applied Mathematics (SIAM), Philadel-
phia, PA, 2015. Algorithms, theory, and parallel implementation.
19. M. Gander and L. Halpern. Optimized Schwarz waveform relaxation methods
for advection reaction diffusion problems. SIAM J. Numer. Anal., 45(2), 2007.
20. M.J. Gander. Overlapping Schwarz for linear and nonlinear parabolic prob-
lems. In Proceedings of the 9th International Conference on Domain decom-
position, pages 97–104, 1996.
![Page 30: Asymptotic convergence rates of Schwarz Waveform ...elorin/MultiSub.pdfMultiscale in Science and Engineering manuscript No. (will be inserted by the editor) Asymptotic convergence](https://reader030.fdocuments.in/reader030/viewer/2022040901/5e721658e210e848dc1f09db/html5/thumbnails/30.jpg)
30 Xavier ANTOINE, Emmanuel LORIN
21. M.J. Gander. Optimal Schwarz waveform relaxation methods for the one-
dimensional wave equation. SIAM J. Numer. Anal., 41:1643–1681, 2003.
22. M.J. Gander. Optimized Schwarz methods. SIAM J. Numer. Anal., 44:699–
731, 2006.
23. M.J. Gander, L. Halpern, and F. Nataf. Optimal convergence for overlapping
and non-overlapping Schwarz waveform relaxation. In Eleventh International
Conference of Domain Decomposition Methods, pages 27–36, 1999.
24. M.J. Gander, F. Kwok, and B. C. Mandal. Dirichlet-Neumann and Neumann-
Neumann waveform relaxation algorithms for parabolic problems. Electron.
Trans. Numer. Anal., 45:424–456, 2016.
25. L. Halpern and J. Szeftel. Optimized and quasi-optimal Schwarz waveform re-
laxation for the one-dimensional Schrodinger equation. Math. Models Methods
Appl. Sci., 20(12):2167–2199, 2010.
26. B. C. Mandal. A time-dependent Dirichlet-Neumann method for the heat
equation. In Domain Decomposition Methods in Science and Engineering
XXI, pages 467–475. Springer International Publishing, 2014.
27. L. Nirenberg. Lectures on linear partial differential equations. American Math-
ematical Society, Providence, R.I., 1973.