Asymptotic convergence rates of Schwarz Waveform ...elorin/MultiSub.pdfMultiscale in Science and...

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

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

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


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.


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


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),


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


.

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),


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))


and

ei+1(τ, ξ+i ) =

(h+i+1(τ)(eεα(τ) − e−εα(τ)) + h−i+1(τ)(eLα(τ) − e−Lα(τ))


.

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 )


)+ ei+1(τ, ξ−i+2)


))eα(τ)(L+ε) − e−α(τ)(L+ε)

.


We then obtain

e(2)i+1(τ, ξ+i ) =

1(eα(τ)(L+ε) − e−α(τ)(L+ε)

)2h−i+1(τ)


)2+ h−i+1(τ)


)2+h−i+1(τ)


)2+ 2h+i+1(τ)


)(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)


)(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)


)2+ h+i (τ)


)2+2h−i (τ)


)(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)


)(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


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).


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〉 .


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

).


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


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)


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) =


(− 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)

+


(− 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)

=


(− 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)

+


(− 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 ‖

)


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)


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


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.


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].


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.


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 .


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.


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.


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.


-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)

,


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.


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.

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