Fractional calculus and non-reflecting

boundary conditions in wave propagation

Xavier ANTOINE 1

Abstract

The aim of this paper is to provide a comprehensive introduction to

the modelling of wave phenomena by linear and nonlinear Schrodinger-

type equations. A special attention is dedicated to the construction

and discretization of truncation boundary conditions to model the ex-

terior infinite domain arising in such situations. Different numerical

simulations show that these techniques are robust in concrete situa-

tions.

1 Introduction

During the last two decades, an increasing attention has been directed to-wards the numerical simulation of linear or nonlinear Schrodinger equations.This is partially due to its central application in quantum mechanics but notonly. Indeed, Schrodinger-type equations are often met in practice as somesimplified models of real-world situations [11]. Among them, let us mentionfor example its application in fluid mechanics, in computational ocean acous-tics, in optical fiber design, in nonlinear electromagnetism laser devices, inseismic migration problems... An example in acoustic wave propagation [12]will be detailled in Section 2. As a result, a clear understanding of the wayof computing the numerical solution to these classes of equations must becarefully investigated.

The temporal part of the equation is often discretized through the stan-dard Crank-Nicolson scheme as introduced and studied by Delfour et al.[8]. It is proved that the resulting schemes for the linear and nonlinearSchrodinger equations in an infinite medium are unconditionnally stable.However, in a realistic situation, a boundary condition must be introduced

1Institut Elie Cartan UMR 7502, Nancy-Universite, CNRS, INRIA.

1

to get a finite computational domain. A classical boundary condition isthe Dirichlet boundary condition. Quite obviously, this Boundary Condition(BC) is not fully satisfactory since it generates some unphysical reflectionsat the boundary. In this sense, this condition must be seen only as an ap-proximation of a kind of expected perfectly absorbing boundary condition,also often called transparent (TBC).

In the one-dimensional linear case and for a free-space potential, the TBCcan be straightforwardly built using the Laplace transform. The resulting BCis then defined by a fractional half-order derivative operator. Since this oper-ator is of memory-type, the discretization of the linear Schrodinger equationcoupled to this TBC is not direct if one wants to preserve the uncondition-nally stable criterion. Therefore, a special attention must be concentratedto the way of designing such a discretization. We will present in Section 3both the construction and discretization of a suitable unconditionnally sta-ble scheme. Next, a few numerical experiments in the 1D linear case for areference solution and the modelling of computational ocean acoustics arereported. In Section 4, we give a formal technique for deriving some bound-ary conditions for the 1D nonlinear Schrodinger (NLS) equation. Unlike theone-dimensional case, these conditions are no longer exact and are generallycalled Artificial Boundary Conditions (ABCs). The numerical discretiza-tion is next discussed and some numerical simulations are presented for thepropagation of a soliton for the nonlinear model. Finally, the last section isdevoted to a conclusion and a short discussion about other related extensionsand problems.

This paper summarizes some results [2, 3, 4, 5] obtained through collabo-rations over the past years with C. Besse (Universite de Lille), S. Descombes(ENS Lyon) and V. Mouysset (ONERA Toulouse & MIP-Toulouse).

2 The Schrodinger equation: a simplified mo-

del in computational ocean acoustics

The linear and nonlinear Schrodinger equations can be met in physics andmechanical systems as an approximation of more complicated models basedfor example on the Helmholtz or Maxwell equations. To exemplify such situa-tions, we outline the derivation of the so-called Standard Parabolic Equation(SPE) in computational ocean acoustics [12] useful as an operational civiland military model in complex environments.

In oceanography, one wants to compute the complex-valued underwa-ter acoustic pressure p(z, r) which is the response to a time-harmonic point

2

source located in the water at (zs, 0) playing the role of an initial data. Here,r > 0 denotes the radial range variable while 0 < z < zb is the depth variable(equal to 0 at the sea surface and zb at the bottom). Both are supposedto be horizontal. More complicated models can be derived integrating somerugosity aspects for instance and more physical modelling like ice-coveredsurfaces. In the far-field zone, the outgoing acoustic pressure

ψ(z, r) =√

k0r p(z, r)e−ik0r (1)

can be approximated by the so-called Standard Parabolic Equation (SPE)

i2

k0ψr = −k−2

0 ρ∂z(ρ∂z)ψ + V (z, r)ψ, r > 0. (2)

This Schrodinger-type equation is a reasonable model to describe the forwardpropagation of a wave located within a cone with the source as origin andwith an aperture limited to 15 degrees. The potential V is related to therefractive index n(z, r) = c0/c(z, r) through

V (z, r) = 1 −N2(z, r) = 1 −[

n(z, r) + iα(z, r)

k0

]2

. (3)

The parameter α describes the attenuation of the ocean (depending on theplace where the simulations occur). These constitutive functions are givenby different models which describe the environment like the effect of salinityof water or the influence of oceanic streams or weather forecast over thewavefield behaviour. The density is set to ρ(z, r), the local sound speed isc(z, r), the reference sound speed is c0 and the related wavenumber is givenby k0 = 2πω/c0, ω denoting the frequency of the emitted sound. A simpleDirichlet BC ψ(z = 0, r) = 0 is generally imposed at the top of the sea (butmore evolved BC could however be also used). People usually employ animpedance boundary condition which replaces the continuity condition

ψ(z−b , r) = ψ(z+b , r),

1

ρwψz(z

−b , r) =

1

ρbψz(z

+b , r),

(4)

with ρw = ρ(z−b , r) the density just above the bottom. According to thesenotations, the impedance boundary condition writes down

ψ(zb, r) = −√

1

2πk0ei π

4ρb

ρw

∫ r

0

ψz(zb, r − τ)√τ

eibτdτ. (5)

We will see in the next section that this last equation is directly connected tothe notion of TBC (because of its definition as a Neumann-to-Dirichlet map)

3

and to fractional derivative operators. We provide some numerical computa-tions in Section 3.4 related to this model in the case of the propagation of aninitial Gaussian beam. Let us mention that Wide-Angle Parabolic Equation(WAPE) can also be built to extend the limited aperture of the SPE to largerangle.

3 The 1D TBC for the linear case: construc-

tion, discretization and simulations

Let us now come back to the one-dimensional linear Schrodinger equationwithout potential

i∂tu+ ∂22u = 0, in Rx × R

∗+t ,

u(x, 0) = u0(x), in Rx.(6)

We will assume here that the intial datum is compactly supported in abounded domain denoted by Ω =]xl; xr[, with boundary: Σ = xl; xr.

3.1 Construction

The construction of the transparent boundary condition can be directly de-rived through simple arguments. Let us restrict our analysis to the rightpoint xr. Introduce the Laplace transform defined by

U(x, τ) = Lu(τ) =

∫ +∞

0

u(x, t)e−τtdt, (7)

denoting by τ the complex Laplace covariable (τ = α + iβ, with α > 0).Then, applying the Laplace transform to system (6) yields the followingspatial differential equation

(∂2x + iτ)U(x, τ) − iu0(x) = 0, (8)

which simplifies as(∂2

x + iτ)U(x, τ) = 0, (9)

in the complementary set of Ω. The TBC is known to be related to theDirichlet-to-Neumann1 (DtN) map from the bounded to the unbounded do-main. Therefore, the solution with finite spatial L2

x(]xr; +∞[)-energy to (9)is given by

U(x, τ) = e−√−iτ(x−xr)Lu(xr, τ). (10)

1Sometimes also called Steklov-Poincare operator!!!

4

Taking the derivative with respect to x and considering the value of theexpression at xr yields the following relation

∂xU(xr, τ) = −e−i π

4√τU(xr, τ). (11)

Using the inverse Laplace transform, the expression (11) can be rewritten as

∂nu+ e−i π

4 ∂1/2t u = 0, for (x, t) ∈ Σ × R

∗+, (12)

defining the half-order derivative operator by the Riemann-Liouville integralformula

∂1/2t ψ(t) =

1√π

d

dt

∫ t

0

ψ(s)√t− s

ds, (13)

and n as the outwardly directed unit normal to Σ.Due to the positiveness of the half-order integration operator, it can be

proven [3] that under some suitable regularity assumptions, the Schrodingerequation set in Ω with TBC (12)-(13) at Σ leads to a well-posed initialboundary-value problem. Moreover, the following energy inequality holds

∀t > 0, ‖u(t)‖L2(Ω) ≤ ‖u0‖L2(Ω) . (14)

3.2 Discretization

The discretization of the TBC (12) is not an easy task. Indeed, the bruteapplication of a quadrature scheme to the Riemann-Liouville integral doesnot yield to a coupled bounded unconditionnally stable time discrete scheme(see e.g. the case of the discretization scheme by Baskakov & Popov [1]).In [3], a way of building an unconditionnally stable scheme is explained byusing a symmetrization principle. Another solution is the following. Let usconsider the Crank-Nicolson discretization of the Schrodinger equation

iun+1 − un

δt+ ∂2

x(un+1 + un

2) = 0 (15)

in the entire spatial domain, denoting the time step by δt. Then, we canintroduce the Z-transform defined by

L(fn) =

+∞∑

n=0

fnz−n = f(z), |z| > Rf, (16)

which is a discrete version of the Laplace transform. Here, (fn)n∈N is a givencomplex-valued signal and Rf the radius of convergence of the series. Using

5

some shifted relations lead to the solution of the differential equation in theunbounded domain

(∂2x +

2i

δt

(1 − z)

(1 + z))f(x, z) = 0 (17)

with bounded outgoing solution

f(x, z) = e−

q

− 2i

δt

(1−z)(1+z)

xf(xr, z). (18)

Deriving the above expression according to x yields the relation at xr

∂nf(xr, z) + e−i π

4

√

2i

δt

√

(1 − z)

(1 + z)f(xr, z) = 0. (19)

If we introduce next the inverse Z-transform and use the discrete convolutionrelation for two complex-valued sequences

L(fn ⋆ gn) = L(fn)L(gn), |z| > max(Rf, Rg), (20)

then we prove that the semi-discrete TBC writes down

∂nun + e−i π

4

√

2i

δt((αn) ⋆ (un))n = 0, at Σ, (21)

where ⋆ designates the discrete convolution of two sequences f = (fn)n∈N

and g = (gn)n∈N and is defined by

(f ⋆ g)n =n

∑

k=0

fkgn−k. (22)

The sequence (αn)n is given by the relations: αk = (−1)kβk for k ∈ N, with

(β0, β1, β2, β3, β4, β5, ...) = (1, 1,1

2,1

2,3

8,3

8, ...). (23)

Some properties of the homography z 7→ (1 + z)/(1 − z) yields followingsome discrete energy estimates to the proof of the unconditionnal stabilityof the global semi-discrete scheme with the time discrete TBC (21). Moreprecisely, we have [2] the following proposition.

Proposition 1. The semi-discrete Crank-Nicolson scheme for the one-di-mensional linear Schrodinger equation with the discrete TBC (21) is givenby the system

iun+1 − un

δt+ ∂2

x(un+1 + un

2) = 0, ∀x ∈ Ω,

∂nun + e−i π

4

√

2i

δt((αn) ⋆ (un))n = 0, at Σ,

u0 = u0, ∀x ∈ Ω.

(24)

6

The sequence of coefficients (αn)n are given above. Moreover, this scheme isunconditionnally stable in the sense of the ‖ · ‖L2(Ω)-norm. Furthermore, thefollowing energy inequality holds

‖uN+1‖L2(Ω) < ‖u0‖L2(Ω), ∀N ≥ 0. (25)

Let us now consider the spatial discretization of system (24). First, aclassical symmetrical variationnal formulation is used. It appears that thenormal derivative trace of the solution un+1 must be evaluated. This issimply done by reformulating the discrete TBC as an impedance boundarycondition. Next, the implementation follows some standard arguments. Inparticular, arbitrary order finite elements could be used. We refer to [2] formore details.

3.3 Comparisons with an explicit solution

We give here some numerical simulations which show the accuracy of theproposed discretization. In the case of the 1D Schrodinger equation withoutpotential, an explicit reference solution, the Gaussian beam, is known andgiven by

u(x, t) =

√

1

4it+ 1e

−x2+ik0x−ik

20t

4it+1 , (26)

where k0 stands for the wavenumber.For the first computation, we consider a gaussian field with a wavenumber

fixed to k0 = 10. The window of computation is ]−5; 5[×]0; 2[. We consider atime step δt = 10−3 andN = 1024 finite elements. The solution is centered atthe origin. We report on Figure 1 and 2 the absolute value of the computedsolution (in log scale) by the proposed discretization (24) with linear andquadratic finite elements, respectively. We see that the reflection is very low.Moreover, the use of quadratic finite elements leads to a complete reductionof the spurious reflections at the boundary.

We have seen in Proposition 1 that the approximate bounded solutionto system (24) has a bounded L2(Ω)-norm. In fact, we should be able toprove that the energy is decaying as numerically shown on Fig. 3. This is inaccordance with the fact that some energy is outgoing to the computationaldomain.

3.4 The example of computational ocean acoustics

We come back in this section to the case of computational acoustics mod-elling where we wish to solve the equation (2). In this case, the transparent

7

Figure 1: Log scale representation of the solution with the ABC discretizedwith the proposed scheme (24) (using linear finite elements).

Figure 2: Log scale representation of the solution with the ABC discretizedwith the proposed scheme (24) (using quadratic finite elements).

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

t

L2 −no

rm

Exact reference solutionApproximate solution computed by using the discrete TBC

Figure 3: Time evolution of the L2(Ω)-norm of the exact and approximatesolutions.

boundary condition writes down as an impedance boundary condition

ψ(zb, r) = − 1√2πk0

ei π

4ρb

ρw

∫ r

0

ψz(zb, r − τ)√τ

eibτdτ, (27)

setting b = k0(N2b − 1)/2. This last condition can be equivalently written as

ψz(zb, r) = −√

2πk0

πe−i π

4ρb

ρw

eibτ∂r

∫ r

0

ψ(zb, τ)√r − τ

eibτdτ. (28)

We see clearly that this condition and equation (12) are closely related.Therefore, some adaptations allows to use the discretization previously de-rived for this type of problem. The initial datum used is a Gaussian beam

ψ(z, 0) =1√w0

[

exp(−(z − zs)2

w20

) − exp(−(z + zs)2

w20

)

]

, (29)

for 0 < z < zs, with w0 = 2/k0.In this example, the vertical variable z lies in the interval 0 < z < 240 m.

We assume that the water density is uniform: ρw = 1.0 g.cm−3. We considera lossless medium and a large density jump (ρb = 2.1 g.cm−3) at the water-bottom interface. Hence, the problem provides an interesting test for thedensity jump in the TBC applied along the bottom surface at zb = 240 m.

9

The source f = 100 Hz is located at a water depth zs = 30 m andthe receiver depth is at zr = 90 m. The sound speed profile is given byc(z) = 1498 + |120 − z|/60 m.s−1, and the sound speed in the bottom iscb = 1505 m.s−1. For our calculation up to a maximum range of 20 km, weuse a reference sound speed c0 = 1500 m.s−1. A uniform computational gridwith depth step ∆z = 2 m and range step ∆r = 5 m is considered.

We present below on Fig. 4 the results of our computations representingthe transmission loss given by

−10 log10(|p|2), (30)

at the receiver depth z = zr for 0 < z < 20 km, where the acoustic pressureis computed from the far-field formula (1).

0 2 4 6 8 10 12 14 16 18 20

30

40

50

60

70

80

90

100

Range r [km]

Tra

nsm

issi

on L

oss

−10

log 10

|p|2

Computation using the discretized TBC

Figure 4: An example of transmission loss calculation using the proposeddiscretization of the impedance boundary condition. For comparisons, wealso produce the results of another computation using the method by Arnoldand Ehrhardt [1].

10

4 A possible approach for the NLS case

4.1 What about time and space potential?

We have seen in the previous section that the TBC for the linear Schrodingerequation can be written at the continuous level in an exact way.

If we consider the same equation as (6) but adding a time variable andspace independent potential V

i∂tu+ ∂22u+ V (t)u = 0, in Rx × R

∗+t ,

u(x, 0) = u0(x), in Rx.(31)

then we can still write the exact TBC via the gauge change

v(x, t) = eiV(t)u(x, t) (32)

with

V(t) =

∫ t

0

V (s)ds. (33)

Indeed, the equation satisfied by v is now a free-potentiel linear Schrodingerequation.

The situation is deeply different if we consider now a time and spacevarying potential V seeking to solve

i∂tu+ ∂22u+ V (x, t)u = 0, in Rx × R

∗+t ,

u(x, 0) = u0(x), in Rx.(34)

In this case, the modification of unknown

v(x, t) = eiV(x,t)u(x, t) (35)

defining

V(x, t) =

∫ t

0

V (x, s)ds. (36)

leads to the variable coefficients linear Schrodinger equation

i∂tv + ∂2xv + A∂xv +Bv = 0, (37)

with initial condition. The two functions A and B are given by the expres-sions

A = 2i∂xV and B = (i∂2xV − (∂xV)2). (38)

Then, we are no longer able to give the exact TBC. In this case, as seen below,more advanced analysis techniques are required to handle the situation. A

11

TBC cannot be built but an approximate condition can be expected. Theresulting condition is called an Artificial Boundary Condition (ABC).

The same conclusion can be drawn for the nonlinear case. Indeed, if oneconsiders the NLS equation

i∂tu+ ∂22u+ V (u)u = 0, in Rx × R

∗+t ,

u(x, 0) = u0(x), in Rx,(39)

with V a suitable nonlinear function, then the TBC cannot be extracted. Anexample of such an equation is given by the cubic NLS where the nonlinearityis: V (u) = q|u|2. In this situation, the TBC can be written as a complexnonlinear DtN map [6]. However, the technique based on inverse scatteringis particularized to a cubic nonlinearity and is therefore not general. We pro-pose here an approximate but general way of deriving a family of increasingorder ABCs. Our strategy is two-fold

• first we consider the linear case with a general potential V dependingon space and time,

• next, we formally replace the potential by the nonlinerity V → q|u|2.

The next subsection is devoted to developing this approach.

4.2 A possible approach using microlocal analysis tech-

niques

Let us define the operator L as

L(x, t, ∂x, ∂t) = i∂t + ∂2x + A∂x +B. (40)

We want to apply the techniques of microlocal analysis to write an asymptoticexpansion of the DtN map associated with the operator L. To do that, weintroduce a special class of pseudodifferential operator calculus related to thework by Lascar [10] and Boutet de Monvel [7] to construct the Nirenberg-like factorization [2] associated with L. We refer to these papers for moretechnical details about this calculus and just give here the main results tounderstand the derivation of the ABCs.

We say that a function f is inhomogeneous of degree m if it satisfies

f(x, t, µ2τ) = τmf(x, t, τ), (41)

for any µ > 0. Using this notation, we can define an inhomogeneous andclassical pseudodifferential operator P of order M , for M ∈ Z, if its total

12

symbol, that we denote here by p = σP , admits an asymptotic expansion ininhomogeneous symbols

pM−j/2

j≥0with decreasing order

p(x, t, τ) ∼+∞∑

j=0

pM−j/2. (42)

In the above expression, functions pM−j/2 are supposed to be inhomogeneousof degree 2M − j, for j in N. The way of understanding the approximation(42) is that

∀m ∈ N, p−m

∑

j=0

pM−j/2 ∈ SM−(m+1)/2. (43)

A symbol p satisfying such a property is denoted by p ∈ SMS . The associated

pseudodifferential operator P = Op(p) defined through the inverse Fouriertransform is quoted by P ∈ OPSM

S . Such examples of operators are givenby the fractional operators

Iα/2t f(t) =

1

Γ(α/2)

∫ t

0

(t− s)α/2−1f(s)ds, (44)

for α ∈ N and setting Γ as the Gamma special function. A straightforwardcalculation shows that the symbol of i−α/2I

α/2t is τ−α/2. As a consequence,

the symbol is inhomogeneous of degree −α and i−α/2Iα/2t ∈ OPS

−α/2S .

Under the previous notations, we can prove the following propositionwhich gives a Nirenberg factorization of L.

Theorem 1. Let L be the operator defined by (40). Then, there exist two in-

homogeneous classical pseudodifferential operator Λ± = Λ(x, t, ∂t) ∈ OPS1/2S ,

smooth according to x and t, such that we get the following factorization inthe sense of Nirenberg

L = (∂x + iΛ−)(∂x + iΛ+) + R. (45)

The error operator R is a smoothing operator which lies in OPS−∞S . The

principal symbol of Λ±, denoted by λ±1/2, is fixed to

λ±1/2 = ∓√−τ . (46)

Furthermore, the total symbol λ± of Λ±

λ± = σΛ± (47)

13

has the following asymptotic expansion in inhomogeneous symbols

λ± ∼+∞∑

j=0

λ±1/2−j/2. (48)

The strenght of this Theorem is related to the fact that a recursive com-putation of the asymptotic expansion of λ± can be explicitly built with thehelp of the symbolic rules associated with the pseudodifferential calculus[10]. This is an important feature since we can have an analytic form of theexpansion of the DtN for our system and therefore obtain some accurate ap-proximate boundary conditions (ABCs) with increasing order by using moreor less terms in the expansion.

More concretely, the four first symbols of λ+ are given by

λ+1/2 = −

√−τ ,

λ+0 = −ia

2,

λ+−1/2 = 0,

λ+−1 = i

∂xV

4τ,

(49)

with a = A. Some arguments based on the reflection of singularities forSchrodinger-type equations [7] show that the theoretical TBC can be reachedthrough the DtN map associated with the outgoing operator Λ+

(∂n

+ iΛ+)v = 0, on Σ × R∗+. (50)

Finally, a concrete approximate ABC is obtained by truncating the infinitetheoretical asymptotic expansion based on λ+. Using this criterion and com-ing back to the true unknown u through an inverse gauge change, we get thefollowing ABCs of order one

∂nu+ e−i π

4 eiV∂1/2t (e−iVu) = 0 (51)

and of order two

∂nu+ e−i π

4 eiV∂1/2t (e−iVu) + i

∂xV

4eiVI

1/2t (e−iVu) = 0, (52)

on Σ × R∗+.

If we now want to derive some associated ABCs for the nonlinear equation,we use the following transformations

V → q|u|2, ∂xV → ∂x(q|u|2). (53)

14

This formal modifications lead to the following respectively first- and second-order ABCs for NLS

∂nu+ e−i π

4 eiU∂1/2t (e−iUu) = 0 (54)

and

∂nu+ e−i π

4 eiU∂1/2t (e−iUu) + iq

∂x|u|24

eiUI1/2t (e−iUu) = 0, (55)

which are valid on Σ × R∗+. The nonlinear integral term U is defined by

U(x, t) = q

∫ t

0

|u(x, s)|2ds. (56)

Around these asymptotics, we have assume that we privileged the validity ofthe DtN approximation in the high frequency spectrum in time. This resultsin a loss of accuracy of the resulting operators for slowly propagating soliton.However, in a lot of applications, the velocity of the solution is sufficientlyhigh to get some accurate results.

If we consider the || · ||0,Ω-norm, then, the well-posedness of the solutionu to the NLS equation in Ω can be proved [4] with the first-order ABC (54).

Proposition 2. Let u0 be an initial datum with compact support Supp(u0)in Ω. Let us denote by u the solution to

i∂tu+ ∂2xu+ q|u|2 = 0, in Ω × R

∗+,

∂nu+ e−i π

4 eiU∂1/2t (e−iUu) = 0, on Σ × R

∗+,u(x, 0) = u0(x), in Ω.

(57)

Then, u satisfies the energy bound

∀t < 0, ∀u ∈ L2(Ω), ||u(t)||0,Ω ≤ ||u0||0,Ω. (58)

In the case of NLS in the entire domain, both the L2-norm and the Hamil-tonian of the system are conserved. We cannot derive here an associatedresult for the Hamiltonian. In the case of the second-order condition, no signcontrol for the corrective term can be obtained. Therefore, we cannot proveany inequality result for the system energy. In [4], other less accurate ABCscan be derived based on the Leibnitz formula for fractional derivatives.

4.3 Discretization scheme

The approximation scheme for system (57) can be obtained by using a Crank-Nicolson scheme. However, for computational cost reasons and simplicity of

15

implementation, it appears that the scheme derived by Duran-Sanz-Serna(DSS) [9] is a better choice for computing the solution to NLS.

Let tn = nδt and un ≈ u(tn). then, a second-order time scheme is givenby

iun+1 − un

δt+ ∂2

x

un+1 + un

2+ q(

|un+1|2 + |un|22

)(un+1 + un

2) = 0, (59)

in Ω. This scheme can be written in a simpler way setting

vn+1 =un+1 + un

2. (60)

Concerning the nonlinear ABC, let us define Ep, for p ≥ 2, as

Ep = exp(iqδt

p−1∑

ℓ=1

|vℓ|2) exp(iqδt

2|vp|2), (61)

and E0 = 1, E

1 = exp(iqδt|v1|2/2). Then, using the quadrature schemes(21)-(23) and a trapezoidal rule for the integral term based on It, we get thefollowing discrete version of the nonlinear artificial operators (54) and (55)

∂nvn+1 + ΛNLS

j,n+1(x, ∂x, |vn+1|)vn+1 = 0, (62)

for j = 1 or 2, with

ΛNLS1,n+1(x, ∂x, |vn+1|)vn+1 = e−i π

4

√

2

δtE

n+1n+1∑

k=0

αkEn+1−kvn+1−k (63)

andΛNLS

2,n+1(x, ∂x, |vn+1|)vn+1 = ΛNLS1,n+1(x, ∂x, |vn+1|)vn+1

+iq

4∂n(|vn+1|)En+1δt

n+1∑

k=0

En+1−kvn+1−k.(64)

It can be proved [4] that the DSS scheme (59) with the discrete ABC (63) isunconditionally stable in the L2(Ω)-norm. Moreover, an energy bound canbe obtained. Again, estimates based on the Z-transform are used like inthe linear case. We refer to [4] where more details are given concerning theproofs. Like the continuous case, no similar results can be obtained for thesecond-order condition.

The practical implementation of the previous scheme involved a fixed-point algorithm. The nonlinear discrete ABCs then appear as some simpleimpedance boundary conditions, very similarly to the linear case, and arehence easy to solve by a finite element scheme. Implementation issues aredetailled in [4].

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4.4 Computational examples: propagation of solitons

We give a few numerical experiments in this section to show the accuracyof the proposed NLS ABCs. A classical test-case consists in considering thesoliton solution for the cubic NLS equation. This solution is known in theone-dimensional case by inverse scattering techniques and writes down

u(x, t) =

√

2a

qsech(

√a(x− ct)) exp(i

c

2(x− ct)) exp(i(a +

c2

4)t). (65)

For our first computations, the soliton is centered at x = 0 and thewindow of computation is ] − 15; 15[×]0; 2[. We fix a time step equal toδt = 10−3 and consider N = 6000 finite elements to divide the spatial gridfor a second-order finite elements scheme. We take q = 1, a = 2, c = 20. Werespectively report on Fig. 5 and 6 the log scale of the absolute value of thesolution over the bounded domain for respectively the first- and second-ordernonlinear ABCs. We see that an improvement arises when the second-ordercondition is used since less reflection occurs. Other tests show that this isalways the case. Moreover, the error that we get with the purely linearboundary condition is larger than the ones presented here with a magnitudeof order 10.

Figure 5: Log scale representation of the solution with the first-order non-linear ABC.

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Figure 6: Log scale representation of the solution with the second-ordernonlinear ABC.

In the last computation, we consider the propagation and interaction oftwo solitons. The first one is centered at x = −6 and has a velocity equalto 15. The second one has a speed of 5 and is centered at x = 7. Thespatial window is ]−20; 20[ and the maximal time is T = 5. The time step isδt = 10−3 and we consider 8000 finite elements with quadratic interpolatingfunctions. We still take q = 1 and fix a = 2. We draw the log scale ofthe modulus of the solution on Figure 7. Even if some reflections occur, thesolution can be reached with a sufficiently acceptable accuracy to observethe reconstruction of the two solitons after their interaction. This is nota trivial numerical computation which cannot be obtain if one considers alinear artificial boundary condition.

5 Conclusion and future aspects

This paper proposed to review of few modelling and numerical questionsthat arise in computational wave propagation for Schrodinger-like equations.We focus essentially on the way of building with some advanced analysistools the transparent and artificial boundary conditions for linear but mostimportantly nonlinear equations. These ABCs lead to well-posed problems

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Figure 7: Log scale representation of the solution with the second-ordernonlinear ABC for the interaction of two solitons.

(when the proof is at hand). A particular attention has been directed towardsthe discretization of the TBC and ABCs which are defined by (nonlinear)integro-differential operators of fractional differential and integral types. Sta-bility results have been stated according to the underlying time discretization.Some numerical results for reference solutions and applications in underwa-ter acoustics problems have been provided. They show that the method isaccurate and leads to suitable computations.

We did not mention anything about the problem in higher dimensions.Some ABCs have been proposed in the linear case with general shaped con-vex boundaries in [2]. The techniques used are mainly based on differentialgeometry of surfaces associated with fractional pseudodifferential operators.The results are summarized in [1] and compared to other possible derivationsfor some special fictitious shapes. In [5], the associated stable discretizationsare studied and numerical tests are provided. The extention is currently be-ing studied in the nonlinear case. A combination of all these results shouldyield a suitable solution for general problems.

Another point which is not discussed here concerns the conception offast and approximate computational schemes to accelerate the evaluation ofthe computation of fractional operators. Some very recent ideas have beendeveloped following different directions to solve this question. A complete

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discussion and details are given in [1].

References

[1] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schadle, A reviewof transparent and artificial boundary conditions techniques for linearand nonlinear Schrodinger equations, work in progress.

[2] X. Antoine and C. Besse, Construction, structure and asymptotic appro-ximations of a microdifferential transparent boundary condition for thelinear Schrodinger equation J. Math. Pures Appl. IX. Ser. 80 (2001),701–738.

[3] X. Antoine and C. Besse, Unconditionally stable discretizationschemes of non-reflecting boundary conditions for the one-dimensionalSchrodinger equation, J. Comp. Phys. 188 (2003), 157–175.

[4] X. Antoine, C. Besse and S. Descombes, Artificial boundary conditionsfor nonlinear Schrodinger equations, SIAM J. Numer. Anal. 43 (2006),2272–2293.

[5] X. Antoine, C. Besse and V. Mouysset, Numerical schemes for the simu-lation of the two-dimensional Schrodinger equation using non-reflectingboundary conditions, Math. Comp. 73 (2004), pp.1779–1799.

[6] A. Boutet de Monvel, A.S. Fokas and D. Shepelsky, Analysis of theglobal relation for the nonlinear Schrodinger equation on the half-line,Lett. Math. Phys. 65 (2003), pp. 199-212.

[7] L. Boutet de Monvel, Propagation des singularites d’equations analoguesa l’equation de Schrodinger, in: Fourier Integral Operators and Par-tial Differential Equations, Lectures Notes in Math., Vol. 459, Springer-Verlag, Berlin, 1975, pp.1–14.

[8] M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of anonlinear Schrodinger equation, J. Comput. Phys. 44 (1981), pp. 277-288.

[9] A. Duran and J.M Sanz-Serna, The numerical integration of relativeequilibrium solutions. The nonlinear Schrodinger equation, IMA J. Nu-mer. Anal. 20 (2000), pp. 235-261.

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[10] R. Lascar, Propagation des singularites des solutions d’equations pseudo-differentielles quasi-homogenes, Ann. Inst. Fourier (Grenoble) 27 (1977),79–123.

[11] C. Sulem and P.L. Sulem, The nonlinear Schrodinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139.Springer-Verlag, New York, 1999.

[12] F.D. Tappert, The parabolic approximation method, in Wave Propa-gation and Underwater Acoustics, Lecture Notes in Physics 70, eds.J.B. Keller and J.S. Papadakis, Springer, New York, 1977, pp. 224–287.

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