AN ASYMPTOTIC PRESERVING SCHEME BASED ON A NEWcarles.perso.math.cnrs.fr/Articles/BeCaMe13.pdf · AN ASYMPTOTIC PRESERVING SCHEME BASED ON A NEW ... the system is always locally well-posed

MULTISCALE MODEL. SIMUL. c© 2013 Society for Industrial and Applied MathematicsVol. 11, No. 4, pp. 1228–1260

AN ASYMPTOTIC PRESERVING SCHEME BASED ON A NEWFORMULATION FOR NLS IN THE SEMICLASSICAL LIMIT∗

CHRISTOPHE BESSE† , REMI CARLES‡ , AND FLORIAN MEHATS§

Abstract. We consider the semiclassical limit for the nonlinear Schrodinger equation. Weintroduce a phase/amplitude representation given by a system similar to the hydrodynamical formu-lation, whose novelty consists of including some asymptotically vanishing viscosity. We prove thatthe system is always locally well-posed in a class of Sobolev spaces, and globally well-posed for afixed positive Planck constant in the one-dimensional case. We propose a second order numericalscheme which is asymptotic preserving. Before singularities appear in the limiting Euler equation,we recover the quadratic physical observables as well as the wave function with mesh size and timestep independent of the Planck constant. This approach is also well suited to the linear Schrodingerequation.

Key words. nonlinear Schrodinger equation, semiclassical limit, numerical simulation, asymp-totic preserving, Madelung transform

AMS subject classifications. 35Q55, 65M99, 76A02, 76Y05, 81Q20, 82D50

DOI. 10.1137/120899017

1. Introduction and main results.

1.1. Motivation. We consider the cubic, defocusing, nonlinear Schrodingerequation (NLS)

(1.1) iε∂tuε +

ε2

2Δuε = |uε|2uε, (t, x) ∈ R+ × R

d,

with WKB-type initial data

(1.2) uε(0, x) = a0(x)eiφ0(x)/ε,

the functions a0 and φ0 being real-valued. The aim of this paper is to construct anasymptotic preserving (AP) numerical scheme for this equation in the semiclassicallimit ε → 0. We seek a scheme which provides an approximation of the solution uε

with an accuracy, for fixed numerical parameters Δt, Δx, that is not degraded as thescaled Planck constant ε goes to zero. In other terms, such a scheme is consistentwith (1.1) for all fixed ε > 0, and when ε → 0 converges to a consistent approximationof the limit equation, which is the compressible, isentropic Euler equation (1.4). Thedifficulty here is that when ε is small the solution uε becomes highly oscillatory withrespect to the time and space variables, and converges to its limit only in a weak sense.

∗Received by the editors November 15, 2012; accepted for publication (in revised form) Septem-ber 4, 2013; published electronically December 5, 2013. The first and second authors were partiallysupported by the French ANR project BECASIM (ANR-12-MONU-0007-04). The first and thirdauthors were partially supported by the ANR-FWF Project Lodiquas (ANR-11-IS01-0003).

http://www.siam.org/journals/mms/11-4/89901.html†Laboratoire Paul Painleve, Universite de Lille 1, Villeneuve d’Ascq 59655, France (Christophe.

[email protected]). This author was partially supported by the French ANR fundings underthe project MicroWave NT09 460489.

‡CNRS & University Montpellier 2, Montpellier 34095, France ([email protected]). Thisauthor was supported by the French ANR project SchEq (ANR-12-JS01-0005-01).

§IRMAR, Universite de Rennes 1 and INRIA, IPSO Project, Rennes 35042, France ([email protected]).

1228

AP SCHEME FOR NLS 1229

In order to follow these oscillations without reducing Δt and Δx at the size of ε, whichmay be computationally demanding, our construction relies on a fluid reformulationof (1.1), well adapted to the semiclassical limit. Note that here the notion of anAP scheme makes sense only so long as the solution to the Euler equation remainssmooth.

In kinetic theory, plasma physics, and radiative transfer, the interest in APschemes has considerably grown in recent years since the pioneer works on the sub-ject [19, 26, 31, 36]. Among a long list of works on this topic, we can mention[33, 28, 11, 40, 7, 20, 25, 16, 21, 39, 23]. For the Schrodinger equation, fewer worksexist. In the stationary case, one can cite [3], which proposes a WKB-type transfor-mation in order to filter out the oscillations in space. In the time-dependent case,the usual numerical methods for solving the nonlinear Schrodinger equation—finite-difference schemes [52, 24, 1, 34], time-splitting methods [47, 51, 9], relaxation schemes[8], or even symplectic methods [49]—are designed in order to guarantee the con-vergence of the discrete wave function uε when ε > 0 is fixed. As it is analyzed,for instance, in [45, 46] for the case of finite-differences or in [37] for Runge–Kuttaschemes, these methods suffer from the oscillations in the semiclassical regime if thetime and space steps are not shrunk. Note, however, that, among these methods,the status of splitting methods is particular. Indeed, in the linear case, it is shownin [5] (see also the recent review article [32]) by Wigner techniques that, if the spacestep Δx has to follow the parameter ε when it becomes small, the time step can bechosen independently of ε (before the appearance of caustics) if we are only interestedin getting a good approximation of the quadratic observables associated to uε. Still,none of these methods is asymptotic preserving in the sense defined above.

The quantum fluid reformulation of the Schrodinger equation provides a naturalframework for the construction of AP numerical methods. The Madelung transform[43] is a polar decomposition of the solution of (1.1), written as

uε(t, x) = aε(t, x)eiφε(t,x)/ε,

where the amplitude aε and the phase φε are real-valued. Inserting this ansatz into(1.1) yields the system of quantum hydrodynamics (QHD), or Madelung equations,satisfied by ρε = (aε)2 and vε = ∇φε:

(1.3)

⎧⎪⎨⎪⎩

∂tρε + div (ρεvε) = 0, ρε|t=0 = (a0)

2,

∂tvε + vε · ∇vε +∇ρε =

ε2

2∇(Δ√ρε√ρε

), vε|t=0 = ∇φ0.

This system takes the form of the compressible Euler equation (with the pressurelaw p(ρ) = ρ2/2), with an additional term testifying for the quantum character of

the equation, the so-called Bohm potential ε2

2 ∇(Δ

√ρε√ρε

). Two comments are in order.

First, the term in ε no longer appears as a singular perturbation and, in the limit ε →0, the quantum pressure disappears, leading formally to the compressible, isentropicEuler equation

(1.4)

{∂tρ+ div (ρv) = 0, ρ|t=0 = (a0)

2,

∂tv + v · ∇v +∇ρ = 0, v|t=0 = ∇φ0,

which is the semiclassical limit of our problem (actually valid before the appearanceof shocks). Second, the fluid-like form of this system enables us to consider manynumerical methods originating from computational fluid mechanics.

1230 CHRISTOPHE BESSE, REMI CARLES, AND FLORIAN MEHATS

For the linear Schrodinger equation, the Madelung formulation is at the heart ofthe method of quantum trajectories (or Bohmian dynamics) (see [53]) which are basedon a Lagrangian resolution of this system. In Eulerian coordinates, the work [22] alsoexploits this formulation to construct AP schemes for (1.3). Unfortunately, the maindrawback of the QHD system is the form of the quantum potential, which becomessingular as the density ρε vanishes (see [13] for a recent survey). Hence, these methodsdo not provide AP schemes in the presence of vacuum. Another point of view wasrecently developed in [15] for the nonlinear equation (1.1), taking advantage of anotherfluid reformulation of the equation—due to Grenier [29]—which tolerates the presenceof vacuum. In the next subsection, 1.2, we present in detail this formulation relyingon a complex amplitude version of the Madelung transform. However, as we willsee, the drawback of this reformulation is that this system may develop singularitieseven for fixed ε > 0, whereas the solution of NLS remains smooth. Then, in the lastsubsection, 1.3, of this introduction, we present a modification in order to remedy thisproblem and preserve the regularity of the solution. The new QHD model that wepropose contains a viscosity term in the equation for the velocity, but is still equivalentto the original equation (1.1).

1.2. Backgrounds on NLS and its semiclassical limit. In this section, werecall some known results on the Cauchy problem for NLS (1.1) and on its semiclassicallimit as ε → 0. For details on this subject, we refer to the textbooks [17] and [12].

The following result is standard in the case ε = 1, and can easily be deducedwhen ε ∈ (0, 1], by scaling arguments.

Proposition 1. Let d � 3 and uε0 ∈ H1(Rd).

1. There exists a unique solution uε ∈ C(R;H1(Rd)) ∩ L8/dloc (R;W

1,4(Rd)) to(1.1), such that uε

|t=0 = uε0. Moreover, the following conservations hold:

Mass:d

dt‖uε(t)‖2L2 = 0.

Momentum:d

dtIm

∫Rd

uε(t, x)∇uε(t, x)dx = 0.

Energy:d

dt

(‖ε∇uε(t)‖2L2 + ‖uε(t)‖4L4

)= 0.

2. If, in addition, uε0 ∈ Hk(Rd) for k ∈ N, k � 2, then uε ∈ C(R;Hk(Rd)).

Note that, if uε = aεeiφε/ε with φε ∈ R, the above three conservation laws become

d

dt‖aε(t)‖2L2 = 0.(1.5)

d

dt

(Im

∫Rd

aε(t, x)∇aε(t, x)dx +1

ε

∫Rd

|aε(t, x)|2∇φε(t, x)dx

)= 0.

d

dt

(‖ε∇aε(t) + iaε(t)∇φε(t)‖2L2 + ‖aε(t)‖4L4

)= 0.(1.6)

Let us now describe the limit ε → 0 for (1.1). The toolbox for studying semiclassi-cal Schrodinger equations contains a variety of methods, depending on the quantitieswe are interested in. If one is only interested in a description of the dynamics ofquadratic observables, such as the mass, current, or energy densities, the Wignertransform is well adapted and has been applied to linear Schrodinger equations andSchrodinger–Poisson systems [27, 42, 56]; on the other hand, it is not adapted to thestudy of NLS [14]. Still, for a description of quadratic observables, the modulated


energy method [10, 55, 41, 2] enables us to treat the nonlinear equation (1.1). Yet, fora pointwise description of the wavefunction uε, WKB techniques are more convenient.We refer to [12] for a presentation of WKB analysis of Schrodinger equations.

In the case of (1.1), also referred to as supercritical geometric optics in thiscontext, the justification of the WKB expansion was done by Grenier [29]. Let usbriefly present his idea. As we said above, the main inconvenience of the QHD system(1.3) is the singularity of the Bohm potential at the zeroes of the function ρε (i.e., atthe zeroes of the wavefunction uε). Looking again for solutions uε under the form

(1.7) uε(t, x) = aε(t, x)eiφε(t,x)/ε,

but now allowing aε to take complex values, Grenier proposed to define aε and φε asthe solutions of the system

(1.8)

⎧⎪⎨⎪⎩

∂taε +∇φε · ∇aε +

1

2aεΔφε = i

ε

2Δaε, aε|t=0 = a0,

∂tφε +

1

2|∇φε|2 + |aε|2 = 0, φε

|t=0 = φ0.

This system, which is formally equivalent to (1.1), can also be expressed in terms ofthe unknowns aε and vε = ∇φε. Differentiating with respect to x the second equationin (1.8), we find, using the fact that vε is irrotational,

(1.9)

⎧⎨⎩ ∂ta

ε + vε · ∇aε +1

2aε div vε = i

ε

2Δaε, aε|t=0 = a0,

∂tvε + vε · ∇vε +∇|aε|2 = 0, vε|t=0 = ∇φ0.

The important remark made in [29] is to notice that the above system is hyperbolicsymmetric, perturbed by a skew-symmetric term. In the limit case ε = 0, and if a0is real-valued and nonnegative, (1.9) is the Euler equation (1.4) written in symmetricform (see e.g., [44, 18]):

(1.10)

⎧⎨⎩ ∂ta+ v · ∇a+

1

2a div v = 0, a|t=0 = a0,

∂tv + v · ∇v +∇|a|2 = 0, v|t=0 = ∇φ0.

This remark enables us to obtain, for small times (but independent of ε ∈ [0, 1]),uniform estimates of the solution (aε, vε) of (1.9) in Sobolev norms and prove theconvergence of these quantities to the solution (a, v) of (1.10), as long as this solutionexists and is smooth. From the numerical point of view, the regular structure of (1.9),in particular when aε vanishes or has very small values, is an advantage compared to(1.3), and is at the basis of the schemes constructed in [15].

1.3. New model and main results. A drawback of (1.9) is that large timeexistence results (for fixed ε > 0) do not seem available. In particular, it is not knownwhether (1.9) still has a smooth solution past the time where the solution to (1.10)has ceased to be smooth (such a time necessarily exists if φ0 and a0 are compactlysupported, regardless of their size, from [44]—see also [54]). A practical consequenceof this fact is that a numerical method based on (1.9), such as the one proposed in [15],may not approximate correctly the original equation (1.1) on arbitrary time intervals,for fixed ε > 0.


To overcome this issue, we use the degree of freedom given by (1.7) in a mannerwhich is slightly different from the approach introduced by Grenier. From (1.7), wehave

iε∂tuε +

ε2

2Δuε − |uε|2uε = −

(∂tφ

ε +1

2|∇φε|2 + |aε|2

)aεeiφ

ε/ε

+ iε

(∂ta

ε +∇φε · ∇aε +1

2aεΔφε − i

ε

2Δaε

)eiφ

ε/ε.

Introduce the viscous term ε2aεeiφε/εΔφε, and reorder the terms so as to consider the

new system

(1.11)

⎧⎪⎨⎪⎩


1

2aεΔφε = i

ε

2Δaε − iεaεΔφε, aε|t=0 = a0,

∂tφε +

1

2|∇φε|2 + |aε|2 = ε2Δφε, φε

|t=0 = φ0.

Proceeding like above, we get the following new system:

(1.12)

⎧⎨⎩ ∂ta

ε + vε · ∇aε +1

2aε div vε = i

ε

2Δaε − iεaε div vε, aε|t=0 = aε0,

∂tvε + vε · ∇vε + 2Re (aε∇aε) = ε2Δvε, vε|t=0 = ∇φ0.

We recover physical observables, respectively particle, current, and energy densities,thanks to the formula

(1.13)

ρε = |aε|2,jε = εIm(aε∇aε) + ρεvε,

eε = |ε∇aε + iaεvε|2 + |aε|4.Remark 2 (linear case). In the case of a linear Schrodinger equation

iε∂tuε +

ε2

2Δuε = V uε,

we will see that the introduction of this artificial diffusion is even more striking thanin the nonlinear framework; see subsection 4.1.

Remark 3 (irreversibility). Our choice for introducing the viscous term in (1.11)makes the system irreversible (in time), while (1.1) is reversible. To solve the Schro-dinger equation backward in time, it suffices to change the signs to⎧⎪⎨

⎪⎩∂ta

ε +∇φε · ∇aε +1

2aεΔφε = i

ε

2Δaε + iεaεΔφε, aε|t=0 = a0,

∂tφε +

1

2|∇φε|2 + |aε|2 = −ε2Δφε, φε

|t=0 = φ0.

Let us now present our main theoretical results on this system. Our first resultconcerns the local in time well-posedness of the Cauchy problem in any dimension,for fixed values of ε � 0.

Theorem 4. Let (a0, φ0) ∈ Hs(Rd) × Hs+1(Rd), with s > d/2 + 1. Then thefollowing hold:

(i) The system (1.12) admits a unique maximal solution (aε, vε) ∈ C([0, T εmax);H

s×Hs).


(ii) There exists T > 0 independent of ε ∈ [0, 1] such that T εmax � T . Moreover,

the L∞([0, T ];Hs ×Hs) norm of (aε, vε) is bounded uniformly in ε ∈ [0, 1].(iii) Defining φε ∈ C([0, T ε

max);Hs+1) by

(1.14) φε(t, x) = φ0(x) −∫ t

0

(1

2|vε(s, x)|2 + |aε(s, x)|2 − ε2 div vε(s, x)

)ds,

then the function

uε = aεeiφε/ε ∈ C([0, T ε

max);Hs)

is the unique solution to (1.1) satisfying (1.2).Remark 5. In fact, the proof of this theorem provides a criterion of global ex-

istence for the solution to (1.12). Indeed, we shall prove that the lifespan T εmax is

independent of s > d/2 + 1 and that

(1.15) T εmax < ∞ =⇒

∫ T εmax

0

(‖(aε, vε)(t)‖W 1,∞ + ‖aε(t)‖2L∞)dt = ∞.

Our second result states the convergence of the solution to (1.12) towards thesolution of the Euler equation, as ε → 0.

Notation 6. For two positive numbers αε and βε, the notation αε � βε meansthat there exists C > 0 independent of ε such that for all ε ∈ (0, 1], αε � Cβε.

Theorem 7. Let s > d/2+3 and (a0, φ0) ∈ Hs×Hs+1. Let T > 0 such that theEuler equation (1.4) has a unique solution (ρ, v) ∈ C([0, T ];Hs ×Hs). Then (1.10)has a unique solution (a, v) ∈ C([0, T ];Hs × Hs). Moreover, for ε > 0 sufficientlysmall, we have T ε

max � T and

‖(aε, vε)− (a, v)‖L∞([0,T ];Hs−2×Hs−2) � ε.

Theorems 4 and 7 are not different from the corresponding results obtained byGrenier in [29] on his system (1.9). We now state a global existence result for fixedε > 0 in dimension one that is not available for (1.9). This result takes advantage ofthe viscous term in the second equation of (1.12).

Theorem 8. Suppose d = 1 and let (a0, φ0) ∈ Hs ×Hs+1, with s � 2. Then forε > 0 fixed, the solution to (1.12) is global in time, in the sense that T ε

max = ∞ inTheorem 4.

The above results show that our approach is a new way to compute the solution to(1.1) for fixed ε > 0 (no approximation is made in (1.11), from Theorem 4), and for alltime at least for d = 1, a case where the solution to (1.11) is global, like the solution to(1.1). As ε → 0, (1.11) yields an approximation of the Euler system, from Theorem 7,so long as the solution to (1.4) remains smooth. After the formation of singularitiesin (1.4), the solution to (1.11) remains smooth for fixed ε > 0 in the case d = 1, butnumerics show that it becomes ε-oscillatory: The solution to (1.11) then no longer hasa strong limit as ε → 0. This aspect was already observed in [15], and is confirmedby the present experiments (see section 3). This can be understood as follows: For“large” time, describing the solution to (1.1) with only one rapid oscillation carriedby a single exponential ceases to be relevant, so the amplitude/phase decompositionno longer simplifies the semiclassical analysis.

This paper is organized as follows. Section 2 is devoted to the proofs of ourthree theorems: the local existence result in subsection 2.1, the semiclassical limit


in subsection 2.2, and the global existence result in subsection 2.3. Section 3 is of adifferent nature and concerns numerics. We describe a second order AP numericalscheme for NLS, based on the reformulation (1.12), and we give the results of numericalexperiments, in dimensions 1 and 2. We show that our scheme is indeed AP beforethe formation of singularities in the Euler equation, which means that the time andspace steps Δt and Δx can be taken independently of ε. After the formation ofsingularities, we show that, in order to get a good approximation of the solution, weneed to take Δt and Δx of order O(ε). At the end of this paper, in section 4, wesketch two extensions of our results: We discuss the case of the linear Schrodingerequation, and then we discuss other nonlinearities.

2. Proofs of the main theorems. This section is devoted to the proofs of ourmain results stated in Theorems 4, 7, and 8.

2.1. Local existence. In this subsection, we show the local existence of asmooth maximal solution to (1.12).

Proof of Theorem 4. Items (i) and (ii): Local existence and uniform estimates.Following [29], introduce the vector-valued unknown function

Uε =

⎛⎜⎜⎜⎜⎜⎝

Reaε

Imaε

vε1...vεd

⎞⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎝

aε1aε2vε1...vεd

⎞⎟⎟⎟⎟⎟⎠ ∈ R

d+2.

In terms of this unknown function, (1.12) reads as

(2.1) ∂tUε +

d∑j=1

Aj(Uε)∂jUε =

ε

2LUε + ε2DUε + ε

d∑j=1

Bj(Uε)∂jUε,

where the (k, �)1�k,��d+2 elements of the matrices Aj(U) ∈ Md+2(R) are given by

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Ajk,k(U) = Uj+2 for k = 1, . . . , d+ 2,

Aj1,j+2(U) = U1/2, Aj

2,j+2(U) = U2/2,

Ajj+2,1(U) = 2U1, Aj

j+2,2(U) = 2U2,

Ajk,�(U) = 0 otherwise.

The linear operator L is given by

L =

⎛⎝ 0 −Δ 0 . . . 0Δ 0 0 . . . 00 0 0d×d

⎞⎠ .

A first important remark is that even though L is a differential operator of ordertwo, it causes no loss of regularity in the energy estimates, since it is skew-symmetric.Also, (2.1) is hyperbolic symmetric (or symmetrizable). Indeed, let

(2.2) S =

(I2 00 1

4Id

).


This matrix is symmetric and positive, and SAj is symmetric for all k,

SAj(U) ∈ Sd+2(R) ∀U ∈ Rd+2.

The diffusive term D is given by

D =

⎛⎝0 0 0 . . . 00 0 0 . . . 00 0 Id×dΔ

⎞⎠ .

Finally, the matrices Bj , defined analogously to Aj , are given by

{Bj

1,j+2(U) = U2, Bj2,j+2(U) = −U1,

Bjk,�(U) = 0 otherwise.

When the right-hand side of (2.1) is zero, local existence of a unique solution in Hs

with s > d/2 + 1 is a consequence of standard quasi-linear analysis for hyperbolicsymmetric systems; see, e.g., [50]. The important point in energy estimates consistsof computing

d

dt〈Λs(SUε),ΛsUε〉 , Λ = (1−Δ)1/2,

and taking advantage of symmetry to obtain, thanks to tame estimates,

d

dt〈Λs(SUε),ΛsUε〉 � C‖Uε‖W 1,∞ 〈Λs(SUε),ΛsUε〉

for some constant C depending only on A and d.In the case of the complete system (2.1), we first recall that as noticed in [29],

the term L does not alter the above energy estimate, since SL is skew-symmetric, sowe have

d

dt〈Λs(SUε),ΛsUε〉 � C‖Uε‖W 1,∞ 〈Λs(SUε),ΛsUε〉+ 2ε2 〈SΛsDUε,ΛsUε〉

+ 2ε

d∑j=1

⟨SΛs

(Bj(Uε)∂jU

ε),ΛsUε

⟩.

In terms of (vε, aε), the first new term reads as

2ε2 〈SΛsDUε,ΛsUε〉 = ε2

2

d∑k=1

∫Δ(Λsvεk) Λ

svεk = −ε2

2

∫|∇Λsvε|2.

For the second new term, we have

2ε⟨SΛs

(Bj(Uε)∂jU

ε),ΛsUε

⟩=

ε

2

∫Λs

(aε2∂jv

εj

)Λsaε1 −

ε

2

∫Λs

(aε1∂jv

εj

)Λsaε2.

By Kato–Ponce commutator estimates [35],∥∥Λs(aεk∂jv

εj

)− aεkΛs∂jv

εj

∥∥L2 � ‖∇aεk‖L∞‖vεj‖Hs + ‖aεk‖Hs‖∇vε‖L∞ .


Then using the Cauchy–Schwarz and Young inequalities,

∣∣∣∣∫

aεkΛs∂jv

εjΛ

saε�

∣∣∣∣ � ‖∇Λsvε‖L2‖aεkΛsaε�‖L2 � δε‖∇Λsvε‖2L2 +1

4δε‖aεkΛsaε�‖2L2 .

Choosing δ > 0 sufficiently small and independent of ε, we infer

(2.3)d

dt〈Λs(SUε),ΛsUε〉+ ε2

4‖∇Λsvε‖2L2 � ‖Uε‖W 1,∞‖Uε‖2Hs + ‖aε‖2L∞‖aε‖2Hs .

The local existence result then follows easily by adapting the standard argumentsfrom [50]. In view of the energy estimate (2.3), we also infer (1.15). The lifespanT εmax of the maximal solution to (1.12) must be expected to actually depend on ε. For

instance, if φ0, a0 ∈ C∞0 (Rd), then from [44] T 0

max < ∞, regardless of the dimensiond. On the other hand, we prove further (proof of Theorem 8) that if d = 1, T ε

max = ∞for all ε > 0.

Remark 9. It may be tempting to ask what remains of the above computationsif the artificial viscosity ε2 in (1.11) is replaced with εθ, for some θ > 0. By resumingthe above computations, it is easy to check that the first two points of Theorem 4remain true provided that θ � 2. This might be a threshold, in the spirit of thework [38], where the competition between vanishing dispersion and vanishing viscositywas initiated, in the case of a scalar conservation law, and a limiting behavior wasemphasized.

Item (iii): uε is solution of the original NLS problem. Define φε by (1.14). Let usprove that (aε, φε) ∈ C([0, T ε

max);Hs×Hs+1) solves the system (1.11). Since (aε, vε) ∈

C([0, T εmax);H

s × Hs), we readily have φε ∈ C([0, T εmax);H

s−1). Moreover, vε isirrotational: Indeed, curl vε satisfies a homogeneous equation (see, e.g., [4, p. 291]),and it is zero at time t = 0, so curl vε ≡ 0. Set Ωε = Dvε − ∇vε, where Dvε standsfor the Jacobian matrix of vε, and ∇vε stands for its transposed matrix. It solves

∂tΩε + vε · ∇Ωε +Ωε ·Dvε +∇vε · Ωε = ε2ΔΩε.

As a consequence, ∇|vε|2 = 2vε · ∇vε. We then deduce from (1.14) and from thesecond equation of (1.12)

∂t (∇φε − vε) = ∇∂tφε − ∂tv

ε = 0,

so we infer from the initial condition vε|t=0 = ∇φ0 = ∇φε|t=0 that v

ε = ∇φε. Therefore,

∇φε ∈ C([0, T εmax);H

s), and hence

φε ∈ C([0, T εmax);H

s+1).

Replacing vε with ∇φε in (1.14) shows that φε solves the first equation in (1.11), andthe second equation in (1.11) is obviously satisfied.

Now, define uε by (1.7). The property uε ∈ C([0, T ε];Hs) is straightforward,since Hs(Rd) is an algebra. It is clear that uε solves (1.1) and (1.2). Since

C([0, T ε];Hs(Rd)) ⊂ C([0, T ε];H1(Rd)) ∩ L8/d([0, T ε];W 1,4(Rd)),

uniqueness stems from Proposition 1.


2.2. Convergence towards the Euler equation. In this subsection, we studythe semiclassical limit of (1.12).

Proof of Theorem 7. The proof proceeds as for [29, Theorem 1.2]. First, Theo-rem 4 yields a solution (a, v) ∈ C([0, T 0

max);Hs ×Hs). Necessarily, T 0

max > T (whereT is an existence time for the Euler equation), for if we had T 0

max � T , then by unique-ness for (1.4), (|a|2, v) = (ρ, v) ∈ L∞([0, T ];Hs ×Hs). The equation for a in (1.10)then shows that a ∈ L∞([0, T ];Hs−1), and hence (a, v) ∈ L∞([0, T ];Hs−1 × Hs),which is impossible if T 0

max � T : This yields the first part of the theorem.To prove the error estimate, introduce the vector-valued unknown U , associated

to (a, v). We know that U ∈ C([0, T ];Hs). Consider the error W ε = Uε−U . It solves

∂tWε +

d∑j=1

(Aj(Uε)∂jU

ε −Aj(U)∂jU)=

ε

2LUε + ε2DUε + ε

d∑j=1

Bj(Uε)∂jUε.

Rewrite this equation as

∂tWε +

d∑j=1

Aj(Uε)∂jWε = −

d∑j=1

(Aj(Uε)−Aj(U)

)∂jU +

ε

2LW ε +

ε

2LU

+ ε2DW ε + ε2DU + ε

d∑j=1

Bj(Uε)∂jUε.

The operator on the left-hand side is the same operator as in (2.1). The term LW ε

is not present in the energy estimates, since it is skew-symmetric. The term εLU isconsidered as a source term: It is of order ε, uniformly in L∞([0, T ];Hs−2). Next, thefirst term on the right-hand side is a semilinear perturbation:∥∥(Aj(Uε)−Aj(U)

)∂jU

∥∥Hs−2 �

∥∥(Aj(Uε)− Aj(U))∥∥

Hs−2 ‖U‖Hs−1

�∥∥(Aj(W ε + U)−Aj(U)

)∥∥Hs−2

� C (‖W ε‖L∞ , ‖U‖Hs−1) ‖W ε‖Hs−2 ,

where we have used tame estimates. Finally, we know that W ε is bounded inL∞([0, T ] × Rd), as the difference of two bounded terms. The terms involving Dand B are treated in a similar way, by adapting the estimates used in the proof ofTheorem 4. We end up with

d

dt

⟨SΛs−2W ε,Λs−2W ε

⟩+

ε2

4‖∇Λs−2W ε‖2L2 � ε‖W ε‖Hs−2 + ‖W ε‖2Hs−2

� ε2 +⟨SΛs−2W ε,Λs−2W ε

⟩,

and the result follows from the Gronwall lemma.

2.3. Global existence result in one dimension. In this section, we consider(1.12) with ε > 0 fixed, in dimension d = 1. To prove a global existence result, weshall need finer estimates than (2.3), in particular for low order derivatives.

Proof of Theorem 8. For s = 0 and s = 1, (1.12) can be refined, and this will beuseful to prove a global existence result.

L2 estimates. Multiplying the second equation in (1.12) by aε, integrating inspace, and considering the real part yields

d

dt‖aε(t)‖2L2 = 0.


We have simply recovered the conservation of mass (1.5).Multiplying now the first equation in (1.12) by vε and integrating in space, we

take advantage of the one dimension:

(2.4)1

2

d

dt‖vε(t)‖2L2 + ε2‖∂xvε(t)‖2L2 �

∫|aε(t, x)|2|∂xvε(t, x)|dx.

H1 estimates. Differentiating (1.12) in space, then multiplying the first equa-tion by ∂xv

ε and the second by ∂xaε, and integrating, we find (thanks to the sym-metrizer)

d

dt

(‖∂xvε(t)‖2L2 + 4‖∂xaε(t)‖2L2

)+ ε2‖∂2

xvε(t)‖2L2

�(‖∂xvε(t)‖L∞ + ‖aε(t)‖L∞ + ‖aε(t)‖2L∞

) (‖vε(t)‖2H1 + ‖aε(t)‖2H1

).(2.5)

Now, we have the tools to prove Theorem 8. In view of Remark 5, it suffices toprove that for any fixed ε > 0,

vε, ∂xvε, aε, ∂xa

ε, |aε|2 ∈ L1loc

([0,∞);L∞(Rd)

).

Since ε > 0 is fixed, we shall omit it in the notations, and fix it equal to 1 in (1.12).Using item (iii) of Theorem 4, and in view of (1.6), we have the a priori estimate

‖a(t)‖L4 � 1 ∀t � 0.

The refined L2 estimate (2.4) for v and the Cauchy–Schwarz inequality yield

1

2

d

dt‖v(t)‖2L2 + ‖∂xv(t)‖2L2 � ‖a(t)‖2L4‖∂xv(t)‖L2 � ‖∂xv(t)‖L2 .

Using Young’s inequality

‖∂xv(t)‖L2 � δ‖∂xv(t)‖2L2 +1

4δ,

we infer, for δ > 0 sufficiently small,

d

dt‖v(t)‖2L2 + ‖∂xv(t)‖2L2 � 1.

Therefore,

‖v(t)‖2L2 +

∫ t

0

‖∂xv(τ)‖2L2dτ � 1 + t.

The Gagliardo–Nirenberg inequality yields∫ t

0

‖v(τ)‖L∞dτ �∫ t

0

‖v(τ)‖1/2L2 ‖∂xv(τ)‖1/2L2 dτ � (1 + t)1/4∫ t

0

‖∂xv(τ)‖1/2L2 dτ

� (1 + t)1/4∫ t

0

(1 + ‖∂xv(τ)‖2L2

)dτ � (1 + t)5/4,

where we have used the Young inequality. In view of (1.6), we infer

‖∂xa(t)‖L2 � 1 + ‖a(t)v(t)‖L2 � 1 + ‖v(t)‖L∞ ,


and the Gagliardo–Nirenberg inequality yields

‖a(t)‖L∞ � ‖a(t)‖1/2L2 ‖∂xa(t)‖1/2L2 � 1 + ‖v(t)‖1/2L∞ .

We infer ∫ t

0

(‖a(τ)‖L∞ + ‖a(τ)‖2L∞)dτ � (1 + t)5/4.

It only remains to prove that ∂xv, ∂xa ∈ L1loc([0,∞);L∞(R)). The H1 estimate (2.5)

yields

∫ t

0

‖∂2xv(τ)‖2L2dτ � 1 +

∫ t

0

(‖∂xv(τ)‖L∞ + 1 + ‖a(τ)‖2L∞) (‖a(τ)‖2H1 + ‖v(τ)‖2H1

)dτ

� 1 +

∫ t

0

(‖∂xv(τ)‖L∞ + 1 + ‖a(τ)‖2L∞) (

1 + ‖v(τ)‖2L∞ + ‖v(τ)‖2H1

)dτ

� 1 +

∫ t

0

(‖∂xv(τ)‖L∞ + 1 + ‖a(τ)‖2L∞) (

1 + ‖v(τ)‖2H1

)dτ,

from Sobolev embedding. Using an integration by parts and the Young inequality, wefind

‖∂xv(τ)‖2L2 � ‖v(τ)‖L2‖∂2xv(τ)‖L2

� δ

‖∂xv(τ)‖L∞ + 1 + ‖a(τ)‖2L∞‖∂2

xv(τ)‖2L2

+‖∂xv(τ)‖L∞ + 1 + ‖a(τ)‖2L∞

4δ‖v(τ)‖2L2 .

Taking δ > 0 sufficiently small, we infer

∫ t

0

‖∂2xv(τ)‖2L2dτ � 1 + (1 + t)

∫ t

0

(‖∂xv(τ)‖L∞ + 1 + ‖a(τ)‖2L∞)dτ

+ (1 + t)

∫ t

0

(‖∂xv(τ)‖L∞ + 1 + ‖a(τ)‖2L∞)2

dτ

� 1 + t2 + (1 + t)

∫ t

0

‖∂xv(τ)‖2L∞dτ + (1 + t)

∫ t

0

‖a(τ)‖4L∞dτ.

We also have∫ t

0

‖a(τ)‖4L∞dτ � t+

∫ t

0

‖v(τ)‖2L∞dτ � t+

∫ t

0

‖v(τ)‖L2‖∂xv(τ)‖L2dτ

� t+√1 + t

∫ t

0

‖∂xv(τ)‖L2dτ � (1 + t)3/2,

and so ∫ t

0

‖∂2xv(τ)‖2L2dτ � (1 + t)5/2 + (1 + t)

∫ t

0

‖∂xv(τ)‖2L∞dτ.


Now the Gagliardo–Nirenberg and Cauchy–Schwarz inequalities yield∫ t

0

‖∂xv(τ)‖2L∞dτ �∫ t

0

‖∂xv(τ)‖L2‖∂2xv(τ)‖L2dτ

�(∫ t

0

‖∂xv(τ)‖2L2dτ

)1/2 (∫ t

0

‖∂2xv(τ)‖2L2dτ

)1/2

�√1 + t

((1 + t)5/2 + (1 + t)

∫ t

0

‖∂xv(τ)‖2L∞dτ

)1/2

� (1 + t)7/4 + (1 + t)

(∫ t

0

‖∂xv(τ)‖2L∞dτ

)1/2

.

The Young inequality implies∫ t

0

‖∂xv(τ)‖2L∞dτ � (1 + t)2,

and hence ∂xv ∈ L2loc([0,∞);L∞) ⊂ L1

loc([0,∞);L∞).Since ∂xa ∈ L1

loc([0,∞);L2), Gagliardo–Nirenberg shows that it suffices to provethat ∂2

xa ∈ L1loc([0,∞);L2) to conclude that ∂xa ∈ L1

loc([0,∞);L∞). For that, weuse again item (iii) of Theorem 4, from which we know that aeiφ ∈ L1

loc([0,∞);H2).Differentiating aeiφ twice, we have, since ∂xφ = v,

‖∂2xa(t)‖L2 � ‖aeiφ‖H2 + ‖v∂xa‖L2 + ‖a∂xv‖L2 + ‖v2a‖L2

� ‖aeiφ‖H2 + ‖v‖L∞‖∂xa‖L2 + ‖a‖L∞‖∂xv‖L2 + ‖v‖2L∞.

From the above estimate, we infer ∂2xa ∈ L1

loc([0,∞);L2), which completes the proofof the theorem.

Remark 10 (higher dimension). Even though we have obviously taken advantageof the one-dimensional setting to use the embedding H1 ⊂ L∞, the true reason whyTheorem 8 is limited to d = 1 lies elsewhere. The refined L2 estimate (2.4) becomes,if d � 2,

1

2

d

dt‖vε(t)‖2L2 + ε2‖∇vε(t)‖2L2 �

∫ (|aε(t, x)|2 + |vε(t, x)|2) |∇vε(t, x)|dx.

Note the appearance of the new term∫ |vε(t, x)|2|∇vε(t, x)|dx: It corresponds to the

fact that∫vε · (vε · ∇vε) is zero if d = 1, but not if d � 2 (in general). This aspect

seems to be the only real limitation to extend Theorem 8 to d � 2.

3. Numerics. In this section, we define a second order numerical scheme for(1.12). Since ε is no longer a singular perturbation parameter in this reformulationof NLS, this scheme is naturally asymptotic preserving for our original problem inthe semiclassical limit ε → 0, as long as the solution of the Euler equation remainssmooth. Since we solve our problem in dimension 1 on a bounded interval and indimension 2 on a rectangle, we add periodic boundary conditions to (1.1) or to (1.12),both formulations remaining equivalent.

3.1. The AP numerical scheme. The nonlinear system to be solved reads as

(3.1)

⎧⎪⎪⎨⎪⎪⎩

∂ta+ div (av) +

(iε− 1

2

)a div v = i

ε

2Δa; a|t=0 = a0,

∂tv +∇( |v|2

2+ |a|2

)= ε2Δv; v|t=0 = v0.


The semidiscretization in time is realized through a Strang splitting scheme. Let usdenote by Δtn = tn− tn−1 the variable time step, such that tn =

∑nk=1 Δtk. On each

time step [tn, tn+1], we split (3.1) into two subsystems and apply the second orderStrang splitting algorithm.

Step 1 for t ∈ [tn, tn + Δtn+1

2 ]:⎧⎨⎩ ∂ta1 = i

ε

2Δa1; a1|t=tn = a|t=tn ,

∂tv1 = ε2Δv1; v1|t=tn = v|t=tn .

Step 2 for t ∈ [tn, tn +Δtn+1]:⎧⎪⎪⎨⎪⎪⎩

∂ta2 + div (a2v2) +

(iε− 1

2

)a2 div v2 = 0; a2|t=tn = a

1|t=tn+Δtn+1

2

,

∂tv2 +∇x

( |v2|22

+ |a2|2)

= 0; v2|t=tn = v1|t=tn+

Δtn+12

.

Step 3 for t ∈ [tn, tn + Δtn+1

2 ]:⎧⎨⎩ ∂ta3 = i

ε

2Δa3; a3|t=tn = a2|t=tn+1

,

∂tv3 = ε2Δv3; v3|t=tn = v2|t=tn+1.

Then, (a3, v3)|t=tn+Δtn+1

2

is an approximation of (a, v)|t=tn+1, solution to (3.1).

This splitting scheme enables us to decouple the fluid part (Step 2) from theparabolic and Schrodinger parts (Steps 1 and 3), which allows a standard treatmentof each step. Note that for ε = 0, we recover exactly the Euler equations (1.10).We denote by (an,ε, vn,ε) the numerical solution at time tn. In the spirit of [30], thefollowing result can be proven.

Proposition 11. Under the assumptions of Theorem 4, there exist ε0 > 0 andC, c0 independent of ε ∈ [0, ε0] such that for all n ∈ N such that tn ∈ [0, T ], where Tis as in Theorem 4(ii), for all Δtn ∈ (0, c0],∥∥∥|an,ε|2 − ρε(tn)

∥∥∥L1(Rd)∩L∞(Rd)

� C(maxn

Δtn)2,∥∥∥εIm(an,ε∇an,ε) + |an,ε|2 vn,ε − jε(tn)

∥∥∥L1(Rd)∩L∞(Rd)

� C(maxn

Δtn)2.

We assume, without loss of generality, that we are in a periodic framework forthe space variable. This allows us to solve Steps 1 and 3 in a spectral way by usingthe fast Fourier transform, whereas for Step 2, we use a Lax–Wendroff scheme (withdirectional splitting in dimension 2). The extension to Dirichlet or Neumann boundaryconditions would require one to consider sine or cosine transforms in place of FFT.The total scheme is consistent with (3.1), and second order in time and space. Sincethe Lax–Wendroff scheme is explicit, a Courant–Friedrich–Lewy condition is necessaryfor stability:

Δtn � CFLΔx

maxj(|vnj |+ |anj |

) ,where vnj denotes the approximation to v(tn, xj), xj being a node of the mesh discre-tization of the computational domain Ω, and Δx is the space discretization parameter.In all our numerical experiments, we take CFL = 0.8.


3.2. Numerical experiments in dimension 1. In order to analyze the behav-ior of the numerical solutions (3.1), we proceed like in [6], and compare the behaviorof numerical physical observables (1.13) computed by our AP numerical scheme to areference solution. Since exact solutions to (1.1) with initial data (1.2) are not avail-able, we numerically generated one thanks to the usual splitting method describedin [6], applied to (1.1)–(1.2) with a meshing strategy that ensures that the time stepΔt � ε and the space step Δx � ε. We consider the initial data

(3.2)a0(x) = e−25(x−0.5)2 ,

v0(x) = − 15∂x ln (e

5(x−0.5) + e−5(x−0.5)),

with x ∈ [−1/2, 3/2].Remark 12 (discontinuity and periodicity). The above example is borrowed

from [6]. The velocity v0 given by (3.2) is not continuous at the boundary, whenconsidered in a periodic setting. First, we remark that this discontinuity will notbe seen when looking at the current density or the wave function itself, since theamplitude decays exponentially to zero before the boundary. A more subtle argumentis the following. In the neighborhood of the boundary, since a is very small at leastfor small times, we are actually solving the viscous Burgers equation on v, with apossibly very small viscosity ε2. However, it is noteworthy that the discontinuity onthe initial data (3.2) is increasing (since v0(3/2−) = − tanh(5) ≈ −1, and v0(3/2+) =v0(−1/2+) = tanh(5) ≈ 1, where the first equality is due to the periodicity), so thesolution of the Riemann problem associated to the nonviscous Burgers equation is ararefaction wave, and this continuous function (for t > 0) is correctly handled by theLax–Wendroff numerical scheme.

In all our simulations, the smallest value of ε is 10−4. We then discretize thex-interval with J = 215 subintervals and the time step is Δt = ε/100. We refer tothe reference solution in a generic way by the notation uε

ref . With the consideredinitial datum, new oscillations appear in the reference solution past a time T ∗ ∼ 0.10,meaning that singularities were created in the limit Euler system (1.10). In Figures 1and 2, we plot particle and current densities ρεref and jεref for ε = 10−4, respectively, attimes t = 0.05 and t = 0.13, before and past the formation of shocks. We present inFigure 3 a zoom close to singularities which shows oscillations of physical observables.

We now emphasize the interest of the viscosity term in our new system (1.12),compared to the system (1.9) introduced by Grenier. Numerical evidence allows usto conjecture that, for fixed ε > 0, the dispersive terms in (1.9) are not sufficientto prevent the formation of singularities. In Figures 4 and 5, we plot the real andimaginary values of the function aε, and the function vε, computed at time t = 0.3105for ε = 1, with fine time and space steps, for our model (1.12) (in blue in the electronicversion) and for the system of Grenier (in red in the electronic version). It appearsclearly that, while the solution of our system remains smooth, the solution of (1.9)has formed a discontinuity on the function v, associated to a discontinuity on thederivative of Im(aε). In Figure 6, we plot the particle and current densities, at thesame time, which enables us to check that both models remain reformulations of thesame NLS equation.

In order to show the AP property of our scheme, we compute relative �1 errorsat time tn using the formula

errρε(tn) = ‖ρεref − ρn,ε‖1/‖ρεref‖1, ρn,εj = |an,εj |2,errjε(tn) = ‖jεref − jn,ε‖1/‖jεref‖1, jn,εj = εIm(an,εj ∇an,εj ) + ρn,εj vn,εj ,


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

Par

itlce

den

sity

(a) Particle density ρref

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.1

−0.05

0

0.05

0.1

0.15

x

Cur

rent

den

sity

(b) Current density jref

Fig. 1. Particle and current densities at time T ∗ = 0.05 and for ε = 10−4.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

Par

ticle

den

sity


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x

Cur

rent

den

sity


Fig. 2. Particle and current densities at time T ∗ = 0.13 and for ε = 10−4.

0.63 0.64 0.65 0.66 0.67 0.68 0.690

0.1

0.2

0.3

0.4

0.5

0.6

x

Par

ticle

den

sity


0.63 0.64 0.65 0.66 0.67 0.68 0.69−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

x

Cur

rent

den

sity


Fig. 3. Zoom around singularity of density and current at time T ∗ = 0.13 and for ε = 10−4.

where the �1 norm is

‖u‖1 = ΔxJ−1∑k=1

|uj|.


−0.5 0 0.5 1 1.5−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Re(

a)

x

(a) Re(aε)

−0.5 0 0.5 1 1.5−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Im(a

)

x

(b) Im(aε)

Fig. 4. Function aε at time t = 0.3105 and for ε = 1, associated to our system (1.12) (red, inthe electronic version, curves) and to the system of Grenier (1.9) (blue, in the electronic version,curves).

−0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

v

x

Fig. 5. Velocity vε at time t = 0.3105 and for ε = 1, associated to our system (1.12) (red, inthe electronic version, curve) and to the system of Grenier (1.9) (blue, in the electronic version,curve).

−0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Par

ticle

den

sity

x

(a) Particle density ρε

−0.5 0 0.5 1 1.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Cur

rent

den

sity

x

(b) Current density jε

Fig. 6. Particle and current densities at time t = 0.3105 and for ε = 1, associated to oursystem (1.12) (red, in the electronic version, curves) and to the system of Grenier (1.9) (blue, inthe electronic version, curves).


10−4

10−3

10−2

10−1

100

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Δ x

Rel

L1E

rr

Slope 1.96

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0001

(a) Error with respect to (w.r.t.) J = 2M

10−4

10−3

10−2

10−1

100

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

ε

Rel

L1E

rr

J=16J=32J=64J=128J=256J=512J=1024J=2048J=4096J=8192

(b) Error w.r.t. ε

Fig. 7. errρε(t = 0.05) for AP scheme.

10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Δ x

Rel

L1E

rrJ

Slope 1.94

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0001

(a) Error w.r.t. J = 2M

10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

101

ε

Rel

L1E

rrJ

J=16J=32J=64J=128J=256J=512J=1024J=2048J=4096J=8192

(b) Error w.r.t. ε

Fig. 8. errjε(t = 0.05) for AP scheme.

We evaluate the error function for different values of J = 2M , where M is an integerhere chosen in [4, 13], and various scaled Planck parameter ε in [10−4, 100]. We plotin Figures 7, 8, 9, and 10 the relative error on physical observables ρε and jε com-puted with our AP schemes at times t = 0.05 and t = 0.13, respectively, before andafter the formation of singularities. We clearly see that the error is proportional to(Δx)2, independently of ε before formation of singularities. Indeed, the mean slopeof the error curves with respect to Δx is close to the value 2 (subfigure (a)). Theindependence with respect to ε appears in subfigure (b) where error curves are flat.After the formation of shocks, the behavior of our scheme is very different and themesh parameters have to be reduced as ε → 0 to get good accuracy.

For a comparison, we present the same study for the classical time splitting schemeapplied to (1.1)–(1.2) in Figures 11, 12, 13, and 14. Thanks to the spectral accuracyof Fourier methods, the error levels are smaller than for our AP scheme which is onlysecond order with respect to time and space variables. However, contrary to our APscheme, the error curves clearly depend on ε, the error being of order O(1) when ε issmaller than 2.10−3. In order to have good accuracy, it is required to take Δx � ε.

We present on Figure 15 the evolution of errρε for our AP scheme with respect to


10−4

10−3

10−2

10−1

100

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Δ x

Rel

L1E

rr

Slope 1.98

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0001


10−4

10−3

10−2

10−1

100

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

ε

Rel

L1E

rr

J=16J=32J=64J=128J=256J=512J=1024J=2048J=4096J=8192

(b) Error w.r.t. ε


10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Δ x

Rel

L1E

rrJ

Slope 1.95

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0001


10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

101

ε

Rel

L1E

rrJ

J=16J=32J=64J=128J=256J=512J=1024J=2048J=4096J=8192

(b) Error w.r.t. ε

Fig. 10. errjε (t = 0.13) for AP scheme.

10−4

10−3

10−2

10−1

100

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Δ x

Rel

L1E

rr

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0001


10−4

10−3

10−2

10−1

100

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

ε

Rel

L1E

rr

J=16J=32J=64J=128J=256J=512J=1024J=2048J=4096J=8192

(b) Error w.r.t. ε

Fig. 11. errρε(t = 0.05) for time splitting scheme.


10−4

10−3

10−2

10−1

100

10−25

10−20

10−15

10−10

10−5

100

105

Δ x

Rel

L1E

rrJ

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0001


10−4

10−3

10−2

10−1

100

10−25

10−20

10−15

10−10

10−5

100

105

ε

Rel

L1E

rrJ

J=16J=32J=64J=128J=256J=512J=1024J=2048J=4096J=8192

(b) Error w.r.t. ε

Fig. 12. errjε(t = 0.05) for time splitting scheme.

10−4

10−3

10−2

10−1

100

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Δ x

Rel

L1E

rr

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0001


10−4

10−3

10−2

10−1

100

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

ε

Rel

L1E

rr

J=16J=32J=64J=128J=256J=512J=1024J=2048J=4096J=8192

(b) Error w.r.t. ε

Fig. 13. errρε(t = 0.13) for time splitting scheme.

10−4

10−3

10−2

10−1

100

10−25

10−20

10−15

10−10

10−5

100

105

Δ x

Rel

L1E

rrJ

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0001


10−4

10−3

10−2

10−1

100

10−25

10−20

10−15

10−10

10−5

100

105

ε

Rel

L1E

rrJ

J=16J=32J=64J=128J=256J=512J=1024J=2048J=4096J=8192

(b) Error w.r.t. ε

Fig. 14. errjε(t = 0.13) for time splitting scheme.


10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

Δ x

Rel

L1E

rr

T=0.080T=0.100T=0.120T=0.125T=0.130T=0.140T=0.150

(a) ε = 50.10−4

10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

Δ x

Rel

L1E

rr

T=0.080T=0.100T=0.120T=0.125T=0.130T=0.140T=0.150

(b) ε = 25.10−4

10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

Δ x

Rel

L1E

rr

T=0.080T=0.100T=0.120T=0.125T=0.130T=0.140T=0.150

(c) ε = 10.10−4

10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

Δ x

Rel

L1E

rr

T=0.080T=0.100T=0.120T=0.125T=0.130T=0.140T=0.150

(d) ε = 1.10−4

Fig. 15. errρε with respect to Δx for various final time and AP scheme.

the discretization parameter Δx for various final times and various values of ε. Theformation time of singularities is close to t∗ = 0.12. We clearly see that before t∗, Δxcan be chosen independently of ε to get a fixed accuracy. This is not, however, thecase past t∗. It suggests that we have to take Δx = o(ε) to get good accuracy.

To make our presentation complete, we present the reconstruction of uε thanksto (1.14) with ε = 0.005. We, therefore, compute the phase φε with amplitude aε andvelocity vε. Since we have access to values of these quantities at every discrete timetn, we approximate the time dependent integral using a simple rectangular quadra-ture. We see that we can recover a very good approximation of the wave function uε

with very few points before the formation of singularities (see Figure 16). Obviously,one needs more points to have a good reconstruction of the wave function past thesingularities (see Figures 17 and 18).

3.3. Numerical experiments in dimension 2. The previous subsection wasdevoted to one-dimensional simulations. We present here the equivalent error analysisfor the two-dimensional case. The computational domain is now the square [−0.5, 1.5]2

discretized with J points in each direction. We still need a reference solution generatedwith a Strang splitting scheme applied to (1.1)–(1.2). We take Jref = 213 = 8192, soΔx ∼ 2.5 × 10−4 and Δt = ε/100. In a first test, the initial datum is related to the


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x

R(u

)

(a) J = 128

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x

R(u

)

(b) J = 256

Fig. 16. Re(uε) at T = 0.05 for ε = 0.005. In blue (in the electronic version), the referencesolution and in red (in the electronic version), the solution computed with our scheme.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

R(u

)

(a) J = 128

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

R(u

)

(b) J = 256


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

R(u

)

(a) J = 512



(a) ρref (b) |jref |

Fig. 19. Contour plot of particle and current densities at time t = 0.05 and for ε = 5.10−4.

(a) ρref (b) |jref |

Fig. 20. Contour plot of particle and current densities at time t = 0.13 and for ε = 5.10−4.

one chosen for the one-dimensional case:

a0(x, y) = e−25 r2 ,

v0(x) = −1

5∇ ln (e5 r + e−5 r),

where r =√(x− 0.5)2 + (y − 0.5)2. We evaluate the error function for different val-

ues of J = 2M , where M is chosen in [4, 11], and various scaled Planck parameter ε in[5.10−4, 100]. The formation time of singularities is, as in the one-dimensional situa-tion, located around 0.10. The reference particle density and the reference modulus ofthe current densities for ε = 5.10−4 are represented on Figures 19 and 20 before andafter formation of singularities. We clearly see that a tiny front appears after shocksin both particle and current densities. The wave function presents strong oscillationswith new ones created past the shock creation (see Figure 21).

The error analysis results are equivalent to the ones obtained for one-dimensionalsimulation. We recover an independence of the error with respect to ε before the


(a) uε at time t = 0.05 (b) uε at time t = 0.13

Fig. 21. Contour plot of real part of uε at times t = 0.05 and t = 0.13 and for ε = 5.10−4.

10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

Δ x

Rel

L1E

rr

Slope 1.95

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0005


10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

ε

Rel

L1E

rr

J=16J=32J=64J=128J=256J=512J=1024J=2048

(b) Error w.r.t. ε


formation of singularities (see Figures 22 and 23). We always need to take finer meshpast the shock formation (see Figures 24 and 25).

The last computation concerns a nonsymmetric solution. The aim of our simu-lations is to show that we recover good qualitative properties, even for nonradiallysymmetric data. We consider here as initial datum a Maxwell distribution with twotemperatures, θ1 = 0.05 and θ2 = 0.015, for the initial amplitude given by

a0 =1√4

1

2π√θ1√θ2

exp

(− (x− 0.5)2

2θ1− (y − 0.5)2

2θ2

).

The initial phase is reduced to φ0 = 0 (note that from (1.11), ∂tφ|t=0 = −|a0|2 �= 0, soφ becomes instantaneously nontrivial). The scaled Planck factor ε is equal to ε = 0.005and the simulations are performed with J = 2048 intervals in each direction. We plotin Figures 26, 27, and 28 the contour plots of ρε, |jε|, and Re(uε) at times t = 0.035and t = 0.08. The oscillatory nature is enhanced at time t = 0.08 mainly in the ydirection.

We recover the good behavior of physical observables before the formation time


10−4

10−3

10−2

10−1

100

10−5

10−4

10−3

10−2

10−1

100

Δ x

Rel

L1E

rrJ

Slope 1.95

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0005


10−4

10−3

10−2

10−1

100

10−5

10−4

10−3

10−2

10−1

100

ε

Rel

L1E

rrJ

J=16J=32J=64J=128J=256J=512J=1024J=2048

(b) Error w.r.t. ε


10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

Δ x

L1R

elE

rr

Slope 1.86

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0005


10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

ε

L1R

elE

rr

J=16J=32J=64J=128J=256J=512J=1024J=2048

(b) Error w.r.t. ε


10−4

10−3

10−2

10−1

100

10−5

10−4

10−3

10−2

10−1

100

101

Δ x

L1R

elE

rrJ

Slope 1.82

ε=1.0000ε=0.6400ε=0.3200ε=0.1600ε=0.0800ε=0.0400ε=0.0200ε=0.0100ε=0.0050ε=0.0025ε=0.0010ε=0.0005


10−4

10−3

10−2

10−1

100

10−5

10−4

10−3

10−2

10−1

100

101

ε

L1R

elE

rrJ

J=16J=32J=64J=128J=256J=512J=1024J=2048

(b) Error w.r.t. ε



(a) ρε(t = 0.035) (b) ρε(t = 0.08)

Fig. 26. Contour plots of particle density ρε for ε = 0.005.

(a) |jε|(t = 0.035) (b) |jε|(t = 0.08)

Fig. 27. Contour plots of current density |jε| for ε = 0.005.

(a) Re(uε)(t = 0.035) (b) Re(uε)(t = 0.08)

Fig. 28. Contour plots of Re(uε) for ε = 0.005.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

y

rho

Strang

Apnls

(a) Slice at x = 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

rho

Strang

Apnls

(b) Slice at y = 0.5

Fig. 29. Slices of ρε(t = 0.035) for ε = 0.005 for Strang and AP schemes.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

y

J

Strang

Apnls


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

x

J

Strang

Apnls


Fig. 30. Slices of |jε|(t = 0.035) for Strang and AP schemes.

of singularities. To distinguish the evolution for each direction (x, y), we present slicesof physical observables along the lines x = 0.5 and y = 0.5. We compare the solutionscomputed by the Strang splitting scheme and our AP scheme at time t = 0.035 inFigures 29 and 30. We cannot notice any differences. Past the formation time ofsingularities, some noticeable differences can be seen on x = 0.5 and y = 0.5 planeslices (see Figures 31 and 32).

4. Extension to other frameworks.

4.1. The linear case: Global solution of the viscous eikonal equation.The advantage of introducing an artificial diffusion is more striking in the linear case.Consider

(4.1) iε∂tuε +

ε2

2Δuε = Vextu

ε; uε|t=0 = a0e

iφ0/ε,

with Vext = Vext(t, x) real-valued. As noticed in [22], using the Madelung transform inthis context is interesting only if vacuum can be avoided. Having the idea of Grenier


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

y

rho

Strang

Apnls


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

x

rho

Strang

Apnls


Fig. 31. Slices of ρε(t = 0.08) for Strang and AP schemes.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

y

J

Strang

Apnls


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

x

J

Strang

Apnls


Fig. 32. Slices of |jε|(t = 0.08) for Strang and AP schemes.

[29] in mind, the counterpart of (1.8) reads as⎧⎪⎨⎪⎩


1

2aεΔφε = i

ε

2Δaε, aε|t=0 = a0,

∂tφε +

1

2|∇φε|2 + Vext = 0, φε

|t=0 = φ0.

We note that the two equations are decoupled: The second equation is of Hamilton–Jacobi type (known as the eikonal equation in this context), and the solution mustbe expected to develop singularities in finite time (see, e.g., [12]), so this approach isdoomed to fail for large time. On the other hand, introducing the artificial diffusionas in (1.11), we obtain

(4.2)

⎧⎪⎨⎪⎩


1

2aεΔφε = i

ε

2Δaε − iεaεΔφε, aε|t=0 = a0,

∂tφε +

1

2|∇φε|2 + Vext = ε2Δφε, φε

|t=0 = φ0.

The system is still decoupled, but the good news is that the diffusion introduced intothe Hamilton–Jacobi equation smooths out the solution. Therefore, this system can


be considered as a good candidate to obtain an AP scheme for the linear Schrodingerequation, regardless of the presence of vacuum.

Instead of giving full details of the analogues of Theorems 4, 7, and 8, as well asof their proofs, we shall simply give a functional framework where the viscous eikonalequation can be solved globally in time. Since we consider the case ε > 0 only, wedrop out the ε2 term in the viscous eikonal equation and consider

(4.3) ∂tφeik +1

2|∇φeik|2 + Vext = Δφeik; φeik(0, x) = φ0(x).

For k � 1, let

SLk = {f ∈ Ck(Rd;R), ∂αf ∈ L∞(Rd) ∀1 � |α| � k}.For k � 2, let

SQk = {f ∈ Ck(Rd;R), ∂αf ∈ L∞(Rd) ∀2 � |α| � k}.The key result is the following lemma.

Lemma 13. Let k � 2 and φ0 ∈ SLk, Vext ∈ L∞loc(R+; SLk). Then (4.3) has a

unique solution φeik ∈ C(R+; SLk−1).Remark 14. No such global result must be expected in the larger class SQk.

Indeed, suppose d = 1, Vext = 0, and φ0(x) = −x2/2. If (4.3) had a solutionφeik ∈ C(R+; QL2), then w = ∂2

xφeik ∈ C(R+;L∞) would solve

∂tw + v∂xw + w2 = ∂2xw; w(0, x) = −1.

The solution of this equation does not depend on x. It is given by w(t, x) = 1/(t− 1),hence a contradiction. However, in the periodic case x ∈ Td, this technical issuedisappears (see Remark 15).

Proof. The first step consists of constructing the gradient of φeik. Differentiating(4.3) in space, we have to solve

(4.4) ∂tv + v · ∇v +∇Vext = Δv; v(0, x) = ∇φ0(x).

This equation is solved locally in time by a fixed point argument by using Duhamel’sformula

v(t) = etΔ∇φ0 −∫ t

0

e(t−s)Δ(v · ∇v)(s)ds −∫ t

0

e(t−s)Δ(∇Vext)(s)ds.

The right-hand side is a contraction on{w ∈ C

([0, T ];W k−1,∞)

, ‖w‖L∞([0,T ];Wk−1,∞) � 2‖∇φ0‖Wk−1,∞},

provided that T > 0 is sufficiently small. This follows from the properties of the heatkernel: for t > 0,

‖etΔf‖L∞ � ‖f‖L∞; ‖etΔ∇f‖L∞ � C√t‖f‖L∞.

The solution to (4.4) is then global, thanks to the maximum principle (see, e.g., [48,Proposition 52.8]), which implies that v− � v � v+, where

∂tv± ∓ ‖∇Vext‖L∞x

= 0; v±(0, x) = ±‖∇φ0‖L∞ .


Once such a solution v ∈ C(R+;Wk−1,∞) is constructed, set

φeik(t) = φ0 −∫ t

0

(1

2|v(s)|2 + Vext(s)− div v(s)

)ds.

We check that ∂t(∇φeik − v) = ∇∂tφeik − ∂tv = 0. Therefore, v = ∇φeik, and φeik

solves (4.3). Rewriting (4.3) as

∂tφeik +1

2|v|2 + Vext = Δφeik; φeik(0, x) = φ0(x),

Duhamel’s formula reads as

φeik(t) = etΔφ0 −∫ t

0

e(t−s)Δ(|v(s)|2)ds−∫ t

0

e(t−s)ΔVext(s)ds,

from which we conclude that φeik ∈ C(R+; SLk−1).Remark 15. In the periodic setting x ∈ Td = (R/(2πZ))d, it is natural to work

in the Wiener algebra

W =

⎧⎨⎩f : Td → C, f(x) =

∑j∈Zd

bjeij·x with (bj)j∈Zd ∈ �1(Zd)

⎫⎬⎭ ,

and for k � 0,

W k = {f : Td → C, ∂αf ∈ W ∀|α| � k}.Then Lemma 13 is easily adapted by replacing SLk with W k.

4.2. Other nonlinearities. One might believe that Theorem 8 is bound to thecubic one-dimensional Schrodinger equation, which is completely integrable, and thatthe two aspects are related. We show that this is not the case, since Theorem 8remains valid for other nonlinearities (with d = 1). Consider now

(4.5) iε∂tuε +

ε2

2Δuε = f

(|uε|2)uε, (t, x) ∈ R+ × Rd.

We suppose that there exists δ > 0 such that f ′(y) � δ for all y � 0.Following [29], the only modification we have to make to recover the results in

section 2.1 consists of replacing the symmetrizer (2.2) with

S =

(I2 00 1

4f ′(a21+a2

2)Id

).

From our assumption on f , S and its inverse S−1 are uniformly bounded, providedthat a is bounded. As a matter of fact, the exact assumption in [29] is f ′ > 0, and allthe results in section 2.1 remain valid under this assumption.

The more precise assumption f ′(y) � δ becomes useful to prove that in dimensiond = 1, the solution is global. The main point to notice is that the conservation ofmass and momentum are the same as for (1.1), like in Proposition 1. On the otherhand, the conservation of energy becomes

d

dt

(‖ε∇uε(t)‖2L2 +

∫Rd

F(|uε(t, x)|2) dx) = 0,


where F is an antiderivative of f ,

F (y) =

∫ y

0

f(r)dr.

By assumption,

F (y) � δ

2y2 + f(0)y,

so the potential energy controls the L4-norm:

‖uε(t)‖4L4 � 2

δ

(∫Rd

F(|uε(t, x)|2) dx− f(0)‖uε(t)‖2L2

)

� 2

δ

(∫Rd

F(|uε(t, x)|2) dx− f(0)‖uε(0)‖2L2

),

where we have used the conservation of mass in the last inequality. This implies anestimate of the form

‖ε∇uε(t)‖2L2 + ‖uε(t)‖4L4 � C,

with C independent of t � 0 and ε ∈ (0, 1]. Therefore, the analysis presented insection 2.3 can be repeated line by line.

Example 16. The assumption f ′ � δ > 0 is satisfied in the following cases:• Cubic-quintic nonlinearity: f(y) = y + λy2, λ � 0.• Cubic plus saturated nonlinearity: f(y) = δy + η y

1+λy , δ, η, λ > 0.

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