Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity

We analyse a class of time discretizations for solving the Gross-Pitaevskii equation at low-regularity on an arbitrary Lipschitz domain Ω⊂ℝ^d, d ≤ 3, with a non-smooth potential. We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space H^r, r ≥ 0 beyond the more typical L^2-error analysis.

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1 Introduction

We consider the Gross-Pitaevskii (GP) equation

 i∂tu(t,x)=−Δu(t,x)+V(x)u(t,x)+|u(t,x)|2u(t,x),(t,x)∈R×Ω (1)

with , , and an initial condition

 u|t=0=u0. (2)

When , we assign boundary conditions which will be encoded in the choice of the domain of the operator . We recall that the linear operator generates a group of unitary operators on . We will deal with mild solutions of (1) and (2) which are given by Duhamel’s formula;

 u(t)=eitΔu0+∫t0ei(t−ζ)Δf(u,¯u,V)(ζ,x)dζ (3)

where we denote the nonlinearity by

 f(u,¯u,V)(t,x)=−i(V(x)u(t,x)+u2(t,x)¯u(t,x)). (4)

Throughout this article we will be interested in studying numerical schemes which approximate the time dynamics of (1) at low-regularity, by means of appropriate approximations of Duhamel’s formula. Namely, we are interested in providing a reliable approximation of (1) (or equivalently of (3)) when the initial data and the potential are non-smooth, in the sense that they belong to Sobolev spaces of low order.

One setting for the Gross-Pitaevskii equation is to describe the dynamics of Bose-Einstein condensates in a potential trap. In many physically relevant situations the potential is assumed to be rough or disordered, and hence the study of (1) in this non-smooth or low-regularity framework is of physical interest ([10], [18], [19]).

Recently much progress has been made in the development of low-regularity approximations to nonlinear evolution equations. First, in the case of periodic boundary conditions a class of schemes called Fourier integrators [12] or resonance based schemes [4] were introduced to approximate the time dynamics of dispersive equations such as NLS, KdV, Boussinesq, Dirac and Klein-Gordon (see [8], [11], [14] , [17], [5]). Very recently higher order extensions of these resonance based schemes were introduced in [4] for approximating in a unified fashion a large class of dispersive equations with periodic boundary conditions. These resonance based schemes were shown to converge in a more general setting, namely under lower regularity assumptions, than classical methods required (see [11], [12] and references therein for a comparative analysis). The name of these schemes are due to their construction which revolves around Fourier based expansions of the solution and of the resonant structure of the equation. We explain the idea behind these resonance based schemes in detail in Section 3.1. These ideas were then extended in [15] to treat more general domains and boundary conditions and allows to deal with a class of parabolic, hyperbolic and dispersive equations in a unified fashion. The resulting schemes were termed low-regularity integrators, or Duhamel’s integrators ([15]). The next natural step in this study of low-regularity approximations to nonlinear PDEs is to introduce a potential term with minimal regularity assumptions on the solution and the potential . The first order low-regularity integrator for (1) was first introduced in the report [1], which also provides a preliminary discussion of the study of low-regularity integrators for (1). The goal of this article is to study the first and second low-regularity schemes for (1), with an emphasis on the error analysis. Using new techniques based on decorated trees series analysis we will present in [2] a general framework for deriving low-regularity schemes up to arbitrary order, extending the constructions presented in this article.

In this article we study a class of low-regularity integrators to solve the Gross-Pitaevskii equation (1) on an arbitrary domain . Further, in the case where the domain is a torus , we state and prove first and second order convergence in any fractional positive Sobolev space , under moderate regularity assumptions on both the solution and the potential . This is a stronger convergence result than the more typical

-convergence analysis. In future work we will extend the Sobolev error estimates in the case of bounded smooth Lipschitz domains with homogeneous Dirichlet boundary conditions. We state our results in the next subsections, starting with the first-order scheme and following with the second order scheme.

1.1 First-order low regularity integrator

In Section 3.2 we construct the following first order low regularity integrator on , which was first stated in [1]. For , we define,

 un+1=Φτnum,1(un):=eiτΔ[un−iτ(unφ1(−iτΔ)V+(un)2φ1(−2iτΔ)¯un)],where u0=u0, (5)

and is a bounded operator on . A different construction based on tree series analysis will be presented in [2] to obtain a similar low-regularity scheme as above. Further, we prove in Section 3.4 the following first order convergence result for the scheme (5), in the case of periodic boundary conditions. For the local-wellposedness result of (1) and (2) we refer to Theorem 2.2 given in Section 2.

Theorem 1.1.

Let , , and

 r1:=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩r+1,  if r>d2,1+d2+ϵ, if 0

where can be arbitrarily small. For every and , let be the unique solution of (1). Then there exists and such that for every time step size the numerical solution given by (5) has the following error bound:

 ∥un−u(nτ)∥Hr≤CTτ,0≤nτ≤T.

Before moving on to the second order scheme and its convergence result we make a few remarks on the regularity assumptions made in the above theorem. A consequence of Theorem 1.1 is that for any initial data and potential in where and (or on the full space ) we have the following global error estimate:

 max1≤nτ≤T∥un−u(tn)∥Hr≤C(sup[0,T]||u(t)||Hr+1,||V||Hr+1)τ.

Namely we only ask one additional Sobolev derivative on the initial data and the potential in order to obtain first-order convergence of our low-regularity scheme (5). This is due to the favorable local error structures that these low-regularity schemes inherit. See ([11], [4], [15]), and references therein for an in depth comparative analysis of these low-regularity schemes with classical methods such as splitting methods, or exponential integrator methods.

Secondly, the error analysis for (1) in the case has not to our knowledge previously been studied, and these are the first convergence results in this regime.

Finally, when , we consider the regularity assumptions required for an -error analysis, and compare them with the existing convergence results for the Gross-Pitaevskii equation (1). When , Theorem 1.1 states,

 ||u(tn)−un||L2≤C(sup[0,T]||u(t)||H1+d4,||V||H1+d4)τ,0≤nτ≤T.

To our knowledge, this is the first convergence result of this type with low-regularity assumptions on both the solution and the potential . Indeed, in the literature -convergence results have been established for smooth potentials. See, for example [9], where the authors showed first-order convergence of a Lie splitting scheme for the linear Schrödinger equation with a potential term where they require to be a -smooth potential. More recently, the authors [7] were able to show first-order convergence to (1) of a Crank-Nicholson scheme while demanding low-regularity assumptions on the potential . Namely, for , where and smooth potential, they obtained first-order convergence of their scheme for -among other assumptions- , which using the PDE boils down to requiring that . In other words, paper [7] relaxes the smoothness assumptions on by requiring more regularity on . In contrast to these results Theorem 1.1 permits low-regularity assumptions on both and . Finally, we refer to [15, Corollary 20] where the authors show first-order convergence in of a low-regularity scheme for the nonlinear Schrödinger equation while asking for the same regularity assumptions on the initial data: .

1.2 Second-order low regularity integrator

In [Section 4, Corollary 4.3] we derive the following second order low regularity integrator on . For , we define,

 un+1=Φτnum,2(un) :=eiτΔun−iτeiτΔ(unφ1(−iτΔ)V+(un)2φ1(−2iτΔ)¯un) (7) −iτ((eiτΔun)φ2(−iτΔ)(eiτΔV)+(eiτΔun)2φ2(−2iτΔ)eiτΔ¯un) +iτeiτΔ(unφ2(−iτΔ)V+(un)2φ2(−2iτΔ)¯u(tn)) −τ22eiτΔ(|un|4un+3un|un|2V−|un|2un¯V+unV2).

We offer in [2] a similar low-regularity scheme as above, using a different construction based on tree series analysis. In addition, in [Section 4.2, Corollary 4.8] we offer yet another derivation of a low-regularity second order scheme for (1). Further, we prove in Section 4.4 the following second-order convergence result for the scheme (7), in the case of periodic boundary conditions.

Theorem 1.2.

Let , , and

 r2:=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩r+2,  if r>d2,2+d2+ϵ, if 0

where can be arbitrarily small. For every and , let ) be the unique solution of (1). Then there exists and such that for every time step size the numerical solution given by (7) has the following error bound:

 max1≤n≤N∥unτ−u(nτ)∥Hr≤CTτ2. (9)

Following the same comments made after Theorem 1.1, we note that in the regime , we ask only for two additional derivatives on the initial data and the potential . This is thanks to the favorable local error structure of these low-regularity integrators (see [4], [15]). Furthermore, we compare once again our result to the -convergence results obtained in the literature; we mention the authors [9] who show second-order convergence of a Strang splitting scheme for a -smooth potential. Whereas, the authors [7] obtain second-order convergence of a Crank-Nicholson scheme for a smooth potential and -among other assumptions- for , which using the PDE boils down to asking . As mentioned previously, we emphasize that in contrast to these results we aim to establish convergence under low-regularity assumptions on both and .

1.3 Outline of the paper

We motivate the construction of the first-order low-regularity integrator in Section 3.1 by first deriving the scheme in the periodic setting. We then generalize to the construction of the low-regularity scheme for an arbitrary domain . In Section 4 we introduce the second-order low-regularity integrator and discuss stability issues which arise. We then propose two different approaches to guarantee the stability of our proposed scheme. Section 3 and 4 also include the local and global error analysis of the first and respectively second low-regularity integrators. In the next section we briefly introduce some notation and nonlinear estimates which are crucial for the local and global error analysis.

2 Notation and nonlinear estimates

We begin by establishing some notation used in the paper, starting by the definition of the norm used throughout the error analysis sections. Our analysis will be made in the periodic fractional Sobolev space,

 Hr(Td):={u=∑k∈Zdukeikx√(2π)d∈L2(Td):|u|2r≜∑k∈Zd|k|2r|uk|2<+∞}

which is endowed with the norm

 ||u||2Hr =||u||2L2(Td)+||(−Δ)r/2u||2L2(Td) =∑k∈Zd(1+|k|2r)|uk|2,

where .

Throughout the remainder of this section we fix , and we restrict the class of initial data and potential to belong to the Sobolev space . We now present some fundamental nonlinear estimates which will be needed in the error analysis sections. We separate our -error analysis into three cases: , , and . First, when , using the Sobolev embedding , we have the following nonlinear estimate:

 ||vw||L2≲||v||Hσ||w||L2. (10)

In the case where we have,

 ||vw||Hr≲||v||Hσ||w||Hr, (11)

while in the regime the above holds with . We provide a proof of the above inequality (11) in the Appendix for both regimes of .

One can easily deduce from the inequalities (10) and (11) the following estimates on the nonlinearity (4);

 ||f(w,¯w,V)||Hr≤Cr,σ(||w||Hσ,||V||Hσ)||w||Hr||f(w,¯w,V)−f(v,¯v,V)||Hr≤Cr,σ(||w||Hσ,||v||Hσ,||V||Hσ)||w−v||Hr, (12)

where denotes a generic constant which depends on the bounded arguments , , and . In the regime the above holds with .

Remark 2.1.

In order to deal with less smooth initial data one cannot make use of the bilinear estimates (10), (11) and one would need to call upon more subtle tools to show appropriate fractional convergence of the scheme. Several works have been made when working on or where low-regularity estimates for very rough data could be obtained by using tools from dispersive PDE such as discrete Strichartz estimates, or Bourgain spaces, see [12], [13], [16]. This refined error analysis is out of scope for this paper.

We finish this subsection by stating the following local well posedness result of a solution to (1) and (2) of the form (3). Indeed, using (12), one obtains from a classical Banach fixed point argument the following result:

Theorem 2.2.

Let . Given any , and there exists and a unique solution to (1).

3 First-order scheme and analysis

In this section we first give the main ideas behind the construction of the first-order low regularity scheme, which was first introduced in [15] when working on an arbitrary domain . We then follow by establishing the first order convergence of our scheme under favorable regularity assumptions on the initial condition and on the potential .

Our low-regularity schemes are built from the discretization of Duhamel’s formula (3). Here we will propose a novel low-regularity integrator for the approximation of Duhamel’s formula (3). We will approximate (3) at the time step , where is the time step size. By iterating Duhamel’s formula (3), we obtain the first order iteration

 u(tn+τ)=eiτΔ[u(tn)−iJ1(τ,Δ,u(tn))]+R1,0(τ,u) (13)

where the principal oscillatory integral (at first order) is given by

 J1(τ,Δ,v)=∫τ0e−iζΔ[V(x)(eiζΔv)+(eiζΔv)2(e−iζΔ¯v)]dζ (14)

and the remainder

 R1,0(τ,u)=∫τ0ei(τ−ζ)Δ[f(u(tn+ζ),¯u(tn+ζ),V)−f(eiζΔu(tn),e−iζΔ¯u(tn),V)]dζ.

We will construct a suitable discretisation of the integral (14) to allow for a low-regularity approximation of (13). The idea is to filter out the dominant parts, which we denote be , of the high frequency interactions in (14) and embed them in the discretisation. The lower-order parts will be approximated and incorporated in the local error analysis.

First, to illustrate the underlying idea and to provide intuition behind the construction of these low-regularity integrators we start by analyzing the case of periodic boundary conditions , with a periodic potential. The ideas presented in the next section were first introduced by the authors [11] for solving a class of semilinear Schrödinger equations. After presenting the periodic case we explain in Section 3.2 the construction of the first-order scheme in the more general case of an arbitrary domain .

3.1 Case of periodic boundary conditions: Ω=T

Assuming that , we can expand in Fourier series . This allows us to express the action of the Schrödinger flow on v, . Similarly assuming we have . In Fourier, the oscillatory integral (14) is then given by,

 J1(τ,Δ,v)=∑l1+l2=l^Vl1^vl2eilx∫τ0eiζR2(l)dζ+∑−k1+k2+k3=k¯^vk1^vk2^vk3eikx∫τ0eiζR1(k)dζ

with the resonance structure,

 R1(k)=2k21−2k1(k2+k3)+2k2k3, and  R2(l)=l21+2l1l2.

We can extract the dominant and lower-order parts from these resonance structures by recalling that and correspond to second-order derivatives in Fourier while the terms (for ) correspond to product of first-order derivatives. We choose,

 R1(k)=Ldom,1(k1)+Llow,1(k1,k2,k3),  R2(l)=Ldom,2(l1)+Llow,2(l1,l2)

with

 Ldom,1(k1)=2k21, Llow,1(k1,k2,k3)=−2k1(k2+k3)+2k2k3, and Ldom,2(l1)=l21,  Llow,2(l1,l2)=2l1l2.

Mapping back into physical space we thus have

 Ldom,1(v)=−2Δv,  Llow,1(v)=2(2|∇v|2v−|∇v|2¯v), Ldom,2(v)=−Δv,  Llow,2(v)=−2∇V∇v.

and

 J1(τ,Δ,v) =∫τ0[eiζLdom,2V]v+[eiζLdom,1¯v]v2+O(ζ(Llow,2+Llow,1)v)dζ (15) =τ[vφ1(iτLdom,2)V+v2φ1(iτLdom,1)¯v]+O(τ2(Llow,2+Llow,1)v)

Hence for a small time step , by plugging the above expression of in (13) and ignoring the lower order terms yields the first-order resonance based discretization

 un+1=eiτΔ[un−iτ(unφ1(−iτΔ)V+(un)2φ1(−2iτΔ)¯un)]. (16)

The above scheme (16) has a favorable local error structure; namely from (15) we see that (formally) this discretization only ask for first order derivatives on the initial data and potential.

We now place ourselves in the general framework , and make use of filtering techniques to recover the first order low-regularity approximation (16) in this general setting. The ideas presented in the next section are inspired by the work of [15].

3.2 General boundary conditions: Ω⊂Rd

The goal of this section is to construct a first order discretization of the oscillatory integral (14) when working on a general domain , and which allows for the improved local error structure (15) established in the preceding section. This is achieved by introducing a properly chosen filtered function which will filter out the dominant oscillatory terms explicitely found in the preceding section.

First, we recall the definition of the commutator for , a function and a linear operator:

 C[H,L](v1,⋯,vn)=−L(H(v1,⋯,vn))+n∑i=1DiH(v1,⋯,vn)⋅Lvi.

We define the filtered function by

 N(τ,s,ζ,Δ,v)=e−isΔ[eisΔe−iζΔV(eisΔv)+(eisΔv)2(eisΔe−2iζΔ¯v)]. (17)

The principal oscillations (14) can be expressed with the aid of the filter function as

 J1(τ,Δ,v)=∫τ0N(τ,ζ,ζ,Δ,v)dζ.

By the fundamental theorem of calculus we have

 J1(τ,Δ,v)=∫τ0N(τ,0,ζ,v)dζ+∫τ0∫ζ0∂sN(τ,s,ζ,v)dsdζ (18)

where

 N(τ,0,ζ,v)=[eiζLdom,2V]v+[eiζLdom,1¯v]v2 (19)

and

 ∂sN(τ,s,ζ,v)=e−isΔC[f,iΔ](eisΔv,eisΔe−2iζΔ¯v,eisΔe−iζΔV),
 C[f,iΔ](u,v,w)=−2i(∇w⋅∇u+|∇u|2v+∇(u2)⋅∇v). (20)

Hence, we recover the discretization of the oscillatory integral (14) together with an improved local error structure of the form (15);

 J1(τ,Δ,v)=τ[vφ1(iτLdom,2)V+v2φ1(iτLdom,1)¯v]+R1,1(τ)

where

 R1,1(τ)=∫τ0∫ζ0e−isΔC[f,iΔ](eisΔv,eisΔe−2iζΔ¯v,eisΔe−iζΔV)dsdζ.
Corollary 3.1.

The exact solution of (1) can be expanded as

 u(tn+τ)=eiτΔ[u(tn)−iτ(u(tn)φ1(−iτΔ)V+(u(tn))2φ1(−2iτΔ)¯u(tn))]+R1(τ,tn)

where the remainder is given by

 R1(τ,tn)=∫τ0ei(τ−ζ)Δ[f(u(tn+ζ),¯u(tn+ζ),V)−f(eiζΔu(tn),e−iζΔ¯u(tn),V)]dζ+∫τ0∫ζ0ei(τ−s)ΔC[f,iΔ](eisΔu(tn),eisΔe−2iζΔ¯u(tn),eisΔe−iζΔV)dsdζ. (21)

The first order low-regularity scheme (5) follows from the above Corollary 3.1 by neglecting the remainder . We next show the first-order error estimates for the scheme (5) by starting by estimating it’s favorable commutator-type local error structure.

3.3 Local error estimates

Proposition 3.2.

Let , , and as in Theorem 1.1, namely

 r1=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩r+1, if r>d2,1+d2+ϵ, if 0

where can be arbitrarily small. Assume there exists such that

 sup[0,T]||u(t)||Hr1≤CT, and  ||V||Hr1≤CT, (22)

then there exists such that for every ,

 ||R1(τ,tn)||Hr≤MTτ2,0≤tn≤T, (23)

where and is given in (21).

Proof.

We write the error term , defined in (21), as the sum of two terms, . We begin by estimating the first term . Using the inequalities (12) on , and the boundedness of on Sobolev spaces we have that for all ,

 ||G1(τ,tn)||Hr≤∫τ0||ei(τ−ζ)Δ[f(u(tn+ζ),¯u(tn+ζ),V)−f(eiζΔu(tn),e−iζΔ¯u(tn),V)]||Hrdζ≤τsupζ∈[0,τ]||f(u(tn+ζ),¯u(tn+ζ),V)−f(eiζΔu(tn),e−iζΔ¯u(tn),V)||Hr≤τCr(sup[0,T]||u(t)||Hσ,||V||Hσ)supζ∈[0,τ]||∫ζ0ei(ζ−s)Δf(u(tn+s),¯u(tn+s),V)ds||Hr≤τ2Cr(sup[0,T]||u(t)||Hσ,||V||Hσ)sups∈[0,τ]||f(u(tn+s),¯u(tn+s),V)||Hr≤Cr(sup[0,T]||u(t)||Hσ,sup[0,T]||u(t)||Hr,||V||Hσ)τ2, (24)

where we use Duhamel’s formula in the third line.

We first note that by definition (6) of we clearly have that , and hence,

 sup[0,T]||u(t)||Hr≤CT.

In the regime , we take in (24), which by the above remark clearly yields the desired bound .

When , we will construct an appropriate which will be used throughout the remainder of the proof when making the analysis in this non-smooth regime.
Let , and let

 σ0=d2+ϵ2. (25)

For , we have that , and hence satisfies

 d2<σ0

Consequently, by recalling the definition (6) of , we have that when . Moreover, in the regime we have the better bound .
Hence, in the regime , since we obtain the desired bound by taking in (24).

We now estimate the second term in the remainder (21). From the explicit expression of the commutator (20) and by making use of the nonlinear estimate (11) we have for all ,

 ||C[f,iΔ](u,v,w)||Hr ≤Cr(||∇w⋅∇u||Hr+|||∇u|2v||Hr+2||u∇u⋅∇v||Hr) (27) ≤Cr(||∇w||Hσ||∇u||Hr+||v||Hσ||∇u||Hσ||∇u||Hr+2||u||Hσ||∇u||Hσ||∇v||Hr) ≤Cr(||u||r+1,||v||r+1,||u||σ+1,||v||σ+1,||w||σ+1).

In the regime , we have and by taking in the above expression it follows that,

 ||C[f,iΔ](u,v,w)||Hr≤Cr(||u||r+1,||v||r+1,||w||r+1)≤Cr(||u||r1,||v||r1,||w||r1).

When , we take in (27), where is defined at (25). Using the fact that and when , we obtain

 ||C[f,iΔ](u,v,w)||Hr≤Cr(||u||r+1,||v||r+1,||u||σ0+1,||v||σ0+1,||w||σ0+1)≤Cr(|