Superconvergence of Strang splitting for NLS in $T^d$

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Superconvergence of Strang splitting for NLS in T

Philippe Chartier, Florian Méhats, Mechthild Thalhammer, Yong Zhang

To cite this version:

Philippe Chartier, Florian Méhats, Mechthild Thalhammer, Yong Zhang. Superconvergence of Strang splitting for NLS inT^d. 2014. �hal-00956714�

Superconvergence of Strang splitting for NLS in T

Philippe Chartier¹, Florian M´ehats², Mechthild Thalhammer³, and Yong Zhang⁴ March 7, 2014

Abstract

In this paper we investigate the convergence properties of semi-discretized approx- imations by Strang splitting method applied to fast-oscillating nonlinear Schr¨odinger equations. In a first step and for further use, we briefly adapt a known convergence result for Strang method in the context of NLS on T^d for a large class of nonlinearities. In a second step, we examine how errors depend on the length of the periodε, the solutions being considered on intervals of fixed length (independent of the period). Our main con- tribution is to show that Strang splitting with constant step-sizes is unexpectedly more accurate by a factorεas compared to established results when the step-size is chosen as an integer fraction of the period, owing to an averaging effect.

Keywords: highly-oscillatory systems, averaging, partial differential equation, Schr¨odinger equation, nonlinear, Strang splitting, splitting schemes.

MSC numbers: 34K33, 37L05, 35Q55.

1 Introduction

In this work, we analyze the convergence ofStrang splittingmethod for nonlinear Schr¨odinger equations of the form

i∂_tu^ε=−1

ε∆u^ε+f(|u^ε|²)u^ε, u^ε(0) =u₀ ∈H^σ(T^d), t≤T, (1) where ∆ is the Laplacian operator acting on the Hilbert space H^σ of functions on T^d = [0,2π]^d with derivatives up to order σ and f is a C^∞-function from R into itself such that f(0) = 0.¹ The operator ∆ being self-adjoint, Stone’s theorem asserts that it generates a strongly continuous one-parameter unitary group e^iτ∆ which, here, is in addition 1-periodic

1INRIA Rennes, IRMAR and ENS Rennes, IPSO Project Team, Campus de Beaulieu, F-35042 Rennes, France. E-mail: Philippe.Chartier@inria.fr

2IRMAR, University of Rennes and INRIA Rennes, IPSO Project Team, Campus de Beaulieu, F-35042 Rennes, France. E-mail: Florian.Mehats@univ-rennes1.fr

3Leopold Franzens Universitt Innsbruck, Institut f¨ur Mathematik, Technikerstraye 13/VII, 6020 Innsbruck, Austria. E-mail: Mechthild.Thalhammer@uibk.ac.at

4Wolfgang Pauli Institute, Nordbergstrasse 15, 1090 Vienna, Austria, E-mail:: sunny5zhang@gmail.com (Yong Zhang)

1The assumptionT^d= [0,2π]^dcan be relaxed, provided the corresponding spectrum of ∆ remains a subset ofωNfor someω >0.

in the variable τ =t/ε, so that on a fixed interval of time, the number of oscillations tend to +∞asεgoes to zero, thus making the problem highly-oscillatory. It is customary for the ease of analysis to reparametrize the time and consider the evolution in the variable τ. In this language (but denoting again byt the new variableτ), the equation reads

i∂tu^ε =−∆u^ε+εf(|u^ε|²)u^ε, u^ε(0) =u0 ∈H^σ(T^d), t≤T /ε, (2) the period of the group e^it∆ is 1, and the equation must be solved on the interval [0, T /ε]

(where interesting dynamics of the solutions appear). Of course, the number of oscillations

⌊T /ε⌋ over the interval of resolution still goes to ∞ for vanishing ε. Albeit the occurrence of ε in front of the nonlinearity, we insist that (2) is not a perturbation problem, owing to the length of the interval of time which scales like 1/ε. In the sequel, we give convergence results for semi-discretized numerical schemes applied to (2), keeping in mind that they are straightforwardly transferred to the original format (as stated in the Abstract). Accordingly, we assume in the whole paper the following:

Hypothesis (H) There exist T > 0 and ε₀ > 0, such that, for any 0 < ε < ε₀, equa- tion (2) has a unique solution u^ε(t,·) ∈ C¹([0, T /ε];H^σ(T^d)) with σ > d/2 + 2 and σ ≥ 4.

Furthermore, there exists a positive constant K such that

∀0< ε < ε₀, ∀0≤t≤T /ε, ku^ε(t)k_H^σ ≤Kku₀k_H^σ. (3) For convenience, we shall denote in the sequel,R= 2Kku₀k_H^σ and

B_ρ^s={u∈H^s,kuk_H^s ≤ρ},

so that under previous hypothesis, the solution u^ε(t) ≡ u^ε(t,·) remains in B_R/2^σ for all t ∈ [0, T /ε]. Note that in our context, the validity of this assumption can be rigorously established (see for instance [3]).

In this paper, we study the approximation properties of a semi-discretization in time.

The numerical scheme that we consider is Strang splitting method, yielding approximations Φ^h(u) =ϕ^h/2_T ◦ϕ^h_V ◦ϕ^h/2_T (u),

whereh is the step size andϕ^h_T and ϕ^h_V are the exact flows of respectively

i∂_tu=−∆u and i∂_tu=εf(|u|²)u. (4) By Stone’s theorem and the gauge invariance, they can be written for all u ∈ H^s and all h∈R as

ϕ^h_T(u) =e^ih∆u and ϕ^h_V(u) =e^{−ihεf(|u|2)}u.

Note that both maps are isometries of L², that is, one has for allu∈L²,kϕ^h_T(u)k_L² =kuk_L² and kϕ^h_V(u)k_L2 =kuk_L2. It is perfectly clear that the first map is also an isometry in H^s for all 0≤s≤σ.

In agreement with the results of [9] derived for NLS in R³ and f(u) = u, the sequence of approximations (Φ^h)ⁿ(u₀) are expected to converge on [0, T /ε] with second-order error estimates of the form

k(Φ^h)ⁿ(u₀)−u(t_n)k_H^σ−4 ≤Consth²

fort_n=nh≤T /εand sufficiently smallh. Note that hereσ anddare arbitrary with the only constraint thatσ > d/2 + 2 and that the convergence is uniform in ε. The aim of this paper is to show that, somehow unexpectedly at first glance, this error estimate can be refined to obtain

k(Φ^h)ⁿ(u0)−u(tn)k_H^σ−2m ≤Const (εh²+h^m)

where 2≤ m≤ ⌊σ/2⌋ depends on the smoothness of the initial value u₀, provided the step size h is chosen in such a way that 1/his an integer.² The precise formulation of this result is given in Theorem 5.5.

The paper is organized more or less along the following inclusions:

B_R/2^σ

| {z }

exact solution



⊂ B_R^σ ⊂B_R^σ−2

| {z }

f unctional bounds



⊂ B_3R/4^σ−2

| {z }

h¹−error, H^s−stability

⊂ H| {z }^σ−4

h²−error

⊂ H| {z }^σ−2m

h²ε−error

In Section 2, we consider the two functions u 7→ F(u) = −if(|u|²)u and (τ, u) 7→ F_τ(u) = e^−iτ∆F e^iτ∆u

which are essential ingredients of further proofs and derive bounds of their derivatives with respect to both u and τ. In addition, we give a Lipschitz estimate for F, which lies at the core of stability estimates derived in Section 3. In the spirit of [9] again, stability holds in H^s-norms for 0 ≤ s ≤ σ−2 provided the numerical solutions remain in H^σ−2. Bounds on derivatives of F and F_τ are then used in Section 4 to analyze the local truncation errors. In Section 5, we first briefly (re)-derive the (uniform in ε) convergence results of [9] (though without resorting to Lie-derivatives) and then analyze in great details the accumulation of errors in the case where 1/h∈ N. The final result is then obtained in two steps, by first studying the errors over one period (see Theorem 5.4) where an averaging effect essentially kills the main error term and then extending the so-obtained estimates to the whole interval (see Theorem 5.5) . Finally, Section 7 presents numerical experiments confirming the occurrence of an additional ε-factor in error bounds. A few more technical results are exposed in Appendix.

2 Bounds on F , F

and their derivatives

The embeddings of H^σ and H^σ−2 into L^∞ implies that H^σ and H^σ−2 are algebras: there exists a constant A >1 such that fors=σ and s=σ−2, one has

∀(u, v)∈H^s×H^s, kuvk_H^s ≤Akuk_H^skvk_H^s.

2Such step sizes are said to be resonant and can lead to exponential error growth (see Weideman and Herbst [12]). However, this possible very long-time instability does not contradict the convergence results given. Instabilities are indeed typically observed on intervals of lengthT /ε² in this scaling.

In this context, anyC^∞-functionGfrom Cto itself, such thatG(0) = 0, satisfies a so-called tame estimate (see [1]): there exists a non-decreasing function χ_G from R⁺ into R⁺, such that fors=σ and s=σ−2, one has

∀u∈H^s, kG(u)k_H^s ≤χ_G(kuk_L^∞)kuk_H^s ≤χ_G(ckuk_H^s)kuk_H^s, (5) where we have used in the second part of the inequalities the continuous embedding ofH^σ−2 and H^σ into L^∞ and denoted c the corresponding constant k · k_L^∞ ≤ck · k_H^σ−2 ≤ck · k_H^σ. More generally, when G(0) is not assumed to vanish, we may consider ˜G(u) = G(u)−G(0) and assert that there exists again a non-decreasing functionχG from R⁺ into R⁺, such that fors=σ and s=σ−2, one has

∀u∈H^s, kG(u)k_H^s ≤(2π)^d/2|G(0)|+χ_G(kuk_L^∞)kuk_H^s. (6) In the sequel, the following functions will play an important role:

F : H^σ → H^σ

u 7→ −if(|u|²)u and

F_τ : [0,1]×H^σ → H^σ

(τ, u) 7→ e^−iτ∆F(e^iτ∆u).

Let us notice that F ≡F₀, so that we will mostly concentrate on estimates for F_τ.

A Lipschitz inequality for F_τ. Regarding F as a function of H^σ−2 into itself, it is not difficult to show that it a Lipschitz continuous function. Owing to Lemma 7.3 in Appendix, it is also a Lipschitz continuous function inH^s-norm for all 0≤s≤σ−2, providedu and v lie in B_R^σ−2. We have indeed

kF(u)−F(v)k_H^s = kf(|u|²)u−f(|u|²)v+f(|u|²)v−f(|v|²)vk_H^s

= kf(|u|²)(u−v)k_H^s +k(f(|u|²)−f(|v|²))vk_H^s

≤ κkf(|u|²)k_H^σ−2ku−vk_H^s +κkvk_H^σ−2kf(|u|²)−f(|v|²)k_H^s

≤ κ Aχ_f(c²kuk²_Hσ−2)kuk²_Hσ−2ku−vk_H^s +αkvk_H^σ−2α(f, AR²)k|u|²− |v|²k_H^s

≤ κ Aχ_f(c²R²)R²ku−vk_H^s+κ R α(f, AR²)kuu¯−u¯v+u¯v−v¯vk_H^s

≤ κ Aχ_f(c²R²)R²ku−vk_H^s+ 2κ²R²α(f, AR²)ku−vk_H^s

≤ Lku−vk_H^s

with³ L=κ R²(Aχ_f(c²R²) + 2κ α(f, AR²)). The map e^ih∆ being an isometry of H^s for all s, it is clear thatFτ is also Lipschitz continuous with the same constantL.

Derivatives with respect to u. Let s = σ or s = σ−2. Function F_τ is C^∞ w.r.t.

u fromH^s into itself, and we have for instance

F_τ^′(u)(v) =e^−iτ∆F^′(e^iτ∆u)(e^iτ∆v)

3Constantsκandα(f, AR²) are defined in Lemma 7.3 in Appendix.

where F^′(u)(v) =−if(|u|²)v−if^′(|u|²)(¯uv+ ¯vu)u. From (6) for f and f^′ and the identity ke^iτ∆uk_H^s =kuk_H^s, we get the estimates for allτ ∈[0,1], all u∈B_R^s and all v∈H^s

kF_τ(u)k_H^s ≤A²χ_f(c²kuk²_Hs)kuk³_Hs ≤M₀ and

kF_τ^′(u)(v)k_H^s ≤ A³kf^′(|e^iτ∆u|²)k_H^skuk²_Hskvk_H^s+Akf(|e^iτ∆u|²)k_H^skvk_H^s

≤ M₁kvk_H^s

for some constantsM₀ and M₁. The second derivative of F takes the form

F^′′(u)(v, w) =−if^′′(|u|²)u(¯uv+ ¯vu)(¯uw+ ¯wu)−2if^′(|u|²)(¯uvw+ ¯vuw+ ¯wuv), so that by similar arguments, we obtain

∀u∈B_R^s,∀(v, w)∈H^s×H^s, kF_τ^′′(u)(v, w)k_H^s ≤M₂kvk_H^skwk_H^s.

Note that the constants M₀,M₁ and M₂ depend on f and R. These estimates can be easily generalized to higher derivatives F_τ^(l),l= 3, . . . ,⌊σ/2⌋ for some constants M_l.

Derivatives with respect to τ. The first derivative of F_τ w.r.t. τ can be computed as follows:

dF_τ(u)

dτ =−ie^−iτ∆∆F(e^iτ∆u) +e^−iτ∆F^′(e^iτ∆u)(ie^iτ∆∆u) =−i∆F_τ(u) +F_τ^′(u)(i∆u).

Note that here,iF_τ^′(u)(∆u)−F_τ^′(u)(i∆u)6= 0, so that the function u7→ ^dF_dτ^τ(u) is a function from H^σ toH^σ−2. The H^σ−2-norm of Fτ(u) for u∈B_R^σ can be estimated as follows

dFτ(u)

dτ

H^σ−2 ≤ k∆Fτ(u)k_H^σ−2 +kF_τ^′(u)(i∆u)k_H^σ−2 ≤M0+M1R.

More generally, thej-th derivative w.r.t. τ reads⁴ d^jF_τ(u)

dτ^j = Xj

l=0

j l

(−i)^j−l∆^j−lF_τ^(l)(u)(i∆u)^l

and is a function fromH^σ intoH^σ−2j, provided σ≥2j. ItsH^σ−2j-norm can be bounded as follows

d^jF_τ(u) dτ^j

H^σ−2j ≤ Xj

l=0

j l

kF_τ^(l)(u)(i∆u)^lk_Hσ−2l

≤ kF_τ(u)k_H^σ + Xj

l=1

j l

kF_τ^(l)(u)(i∆u)^lk_Hσ−2

≤ (1 +R)^j max

l=0,...,jM_l.

4We sometimes use the notationF^(l)(u)(w^j) as a shorthand forF^(l)(u) (w, . . . , w)

| {z }

jtimes .

The bounds of local truncation errors will finally require to estimate d

dτF_τ^′(u)(v) =−i∆F_τ^′(u)(v) +F_τ^′(u)(i∆v) +F_τ^′′(u)(v, i∆u).

Foru and v inB^σ_R, the following bound stems directly from previous inequalities

dF_τ^′(u) dτ

H^σ−2 ≤ 2M1R+M2R².

H^σ-H^s inequality for derivatives of F_τ. Proceeding as for F_τ, it is straightforward to show that derivatives of Fτ w.r.t. to u satisfy a Lipschitz estimate on B_R^σ: there exists a constant L>0 such that, for all 0≤2j≤σ,

∀w∈B^σ−2_R , ∀(u, v)∈B_R^σ ×B_R^σ, kF^(j)(u)(w^j)−F^(j)(v)(w^j)k_H^σ−2 ≤ Lku−vk_H^σ−2. It then follows that

∀(u, v)∈B_R^σ ×B_R^σ, d^jF_τ(u)

dτ^j −d^jF_τ(v) dτ^j

H^σ−2j ≤ L₂ku−vk_H^σ

for some constant L₂. We finally collect in next proposition the findings of this Section.

Proposition 2.1. Let m = ⌊σ/2⌋. For s ∈ {σ, σ −2}, the function (τ, u) 7→ F_τ(u) = e^−iτ∆F(e^iτ∆u) is a well-defined function from[0,1]×H^s intoH^s, isC^∞ w.r.t. u, and for all 0≤j≤m, it isj-times differentiable with values inH^σ−2j. Moreover, there exists a positive constant M such that for all τ ∈[0,1], all u∈B_R^s and all (v₁, v₂)∈(H^s)² one has

kFτ(u)kH^s ≤M, kF_τ^′(u)(v1)kH^s ≤Mkv1kH^s, kF_τ^′′(u)(v1, v2)kH^s ≤Mkv1kH^skv2kH^s, (7) and for all τ ∈[0,1] and all (u, v) ∈(B_R^σ)², one has

d^jF_τ(u) dτ^j

H^σ−2j ≤M and dF_τ^′(u) dτ

H^σ−2 ≤M. (8) Finally, there exist positive constants L and L₂ such that for all τ ∈[0,1]

∀0≤s≤σ−2, ∀(u, v)∈B_R^σ−2×B_R^σ−2, kF_τ(u)−F_τ(v)k_H^s ≤Lku−vk_H^s, (9) and for all 0≤j≤m, all τ ∈[0,1]

∀(u, v)∈B_R^σ ×B_R^σ, d^jF_τ(u)

dτ^j −d^jF_τ(v) dτ^j

H^σ−2j ≤ L₂ku−vk_H^σ. (10) The constants M, L and L₂ depend on f, R andσ.

3 Stability estimates

In order to estimate local errors in a neighborhood of the exact solution, we need to ensure that the numerical solutions lie in B_R^σ−2. Besides, stability estimates in various norms are required in the study of the error accumulation. These points are addressed in next lemma.

Lemma 3.1. Consider the map u7→A^h₁(u) := Φ^h(u)−e^ih∆u and let h₀ = ^log(4/3)_Lε

0 . If v and ware two functions of B_3R/4^σ−2, then for all 0≤h < h0, Φ^h(v) andΦ^h(w)are inB_R^σ−2 and the following statements hold for all 0≤s≤σ−2

kΦ^h(v)−Φ^h(w)kH^s ≤e^εLhkv−wkH^s and kA^h₁(v)−A^h₁(w)kH^s ≤εhe^εLhkv−wkH^s. Proof. Denoting v(t) =ϕ^t_V(v) and w(t) =ϕ^t_V(w), we have

˙

w(t) =εF(w(t)) and ˙v(t) =εF(v(t)) so that

kv(t)−w(t)k_H^s ≤ kv−wk_H^s+ε Z t

0

F(v(τ))−F(w(τ))

H^sdτ.

As long as w(t) and v(t) remain in B^σ−2_R , we have, according to (9)

F(v(τ))−F(w(τ))

H^s ≤ Lkv(τ)−w(τ)k_H^s so that, by Gronwall lemma

kv(t)−w(t)k_H^s ≤e^εLtkv−wk_H^s.

In particular, takingw= 0, we havekv(h)k_H^σ−2 ≤ ^3R₄ e^εLh≤R. Now, we immediately get kϕ^h_V(v)−v−ϕ^h_V(w) +wk_H^s ≤ε L h e^εLhkv−wk_H^s,

and both statements then follow from the fact thate^ih2∆ is an isometry of H^s.

4 Local truncation errors

Denoting t_n=nh≤T /ε andu^ε_n=u^ε(t_n), we define the local truncation errors as follows δⁿ(ε, h) = Φ^h(u^ε_n)−u^ε_n+1.

Comparing the Taylor expansions of u^ε(t_n+h) and Φ^h(u^ε_n) and using Proposition 2.1, we can now derive bounds forδⁿ(ε, h).

4.1 Expansion of the exact solution

The exact solution at timet+h can be written with the Duhamel formula as u^ε(t+h) =e^ih∆u^ε(t) +εe^ih∆

Z h

0

F_τ(e^−iτ∆u^ε(t+τ))dτ, or equivalently for v^ε(t) =e^−it∆u^ε(t)

v^ε(t+h) =v^ε(t) +ε Z h

0

F_t+τ(v^ε(t+τ))dτ. (11) We then use the following expansion of Fτ(˜u+ ˜v)

F_τ(˜u+ ˜v) =F_τ(˜u) +F_τ^′(˜u)(˜v) + Z ₁

0

(1−ξ)F_τ^′′(˜u+ξv)(˜˜ v²)dξ.

Inserting the expression ofv^ε(t+h) into the right-hand side of (11) and denotingv=v^ε(t) for the sake of brevity, we obtain

v^ε(t+h) = v+ε Z h

0

Ft+τ

v+ε

Z τ

0

Ft+τ1(v^ε(t+τ1))dτ1

dτ

= v+ε Z h

0

F_t+τ(v)dτ+ε² Z h

0

Z τ

0

F_t+τ^′ (v)F_t+τ1(v^ε(t+τ₁))dτ₁dτ + ε³

Z _h

0

Z ₁

0

(1−ξ)F_t+τ^′′

(1−ξ)v+ξv^ε(t+τ)

dξ Z ^τ

0

F_t+τ1(v^ε(t+τ₁))dτ₁2

dτ.

The second-order term in εcan be further expanded as Z h

0

Z τ

0

F_t+τ^′ (v)F_t+τ1(v)dτ₁dτ +ε

Z _h

0

Z _τ

0

F_t+τ^′ (v) Z ₁

0

F_t+τ^′ ₁

(1−ξ)v+ξv^ε(t+τ₁)

dξ Z ^τ1

0

F_t+τ2(v^ε(t+τ₂))dτ₂ dτ₁dτ.

Finally, we have with u=u^ε(t) u^ε(t+h) =e^ih∆u+εe^ih∆

Z _h

0

F_τ(u)dτ+ε²e^ih∆

Z _h

0

Z _τ

0

F_τ^′(u)F_τ1(u)dτ₁dτ+ε³e^ih∆E₃(u, ε, h), withE₃(u, ε, h) =E_3,a(u, ε, h) +E_3,b(u, ε, h) where

E_3,a(u, ε, h) = Z h

0

Z 1

0

(1−ξ)F_τ^′′

(1−ξ)u+ξe^−iτ∆u^ε(t+τ)

dξ Z ^τ

0

F_τ1(e^−iτ1∆u^ε(t+τ₁))dτ₁2

dτ,

E_3,b(u, ε, h) = Z _h

0

Z _τ

0

F_τ^′(u) Z ₁

0

F_τ^′₁

(1−ξ)u+ξe^−iτ1∆u^ε(t+τ₁)

dξ Z ^τ1

0

F_τ2(e^−iτ2∆u^ε(t+τ₂))dτ₂ dτ₁dτ.

Since u^ε(t) remains in B_R/2^σ ⊂B_R^σ ⊂B_R^σ−2, we have by Proposition 2.1 kE₃(u^ε(t), ε, h)k_Hσ−2 ≤ 1

3M³h³. (12)

4.2 Expansion of numerical solutions

Assume here that u∈B_3R/4^σ−2 and 0< h≤h₀ where h₀ is defined in Lemma 3.1. Using that d

dhϕ^h_V(u) =εF(ϕ^h_V(u)) we get immediately

ϕ^h_V(u) = u+εhF(u) +1

2(εh)²F^′(u)F(u) + ε³

Z _h

0

(h−τ)² 2

F^′′(ϕ^τ_V(u))(F(ϕ^τ_V(u)))²+F^′(ϕ^τ_V(u))F^′(ϕ^τ_V(u))F(ϕ^τ_V(u)) dτ, which we may write as

ϕ^h_V(u) = u+εhF(u) + 1

2(εh)²F^′(u)F(u) +ε³E3,V(u, ε, h) (13) with

E_3,V(u, ε, h) = Z _h

0

(h−τ)² 2

F^′′(ϕ^τ_V(u))(F(ϕ^τ_V(u)))²+F^′(ϕ^τ_V(u))F^′(ϕ^τ_V(u))F(ϕ^τ_V(u)) dτ.

The Strang splitting solution can then be expanded as Φ^h(u) = e^ih/2∆

e^ih/2∆u+εhF(e^ih/2∆u) +1

2(εh)²F^′(e^ih/2∆u)F(e^ih/2∆u) +ε³E_3,V(e^ih/2∆u, ε, h)

= e^ih∆

u+εhF_h/2(u) + 1

2(εh)²F_h/2^′ (u)F_h/2(u)

+ε³e^ih/2∆E_3,V(e^ih/2∆u, ε, h). (14) Note that owing to Lemma 3.1, ϕ^τ_V(u) remains in B_R^σ−2 for 0 ≤ τ ≤ h so that, by using Proposition 2.1,

∀u∈B_3R/4^σ−2, ∀0< h≤h₀, kE_3,V(u, ε, h)k_H^σ−2 ≤ M³

3 h³. (15)

4.3 Bounds of local truncation errors The local truncation error finally reads

δⁿ(ε, h) = εe^ih∆

hF_h/2(u^ε_n)− Z _h

0

Fτ(u^ε_n)dτ + ε²e^ih∆1

2h²F_h/2^′ (u^ε_n)F_h/2(u^ε_n)− Z h

0

Z τ

0

F_τ^′(u^ε_n)F_τ1(u^ε_n)dτ₁dτ + ε³

e^ih/2∆E_3,V(e^ih/2∆u^ε_n, ε, h)−e^ih∆E₃(u^ε_n, ε, h)

(16) and we have to estimate each term individually.

First-order term in ε. We use the second-order Peano kernelκ2 of the midpoint rule hF_h/2(u)−

Z h

0

Fτ(u)dτ =h³ Z 1

0

κ2(τ) d² dθ²F_θ(u)

θ=τ h

dτ

whereκ₂ is a scalar continuous function, so that, owing to (8) (note that σ≥4)

hF_h/2(u)− Z h

0

F_τ(u)dτ

H^σ−4 ≤M Z ¹

0

|κ₂(τ)|dτ

h³. (17)

Second-order term in ε. We here use that F_τ1(u) =F_h/2(u) +

Z _τ1

h/2

d

dθF_θ(u)dθ and F_τ^′(u) =F_h/2^′ (u) + Z _τ

h/2

d

dθF_θ^′(u)dθ and insert these expressions into the double integral term to get

Z _h

0

Z _τ

0

F_τ^′(u)F_τ1(u)dτ₁dτ = h²

2 F_h/2^′ (u)F_h/2(u) +r₁ where

r₁ = Z _h

0

Z _τ

0

F_h/2^′ (u) Z _τ1

h/2

d

dθF_θ(u)dθdτ₁dτ+ Z _h

0

Z _τ

0

Z _τ

h/2

d

dθF_θ^′(u)dθF_h/2(u)dτ₁dτ +

Z h

0

Z τ

0

Z τ

h/2

d

dθF_θ(u)dθ Z τ1

h/2

d

dθF_θ(u)dθdτ₁dτ so that

kr1k_H^σ−2 ≤ 1

4M²h³+ 1

32M²h⁴ ≤ 1

4M²h³(1 +h0/8).

Third-order term in ε. Collecting previous estimates of this section and estimates (15) and (12), we obtain

∃C >0, ∀0< h≤h₀, kδⁿ(ε, h)k_Hσ−4 ≤Cεh³. (18) We end up this section by noticing that the following bound also holds (using the first-order Peano kernel)

∃C >˜ 0, ∀0< h≤h₀, kδⁿ(ε, h)k_H^σ−2 ≤Cεh˜ ², (19) and for later use in our refined analysis of errors, we observe that

δⁿ(ε, h) = εe^ih∆Λ_h(u^ε_n) +ε²R_h(u^ε_n) (20) with

Λ_h(u^ε_n) =hF_h/2(u^ε_n)− Z h

0

F_τ(u^ε_n)dτ, (21)

and

∀0< h≤h₀, kΛ_h(u^ε_n)k_H^σ−4 ≤Ch³ and kR_h(u^ε_n)k_H^σ−2 ≤Ch³.

5 Global error estimates

The basic ingredient of the proofs of this section is the following telescopic identity (Φ^h)ⁿ(u₀)−u^ε_n=

Xn

l=1

(Φ^h)^n−l◦Φ^h(u^ε_l−1)−(Φ^h)^n−l(u^ε_l) .

We proceed in two steps: in the first one, we obtain ε-independent error estimates on the whole interval [0, T /ε] (in agreement with [9]) and in the second one we use these estimates in a more refined analysis, on one period and then on the whole interval.

5.1 H^σ−4-convergence

The following theorem is the formulation in our context of a result from [9].

Theorem 5.1. Leth₁ = min(h₀, ^RL

4 ˜C(e^LT−1))whereh₀is defined in Lemma 3.1. The numerical solution given by the Strang splitting scheme for equation (2) with 0 ≤ h ≤ h₁ satisfies a second-order error bound in H^σ−4

∃C >0, k(Φ^h)ⁿ(u₀)−u^ε(t_n)k_H^σ−4 ≤Ce^LT −1

L h² for t_n=nh≤T /ε. (22) The constants L and C depend on σ,R and f, but are independent of ε.

Proof. As long as the numerical approximations (Φ^h)^j(u^ε(t_k)) remain inB_3R/4^σ−2 for 1≤j+k≤ n, stability estimates of Lemma 3.1 hold and the telescopic identity leads to

k(Φ^h)ⁿ(u₀)−u^ε(t_n)k_H^σ−2 ≤ Xn

l=1

e^εLh(n−l)kδ^l−1(ε, h)k_H^σ−2

wherekδ^l−1(ε, h)k_H^σ−2 can be bounded by εCh˜ ² (see (19)). Hence, since nh≤T /ε, we have straightforwardly

k(Φ^h)ⁿ(u₀)−u^ε(t_n)k_H^σ−2 ≤C˜ e^nεLh−1

L h≤C˜ e^LT −1 L h.

The boundedness in H^σ−2 required by the stability lemma is ensured by induction by the previous error bound provided ˜C^eLT_L⁻¹ h ≤ ^R₄. Now, Lemma 3.1 applies and gives stability inH^σ−4, so that owing to estimate (18), we get the second-order convergence inH^σ−4.

5.2 H^σ−2m-convergence on one period

We now examine more closely the errors after one period. Let us first notice that, according to Theorem 5.1, (Φ^h)^l∈B_3R/4^σ−2 for all 0≤l≤n withnh≤T /ε and hence (Φ^h)^l ∈B_3R/4^s for all 0≤s≤σ−2.

Lemma 5.2. The following estimate holds

∀0≤t≤1, ku^ε(t)−e^it∆u₀k_H^σ ≤εM. (23) Proof. The termsu^ε(t) ande^it∆u₀ are the exact solutions of the differential equations

˙

u^ε =i∆u^ε+εF(u^ε), u^ε(0) =u₀ and v˙^ε=i∆v^ε, v^ε(0) =u₀. By Duhamel formula and Proposition 2.1, we have immediately

ku^ε(t)−v^ε(t)k_H^σ =ε Z t

0

e^i(t−τ)∆F(u^ε(τ))dτ

H^σ ≤εtM ≤εM.

Lemma 5.3. Given u and v, assume that both sequences (Φ^h)^l(u)

l and (Φ^h)^l(v)

l lie in B_3R/4^σ−2 for all0≤l≤nwithnh≤1and0< h≤h₁ whereh₁is defined in Theorem 5.1. Then the map u7→A^h_l(u) := (Φ^h)^l(u)−e^ilh∆u satisfies a Lipschitz condition for all 0≤s≤σ−2 of the form

kA^h_l(u)−A^h_l(v)k_H^s ≤εLe^εLku−vk_H^s. Proof. A telescopic identity gives

(Φ^h)^l(u)−e^ilh∆u = Xl

k=1

e^ih(l−k)∆

Φ^h−e^ih∆

◦

(Φ^h)^(k−1)(u) .

Hence,

kA^h_l(u)−A^h_l(v)k_H^s ≤ Xl

k=1

A^h₁((Φ^h)^(k−1)(u)−A^h₁((Φ^h)^(k−1)(v))

H^s

and according to Lemma 3.1 , we have kA^l_h(u)−A^l_h(v)kH^s ≤

Xl

k=1

εLhe^Lεh(Φ^h)^(k−1)(u))−(Φ^h)^(k−1)(v)

H^s

≤ Xl

k=1

εLhe^kLεhku−vk_H^s ≤εLe^εLku−vk_H^s.

Theorem 5.4. Let m=⌊σ/2⌋. The numerical solution given by the Strang splitting scheme for equation (2) with step size0< h≤h₁ where h₁ is defined in Theorem 5.1, has a second- order error bound in H^σ−2m of the form

∃C >ˆ 0, k(Φ^h)ⁿ(u0)−u^ε(1)k_H^σ−2m ≤C(εˆ ²h²+εh^m) for nh= 1. (24) The constant Cˆ depends on σ, R and f, but is independent of ε.

Proof. The proof proceeds in two steps.

Identification of the ε-error term. Replacing (Φ^h)^(n−l) by e^ih(n−l)∆ in the telescopic identity we get

(Φ^h)ⁿ(u₀)−u^ε(1) = Xn

l=1

e^ih(n−l)∆δ^l−1(ε, h) +r where

r = Xn

l=1

A^h_n−l

Φ^h(u^ε_l−1)

−A^h_n−l(u^ε_l) .

According to previous lemma and using (18), we can estimate r as follows:

krk_H^σ−4 ≤εLe^εL Xn

l=1

kδ^l−1(ε, h)k_H^σ−4 ≤ε²Le^εLCh². In addition, according to (20), we have

Xn

l=1

e^ih(n−l)∆δ^l−1(ε, h) =ε Xn

l=1

e^{ih(n−l+1)∆}Λ_h(u^ε_l−1) + ˜r where

˜ r=ε²

Xn

l=1

e^ih(n−l)∆R_h(u^ε_l−1)

can again be bounded byCε²h² inH^σ−2-norm. Finally, taking into account that (see Lemma 5.2)

ku^ε_l−1−e^i(l−1)h∆u0kH^σ ≤M ε

and that Λ_h is Lipschitz continuous with a Lipschitz constant of the form h³L˜₂ (see (21-17) and Proposition 2.1) we have

(Φ^h)ⁿ(u₀)−u^ε(1)−ε Xn

l=1

e^{i(n−l+1)h∆}Λ_h(e^i(l−1)h∆u₀)

H^σ−4 ≤ Const ε²h².

Estimate of the ε-error term. From previous analysis, the main error is concentrated in the term

Xn

l=1

e^{i(n−l+1)h∆}Λ_h(e^i(l−1)h∆u₀)

which is of order εh². For a finer estimation, we now proceed as follows:

Xn

l=1

e^{i(n−l+1)h∆}Λ_h(e^i(l−1)h∆u₀) =

n−1X

l=0

hF_(lh+h/2)(u₀)− Z ₁

0

F_τ(u₀)dτ

=

n−1X

l=0

hF_(lh+h/2)(u₀)− Z 1

0

F_(τ+h/2)(u₀)dτ

= e^−ih/2∆E_Rie(e^ih/2∆u₀, h)

where ERie denotes the error in the approximation by Riemann sums according to Lemma 7.1 and where we have right away taken into account thate^inh∆is the identity operator and thatτ 7→F_τ(u₀) is 1-periodic. From Lemma 7.1, we thus have

Xn

l=1

e^{i(n−l+1)h∆}Λ_h(e^i(l−1)h∆u0)

H^σ−2m ≤C_m^Rie

d^j dτ^jFτ

H^σ−2m

h^m

withC_m^Rie= 2_(2π)^ζ(m)m. Together with estimate (8) of Proposition 2.1, this completes the proof.

5.3 H^σ−2m-convergence on the whole interval

It is now straightforward to get error estimates on the whole interval by considering ˆΦ(u) = (Φ^h)ⁿ(u) as an integrator with step size 1, for which the “local error” is of size ˆCε(εh²+h^m) and which is obviously Lipschitz continuous. A telescopic identity with ˆΦ iteratedN =⌊T /ε⌋

times then leads to an error of size ˆCT(εh²+h^m). Finally, the solution at intermediate points (i.e. within an interval [n, n+ 1] or [N, T /ε]) can then be obtained by composing ˆΦ and Φ^h. Theorem 5.5. Letm=⌊σ/2)⌋. The numerical solution given by the Strang splitting scheme for equation (2) with step size h > 0 such that 1/h ∈ N has a second-order error bound in H^σ−2m of the form

∃C >ˆ 0, k(Φ^h)ⁿ(u₀)−u^ε(1)k_H^σ−2m ≤CTˆ (εh²+h^m) for nh≤T /ε. (25) The constant Cˆ depends on σ, R and f, but is independent of ε.

6 Extension to general splitting methods

Assume now that

Φ^h(u) =e^ihα1∆◦ϕ^β_V^1h◦ · · · ◦e^ihαr∆◦ϕ^β_V^rh, is a general splitting method of orderp. More precisely, suppose that

δⁿ(ε, h) = Φ^h(u^ε_n)−u^ε_n+1

can be written as in (20)

δⁿ(ε, h) =εe^ih∆Λ_h(u^ε_n) +ε²R_h(u^ε_n) with

kΛ_h(u^ε_n)k_H^σ−2p ≤Ch^p+1 and kR_h(u^ε_n)k_H^σ−2p+2 ≤Ch^p+1.

Then, it is possible to identify through an ε-expansion the term Λ_h. One has indeed Λ_h(u) =

Xr

j=1

β_jhe^{ih(γj−1)∆}F

e^{ih(1−γj)∆}u

− Z _h

0

F_τ(u)dτ,

where γ_j = Pj

k=1α_k and γ_r = 1 by a first-order condition on the splitting method. By consistency again,Ps

j=1βj = 1, and one has Λ_h(u) =

Xr

j=1

β_j

hF_(γj−1)h(u)− Z _h

0

F_τ(u)

so that Xn

l=1

e^{i(n−l+1)h∆}Λ_h(e^i(l−1)h∆u₀) = Xr

j=1

β_j

n−1X

l=0

hF_(lh+γjh)(u₀)− Z 1

0

F_τ(u₀)dτ

!

= Xr

j=1

β_j

e^−iγjh∆E_Rie(e^iγjh∆u₀, h)

Higher-order conditions on the splitting method then lead to the following expression Λ_h(u) =h^p+1

Z ₁

0

κ_p(τ) d^pF_θ dθ^p

θ=τ h

(u)dτ

so that Λ_h is Lipschitz continuous with a Lipschitz constant of the ˜L₂h^p+1. The proof derived for Strang splitting can then be readily adapted leading to the estimate

∃C >ˆ 0, k(Φ^h)ⁿ(u₀)−u^ε(1)k_H^σ−2m ≤CTˆ (εh^p+h^m) for nh≤T /ε, under the additional assumption that σ≥2p.

7 Numerical experiments

In this section, we present numerical results of Strang and another fourth-order splitting methods when applied to NLS equation with initial conditions as follows:

i∂_tu^ε=−∂_xxu^ε+ε2 cos(2x)|u^ε|²u^ε, 0≤t≤T /ε, x∈T_2π (26) u^ε(0, x) =u0(x) = cos(x) + sin(x), x∈T_2π. (27)