Strong Convergence Rate of Finite Difference Approximations for Stochastic Cubic Schrödinger Equations

# Strong Convergence Rate of Finite Difference Approximations for Stochastic Cubic Schrödinger Equations

## Abstract

In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schrödinger equations driven by a multiplicative -Wiener process. Beyond the uniform boundedness of moments for high order derivatives of the exact solution, the key requirement of our approach is the exponential integrability of both the exact and numerical solutions. By constructing and analyzing a Lyapunov functional and its discrete correspondence, we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution, which immediately yields the well-posedness of both the continuous and discrete problems. The latter exponential integrability is obtained through a variant of a criterion given by [Cox, Hutzenthaler and Jentzen, arXiv:1309.5595]. As a by-product of this exponential integrability, we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation-type result on the dependence of the noise with first order strong convergence rate.

###### keywords:
stochastic cubic Schrödinger equation, strong convergence rate, central difference scheme, exponential integrability, continuous dependence
###### Msc:
[2010] 60H35, 60H15, 60G05
12

## 1 Introduction and main idea

There is a general theory on strong error estimations for stochastic partial differential equations (SPDEs) with Lipschitz coefficients (see e.g. ACLW16 (); BJK16 (); CHL16 (); CHL15a () and references therein), where one usually adopts the semigroup or equivalent Green’s function framework. For SPDEs with non-Lipschitz but monotone coefficients, one can use the variational framework to derive strong convergence rates of numerical approximations (see e.g. KLL15 ()). Unfortunately, the monotonicity assumption is too restrictive in the sense that the coefficients of the majority of nonlinear SPDEs from applications, including stochastic Navier-Stokes equations, stochastic Burgers equations, Cahn-Hilliard-Cook equations and stochastic nonlinear Schrödinger equations etc., do not satisfy the monotonicity assumption. Since these SPDEs can not be solved explicitly, one needs to develop effective numerical techniques to study them. Depending on the particular physical model, it may be necessary to design numerical schemes for solutions of the underlying SPDEs with strong convergence rates, i.e., rates in for some . We point out that strong convergence rates are particularly important for efficient multilevel Monte Carlo methods (see e.g. BFRS16 (); Keb05 ()).

Our main purpose in this paper is to derive the strong convergence rate of a representative, spatial approximation for the one-dimensional stochastic cubic Schrödinger equation

 idu+(Δu+λ|u|2u)dt =u∘dW(t)in(0,T]×O

with the initial datum under homogenous Dirichlet boundary condition. Here or corresponds to focusing or defocusing cases, is a fixed real number, and is a -Wiener process on a stochastic basis , i.e., there exists an real-valued, orthonormal basis of and a sequence of mutually independent, real-valued Brownian motions such that for . For convenience, we always consider the equivalent Itô equation

 du=(iΔu+iλ|u|2u−12FQu)dt−iudW(t)in(0,T]×O (1)

with the initial datum , where .

The stochastic cubic Schrödinger equation (1) has been studied in e.g. DDM02 (); RGYBC95 () to motivate the possible role of noise to prevent or delay collapse formation. Many authors concern numerical approximations for Eq. (1); see e.g. CHP16 (); BD04 (); DD06 () and references therein. To deal with the nonlinearity, one usually apply the truncation technique. However, this method only produces convergence rates in certain sense such as in probability or pathwise which is weaker than strong sense. In this paper, by deducing the uniform boundedness for moments of high order derivatives of the exact solution and exponential moments of both the exact and numerical solutions, we obtain the strong error estimate of the central difference scheme (see (6) below). To the best of our knowledge, this is the first result deriving strong convergence rates of numerical approximations for Eq. (1). For other types of SPDEs, we are only aware that Dor12 () analyzes the spectral Galerkin approximation for the two-dimensional stochastic Navier-Stokes equation using a uniform bound for exponential moments of the solution derived in HM06 (), and that HJ14 () studies spectral Galerkin approximations for the one-dimensional Cahn-Hillard-Cook equation and stochastic Burgers equation through exponential integrability characterized in CHJ13 ().

Before proposing our main idea, we introduce some frequently used notations and related technical analytic tools.

### 1.1 Notations and analytic tools

1. We denote for a given . Let be the uniform partition of the interval with the step size . For a grid function , we denote for simplicity. We use and to denote the forward difference operator and backward difference operator, respectively, i.e.,

 δ+f(l):=f(l+1)−f(l)h,δ−f(l):=f(l)−f(l−1)h.
2. Denote by and the continuous and discrete Hilbert spaces with inner products

 ⟨f,g⟩L2:=R[∫O¯¯¯f(x)g(x)dx],⟨f,g⟩h:=N+1∑l=0R[¯¯¯f(l)g(l)]h.

We also use the discrete - and -spaces with norms

 ∥f∥l4h:=(N+1∑l=0|f(l)|4h)14,∥f∥l∞h:=supl∈ZN+1|f(l)|.
3. Denote by the space of Hilbert-Schmidt operators from to another separable Hilbert space , endowed with the norm

In particular, is denoted simply by . Here and what follows, is an positive integer and denotes the usual Sobolev space , which consists of functions such that exist and are square integrable for all , with the additional boundary condtion on for .

4. To bound the -norm, we need the Gagliardo-Nirenberg inequality

 ∥f∥2L∞ ≤2∥f∥L2⋅∥∇f∥L2,f∈H1 (2)

and its discrete correspondence

 ∥fh∥2l∞h ≤2∥f∥h∥δ+f∥h,f∈l2h, f(0)=f(N+1)=0, δ+f∈l2h. (3)

We also use the Sobolev embedding :

 ∥f∥L∞≤C0∥f∥H1,f∈H1 (4)

for some constant and the fact that is an algebra, i.e., for any , there exists a constant such that

 ∥fg∥Hs≤C∥f∥Hs∥g∥Hs. (5)

Throughout the paper is a generic constant, independent of the discretization parameter , which will be different from line to line.

### 1.2 Main idea

Our main aim is to derive the convergence of the spatial central difference scheme

 duh(l) =(iδ+δ−uh(l)+iλ|uh(l)|2uh(l)−12FQ(l)uh(l))dt−iuh(l)dW(t,l) (6)

towards Eq. (1) with an algebraic rate in strong sense. Define for . The initial datum of Eq. (6) is the grid function . It is clear that Eq. (1) and Eq. (6) possess the continuous and discrete charge conservation laws, respectively, i.e., for all it holds a.s. that

 ∥u(t)∥2L2=∥u0∥2L2,∥uh(t)∥2h=∥uh0∥2h. (7)

The exact solution of Eq. (1), at the grid points, satisfies

 du(l) =(iδ+δ−u(l)+iRh(l)+iλ|u(l)|2u(l)−12u(l)FQ(l))dt−iu(l)dW(t,l).

Denote by the difference between the exact solution and the numerical solution defined at the grid points , . Applying Itô formula to the functional , using the continuous and discrete Gagliardo-Nirenberg inequalities (2)–(3) and charge conservation laws (7) (more details see Theorem 4.1), we obtain

 ∥ϵh(t)∥2h≤∫t0∥Rh∥2l∞hdr+∫t0(1+2∥u0∥L2∥∇u∥L2+2∥uh0∥h∥δ+uh∥h)∥ϵh∥2hdr. (8)

Taking expectation, as in the deterministic case, leads to

 E[∥ϵh(t)∥2h]≤∫t0E[∥Rh∥2l∞h]dr +∫t0E[(1+2∥u0∥L2∥∇u∥L2+2∥uh0∥h∥δ+uh∥h)∥ϵh∥2h]dr.

Due to the appearance of the nonlinear term in the last integral above, the classical Gronwall inequality is not available and one could not derive a strong convergence rate. These difficulties are common features to obtain strong convergence rates for numerical approximations appearing in many situations, see e.g. CHP16 (); DD06 () for stochastic nonlinear Schrödinger equations and BJ13 (); KLM11 () for other SPDEs with non-monotone coefficients.

Our main idea is applying Gronwall-Bellman inequality to (8) before taking expectation. Then using Hölder and Minkowski inequalities, we have

 (E[supt∈[0,T]∥ϵh(t)∥2h])12 ≤C(E[supt∈[0,T]∥Rh∥4l∞h])14 ∥∥∥exp(∫T0∥u0∥L2∥∇u∥L2dr)∥∥∥L8(Ω) ∥∥∥exp(∫T0∥uh0∥h∥δ+uh∥hdr)∥∥∥L8(Ω). (9)

In order to obtain the strong convergence rate for our scheme (6), we need to estimate the three terms appearing on the right hand side of (1.2). The first expectation produces the strong convergence rate with a multiple given by -moments of the solution under -norm, which is proved to be uniformly bounded in Theorem 2.1 and Corollary 1. To control the last two exponential moments with the random initial datum , we apply a variant of a criterion given by CHJ13 () on exponential integrability of a Hilbert-valued stochastic process which is the strong solution of a stochastic differential equation in Hilbert space (see Lemma 3.1 and Theorem 3.1).

Meanwhile, we derive the continuous dependence, in with or , for the solution of the stochastic cubic Schrödinger equation (1) on both the initial data and the noises with explicit rates (see Corollaries 3 and 4). Similar continuous dependence on the initial data for the numerical solution of the central difference scheme (6) can also be obtained. Such continuous dependence property is not a trivial property for the solutions of SPDEs with non-Lipschitz coefficients. We also illustrate that this continuous dependence property is very crucial, besides for theoretical analysis such as in (DZ14, , Chapter 9.1), for numerical computation because there exist round-off errors in computer simulations.

The rest of the paper is organized as follows. In Section 2, we bound the -moments for high order derivatives of the exact solution and discrete first derivative of the numerical solution. The uniform bound on exponential moments of energy functionals of solutions as well as continuous dependence on initial data and noises is proved in Section 3. The results in Section 2 and Section 3 are used in Section 4 to derive the strong convergence rate of the central difference scheme (6).

## 2 Well-posedness and regularity

In this section, we first prove the moments’ uniform boundedness for the solution of the stochastic cubic Schrödinger equation (1) by analyzing the Lyapunov functional defined by (10), which is necessary for proving the global well-posedness of Eq. (1). This uniform boundedness is also useful to derive a strong convergence rate of the central difference scheme (6). Then we show, with the help of the discrete energy functional defined by (24), a priori estimate and thus the well-posedness of the discrete equation (6).

### 2.1 A priori estimation of the exact solution

For or , it is known that the stochastic cubic Schrödinger equation (1) possesses a unique mild solution which is in a.s. under some assumptions on and (see BD03 () for , and CHP16 () for in the defocusing case). Our main result in this part is a priori estimation of in -norm with integer for both focusing and defocusing cases. This will enables us to bound the term appearing in (1.2). We remark that our arguments can also be applied to the whole line.

Throughout this section, we assume that the initial datum is -measurable and belongs to a.s. for certain . To control the nonlinear term in Eq. (1), we introduce the Lyapunov functional

 (10)

By the inequality (5) and integration by parts, we have

 ∣∣∣E[f(u0)]∣∣∣ (11)

Simple calculations yield that the first and the second order derivatives of are

 Df(u)(v) (12) D2f(u)(v,w) −2λ⟨(−Δ)s−1w,uR[¯¯¯uv]⟩L2−2λ⟨(−Δ)s−1u,uR[¯¯¯vw]⟩L2 −2λ⟨(−Δ)s−1u,vR[¯¯¯uw]⟩L2−λ⟨(−Δ)s−1w,|u|2v⟩L2 −2λ⟨(−Δ)s−1v,uR[¯¯¯uw]⟩L2−λ⟨(−Δ)s−1v,|u|2w⟩L2, (13)

where . Applying the Itô formula to the functional defined by (10), we get

 f(u(t))−f(u0) =∫t0Df(u)(iΔu+iλ|u|2u−12FQu)dr +12∫t0tr[D2f(u)(−iuQ12)(−iuQ12)∗]dr +∫t0Df(u)(−iu)dW(r)=:I1(t)+I2(t)+I3(t). (14)

Substituting the expressions (12) and (13) of and into and , respectively, we obtain

 I1(t) =2∫t0⟨∇su,∇s(iΔu+iλ|u|2u−12FQu)⟩L2dr −λ∫t0⟨(−Δ)s−1u,u(i¯¯¯uΔu−iuΔ¯¯¯u−|u|2FQ)⟩L2dr −λ∫t0⟨(−Δ)s−1u,|u|2(iΔu+iλ|u|2u−12FQu)⟩L2dr −λ∫t0⟨(−Δ)s−1(iΔu+iλ|u|2u−12FQu),|u|2u⟩L2dr,

and

 I2(t) −2λ∫t0tr[((−Δ)s−1¯¯¯u)uR[(−iuQ12)∗⊗(−iuQ12)]]dr

Our main result in this subsection is the following a priori estimation of algebraic moments for high order derivatives of the solution of Eq. (1).

###### Theorem 2.1

Let or and . Assume that

 u0∈s⋂m=2L3s−mp(Ω;Hm)∩1⋂m=0L3s−m−15p(Ω;Hm) (15)

and . Then there exists a constant such that

 supt∈[0,T]E[∥u(t)∥pHs]≤C. (16)

Proof Let . We first estimate in (2.1). Denote the four integrals in successively by , , and . Integration by parts yields that and can be rewritten as

 I11(t)=I111(t)+I112(t),I12(t)=I121(t)+I122(t),

where

 I111(t) =−2∫t0⟨(−Δ)s−1u,iλΔ(|u|2u)⟩L2dr, I112(t) =−∫t0⟨∇su,∇s(FQu)⟩L2dr, I121(t) I122(t) =λ∫t0⟨(−Δ)s−1u,|u|2FQu⟩L2dr.

The Cauchy-Schwarz inequality and the inequality (5) yield that

 ∣∣∣E[I112(t)]∣∣∣+∣∣∣E[I122(t)]∣∣∣ ≤C(∫t0E[∥u(r)∥4Hs−1]dr+∫t0E[|u(r)|2Hs]dr).

The term is divided into two equal parts which can balance and . More precisely, inserting the identities and , we obtain

 I111(t)+I121(t)+I14(t) =(I111(t)2+I121(t))+(I111(t)2+I14(t)) =−λ∫t0⟨(−Δ)s−1u,3i|u|2Δu⟩L2dr −λ∫t0⟨(−Δ)s−1u,4i|∇u|2u+2i(∇u)2¯¯¯u⟩L2dr −λ∫t0⟨(−Δ)s−1(|u|2u),−12FQu⟩L2dr=:Ia(t)+Ib(t)+Ic(t).

Applying integration by parts, Leibniz formula and the fact that , we have

 ⟨(−Δ)s−1u,i|u|2Δu⟩L2 =⟨∇su,i∇s−2(|u|2Δu)⟩L2

Then by the Sobolev embedding (4) and the inequality (5) we get for ,

 ∣∣⟨(−Δ)s−1u,i|u|2Δu⟩L2∣∣≤C|u|Hss−3∑j=0∥∇s−2−j|u|2∥L∞∥u∥Hj+2≤C|u|Hs∥u∥3Hs−1.

The above estimate is also valid for , since . This implies that

 ∣∣∣E[Ia(t)]∣∣∣≤C(∫t0E[∥u(r)∥6Hs−1]dr+∫t0E[|u(r)|2Hs]dr).

Applying Hölder inequality, integration by parts and the inequality (5), we obtain for ,

 ∣∣∣E[Ib(t)]∣∣∣ ≤C(∫t0E[∥u(r)∥6Hs−1]dr+∫t0E[|u(r)|2Hs]dr).

When , by using the Sobolev embedding (4), the Gagliardo-Nirenberg inequality (2) and the Young inequality, we get

 ∣∣∣E[Ib(t)]∣∣∣ ≤∫t0E[|u|H2∥u∥L∞∥∇u∥L∞∥∇u∥L2]dr ≤C(∫t0E[∥u(r)∥10Hs−1]dr+∫t0E[|u(r)|2Hs]dr).

Similar arguments imply that

 ∣∣∣E[Ic(t)]∣∣∣

As a result, there exists a constant such that

 ∣∣∣E[I111(t)+I121(t)+I14(t)]∣∣∣ ≤C(1+∫t0E[∥u(r)∥10Hs−1]dr+∫t0E[|u(r)|2Hs]dr).

For , using integration by parts and the inequality (5), we have

 ∣∣∣E[I13(t)]∣∣∣ ≤C(E[|Ia(t)|]+∫t0E[∣∣∣⟨(−Δ)s−1u,i|u|4u+12FQ|u|2u⟩L2∣∣∣]dr) ≤C(1+∫t0E[∥u(r)∥6Hs−1]dr+∫t0E[|u(r)|2Hs]dr).

Combining the estimations of to , we have

 ∣∣∣E[I1(t)]∣∣∣ ≤C(1+∫t0E[∥u(r)∥6Hs−1]dr+∫t0E[|u(r)|2Hs]dr) (17)

for and

 ∣∣∣E[I1(t)]∣∣∣ ≤C(1+∫t0E[∥u(r)∥10H1]dr+∫t0E[|u(r)|2H2]dr). (18)

Now we turn to the estimations of and in (2.1). Using the Cauchy-Schwarz inequality and the inequality (5), we get

 ∣∣∣E[I2(t)]∣∣∣ ≤C(1+∫t0E[∥u(r)∥6Hs−1]dr+∫t0E[|u(r)|2Hs]dr). (19)

On the other hand, owing to the property of Itô integral, we have

 ∣∣E[I3(t)]∣∣ =0. (20)

For , we apply the Itô formula to and obtain

 fp2(u(t)) =fp2(u0)+p2∫t0fp2−1(u)Df(u)(iΔu+iλ|u|2u−12FQu)dr +p(p−2)8∫t0tr[fp2−2(u)Df(u)(−iuQ12)Df(u)(−iuQ12)∗]dr +p2∫t0fp2−1(u)Df(u)(−iu)dW(r). (21)

It follows from the inequality (5) and the Cauchy-Schwarz inequality that

 fp2−1(u)≤C(∥u∥2(p−2)Hs−1+|u|p−2Hs).

By the estimations (17) and (18) of , it holds a.s. that

 ∣∣Df(u)(iΔu+iλ|u|2u−12FQu)∣∣ ≤C(∥u∥6Hs−1+|u|2Hs)

for and

 ∣∣Df(u)(iΔu+iλ|u|2u−12FQu)∣∣ ≤C(∥u∥10H1+|u|2H2).

Then by the Young inequality, we get an estimate for the first integral in (2.1):

 E[∣∣∣∫t0fp2−1(u)Df(u)(iΔu+iλ|u|2u−12FQu)dr∣∣∣] ≤C∫t0(E[∥u∥2p+2Hs−1]+E[∥u∥2p−4Hs−1|u|2Hs]+E[∥u∥6Hs−1|u|p−2Hs]+E[|u|pHs])dr

for and

 E[∣∣∣∫t0fp2−1(u)Df(u)(iΔu+iλ|u|2u−12FQu)dr∣∣∣] ≤C(1+∫t0E[∥u(r)∥5pH1]dr+∫t0E[|u(r)|pH2]dr).

Similar arguments can be applied to other terms in (2.1). Thus we obtain

 E[|u(t)|pHs] ≤C(E[fp2(u(t))]+E[∥u∥2pHs−1]) ≤C(1+E[∥u0∥pHs]+E[∥u0∥2pHs−1] +∫t0E[∥u(r)∥3pHs−1]dr+∫T0E[∥u(r)∥pHs]dr)

for and

 E[|u(t)|pH2] ≤C(1+E[∥u0∥pH2]+E[∥u0∥2pH1]

Gronwall inequality then yields that

 E[|u(t)|pHs] ≤C(1+E[∥u0∥pHs]+E[∥u0∥2pHs−1]+∫T0E[∥u(r)∥3pHs−1]dr)

for and that

 E[|u(t)|pH2] ≤C(1+E[∥u0∥pH2]+E[∥u0∥2pH1]+∫T0E[∥u(r)∥5pH1]dr).

Similar arguments as in (BD03, , Theorems 4.1 and 4.6) lead to

 supt∈[0,T]E[∥u(t)∥5pH1]<∞

provided that . This implies that

 supt∈[0,T]E[∥u(t)∥pH2]<∞

provided that . For , when , it holds that

 E[|u(t)|pH3] ≤C(1+