Convergence of an Embedded Exponential-Type Low-Regularity Integrators for the KdV Equation without Loss of Regularity

Yongsheng Li, Yifei Wuand Fangyan Yao

1 School of Mathematical Sciences, South China University of Technology, Guangzhou, Guangdong 510640, China

2 Center for Applied Mathematics, Tianjin University, Tianjin 300072,China

Abstract. In this paper, we study the convergence rate of an Embedded exponential-type low-regularity integrator (ELRI) for the Korteweg-de Vries equation. We develop some new harmonic analysis techniques to handle the“stability” issue. In particular, we use a new stability estimate which allows us to avoid the use of the fractional Leibniz inequality,

Key words: The KdV equation,numerical solution,convergence analysis,error estimate,low regularity, fast Fourier transform.

1 Introduction

The Korteweg-de Vries (KdV) equation arises as a model equation from the weakly nonlinear long waves and describes the propagation of shallow water waves in a channel [23]. It has taken a wide range of applications in a diverse field of the industries, especially in terms of application and technology. In this paper, we consider the KdV equation with periodic boundary conditions

where T=(0,2π), u=u(t,x):R×T→R is the unknown and u∈H(T) with some 0≤s<∞is a given initial data.

Many authors have studied the initial value problem of the KdV equation both on the real line and in the period case, and established the global well-posedness in Hfor s ≥?1; see [4,18,20]. The numerical solution of the KdV equation has been important in a wide range of fields. One interesting question in the numerical solution of the KdV equation is how much regularity is required in order to have certain desired convergence rates. Correspondingly, many numerical methods and numerical analysis were developed to address this question,including finite difference methods [5,17,21,37], finite element methods [1,6,38], operator splitting [14–16,36], spectral methods [7,28,29,35], discontinuous Galerkin methods [26,42] and exponential integrators [2,11,12].

Among the many numerical time integration methods for time-dependent partial differential equations (PDEs), the splitting methods are very popular in many classic studies. We refer the readers to [8,13,30] for an extensive overview of splitting methods. As far as we know, operator splitting methods for the KdV equation (often referred to as fractional-step methods) first appeared in [36] and were analysed rigorously in[15]. Operator splitting methods have been developed into a systematic approach for constructing time-stepping methods for evolutionary PDEs. In particular, Holden et al. [14,16] proved that the Godunov and Strang splitting methods for the KdV equation converge with the first-order and the second-order rates in Hwith γ ≥1, if the initial datum belong to Hand H, respectively. For the nonlinear Schr¨odinger equation (NLS), Lubich [27] proved that for the initial data in H, the Strang splitting scheme provides the first-order and the secondorder convergence in Hand L, respectively. In addition to the splitting method,exponential integrators is also a very effective numerical method for solving partial differential equations including hyperbolic and parabolic problems [9,10]. In particular, Hochbruck and Ostermann [11] presented some typical applications that illustrate the computational benefits of exponential integrators.

These classical numerical methods greatly promote the development of numerical computation. However, it needs a large enough regularization requirement to achieve optimal convergence. Thus more and more attention has been paid recently to so-called low-regularity integrators (LRIs) that are based on the exponential integrators. This novel method has been used to obtain numerical solutions of many kinds of equations and obtained relatively ideal results. For the cubic nonlinear Schr¨odinger equation,the first-order convergence rate in Hhave been achieved under H-data by Ostermann and Schratz [33]. The second-order convergence rate in His proved under H-data in one dimension and H-data in high dimensions [22], respectively. More recently, Wu and Yao [39] proposed a new scheme which provides the first-order accuracy without loss of regularity in one dimensional case, that is, the first-order convergence in H(T) for H(T)-data. Moreover, the algorithm in [39] almost preserves the mass of the numerical solution. Further, Li and Wu [25] constructed a fully discrete low-regularity integrator which has firstorder convergence (up to a logarithmic factor) in L(T) in both time and space for H(T)initial data. For the KdV equation,Hofmanov′a and Schratz[12]proposed an exponential-type integrator and proved the first-order convergence in Hfor initial data in H. Then based on a classical Lawson-type exponential integrator, Ostermann and Su [34] proposed a Fourier pseudospectral method to prove first-order convergence in both space and time under a mild Courant-Friedrichs-Lewy condition. Based on the scheme that was proposed in[12],Wu and Zhao[40]obtained the second-order convergence result in Hfor initial data in Hfor the KdV equation.

Very recently, Wu and Zhao [41] further improved these results and established first-order and second-order convergence in Hunder H-data and H-data respectively by introducing the Embedded exponential-type low-regularity integrators(ELRIs). That is, for any γ>,

with u=u, where

Here P is the orthogonal projection onto mean zero functions

The purpose of this investigation is to obtain the fractional order convergence without any derivative loss based on the same low-regularity integrator. More precisely,we are aiming to show that for any γ>,

This is regarded as the local error estimate. It is derived from some bilinear estimates based on the ingenious harmonic analysis, see Lemma 2.3 below.

Second, the handling of “stability” issues. As in the standard way, one shall prove the stability estimate:

To prove(1.7),it reduces to show another type of the fractional Leibniz inequality involving on the linear flow:

Thanks to the presence of the linear flow ex, the inequality can be obtained from the smooth effect.

Applying (1.8), the stable part can be controlled by

with other easy treated terms. Then we use the convergence result(1.2)to establish the desired estimate (1.7).

Now,we state the convergence theorem of the presented(semi-discretized) ELRI method given in (1.3).

This research study fills a gap in the literature that optimal fractional order convergence without any derivative loss.

The paper is organized as follows. In Section 2,we give some notations and some useful lemmas. In Section 3, we present the local error and stability estimates and prove Theorem 1.1.

2 Preliminary

2.1 Some notations

Sometimes, the subscript Z is omitted. The Fourier transform of a function f on T is defined by

and thus the Fourier inversion formula is

Then the following usual properties of the Fourier transform hold:

The Sobolev space H(T) for s≥0 has the equivalent norm,

where we denote the operator

For simplicity, we denote

Then if ξ=ξ+···+ξ, we have that

For convenience, in the following, we shall assume the zero-mode/average of the initial value of(1.1)is zero,that is, ?u(0)=0. Otherwise, we may consider to replace u with

2.2 Some preliminary estimates

First,we will frequently apply the following Kato-Ponce inequality(simple version),which was originally proved in [19] and an important progress in the endpoint case was made in [3,24] very recently.

Lemma 2.1 (Kato-Ponce inequality). The following inequalities hold:

Based on the above inequalities, we can derive two lemmas as follows, which have been proved in [40,41].

Lemma 2.2. The following inequalities hold:

Lemma 2.3. The following inequalities hold:

Next we present a lemma that plays an important role in the stability estimate.

Proof. By Plancherel’s identity, we get derictly

Note that the right term of the above inequality is equal to

Note that

Therefore it gives that

Since |ξ|≥2|ξ|, we have that |ξ|～|ξ|. The right term of the above inequality is controlled by

and for simplicity, we denote A(f)=A(f,f,f) below.

Then we have the following estimate.

Lemma 2.5. Let γ>1, and f,f,f∈H, then for any 0≤t≤τ,

By symmetry, without loss of generality, we assume that |ξ|≥|ξ|≥|ξ|, then

We get the desired result.

3 Convergence result

3.1 The low-regularity integrator

By introducing the twisted variable v:=exu and the Duhamel’s formula at t=nτ with τ>0 the time step:

As presented in [41], we rewrite vas

where Fis defined as

and Ais defined in (2.6). Let

3.2 Local error

The main result in this subsection is

Hence, by Lemma 2.3(ii), we have

From (3.1), we write

From the above estimates, we conclude that

From the definition of Fin (3.3), the following equality holds:

Hence, by Lemma 2.1, we have

Put (3.5) into the above inequality to get

Together with (3.8)–(3.10), we prove the lemma.

3.3 Stability

The main result in this subsection is

Proof. Note that

where

For short, we denote f=v?v(t), then

Hence we get

For (3.11), by Lemma 2.2(ii), we have

Then put the above inequality into (3.14) to yield that

Combining with the above estimates, we conclude that

This gets the desired result.

3.4 Proof of Theorem 1.1

Now, combining the local error estimate and the stability result, we give the proof of Theorem 1.1. From Lemma 3.1 and Lemma 3.2, there exits a constant C>0,such that for 0<τ ≤1, we have

which proves Theorem 1.1.