Two novel linear-implicit momentum-conserving schemes for the fractional Korteweg-de Vries equation

Abstract

We propose two conservative linear-implicit schemes for the space fractional Korteweg-de Vries (fKdV) equation. One is the linear-implicit Crank–Nicolson scheme and the other is the linear-implicit leap-frog scheme. In order to obtain a high order discretization in the space direction, we adopt the Fourier pseudospectral method. The Crank–Nicolson scheme and leap-frog scheme are used for temporal discretization, and those two schemes are efficient in practical computations because of their linear property. Furthermore, we analyse the uniqueness, boundness, convergence of the two schemes. Numerical experiments are presented to validate the theoretical analysis.

Introduction

The Korteweg–de Vries (KdV) equations are commonly used to describe the ion acoustic wave phenomena in plasma dynamics and the evolution in time of long, unidirectional, nonlinear shallow water waves. One of the important applications of the KdV equation is the study on solitonic behaviour of the acoustic waves on a crystal lattice. Washimi and Taniuti [1] pioneered to establish the theory that the KdV equation governs the propagation of small-amplitude ion-accoustic waves.

During the last few decades, fractional equations have attracted much attention in fractional quantum mechanics and condensed matter physics [2], [3], [4]. For the fKdV equations, Linares et al. [5] proved the (conditional) orbital stability of solitary waves solutions in the L² subcritical case. Durán [6] constructed a numerical method which can improve the convergence rate and the computation performance for limiting values of the fractional parameter. Other studies on the Cauchy problem, the existence of solitary wave solutions, the stability properties of the ground states can be found in Refs. [5], [7]

Besides the fKdV equation, other fractional equations have also been studied numeriously. Wang et al. [8] developed Crank–Nicolson difference method to solve the coupled fractional nonlinear Schrödinger equations. In [9], [10], some linear energy conservation schemes were proposed to overcome the nonlinear difficulty for the fractional Schrödinger equations. Wang and Xiao [11], [12] constructed two schemes which preserve mass and energy conservation laws by using the Crank–Nicolson/leap-frog methods in the time direction to solve the fractional KGS equation. Duo and Zhang [13] developed three Fourier spectral schemes which satisfy mass conservation, including the split-step Fourier spectral, the Crank–Nicolson Fourier spectral, and the relaxation Fourier spectral methods to solve the fractional nonlinear Schrödinger equation.

In literatures, the ability to preserve some physical and mathematical properties of the original differential equations is a criterion to judge the effectivity of a numerical simulation. L.Vázquez explained the reason why it is physically necessary to develop methods to preserve physical properties in [14]. It is remarkable that “structure preservation” not only refers to the conservation of physical quantities including energy, momentum, mass and etc, but also refers to the capability of preserving mathematical features such as the solutions’ positivity [15], monotonicity [16] and boundedness [17]. It is worth noting that some researchers have done a lot of work on structure-preserving algorithms in recent years. For traditional KdV equations, there are so many investigations on structure-preserving schemes. Wang et al. [18] investigated multi-symplectic Euler Box Scheme for the KdV equation. The stochastic multi-symplectic methods were also investigated for the conservation in [19]. Zhang et al. [20] applied the average vector field method to solve KdV equation. Hamiltonian Boundary Value Method was used to construct a scheme which can preserve Hamiltonian in [21]. But there are few studies about the conservative schemes for the space fKdV equation especially about the momentum conservation. Therefore this motivates us to construct momentum-conserving schemes of the space fKdV equation and analyse the properties of the solution.

In this paper, we consider the space fKdV equation with the fractional Laplacian as follows

$$\begin{array}{c}\hfill \{\begin{array}{c}{u}_t-c_1{(-▵)}^{\frac{\alpha }{2}}{u}_x+c_{2}u{u}_x=0,x\in \Omega =[a,b],\hfill \\ u(x,0)=u_0\left(x\right),u(a,t)=u(b,t).\hfill \end{array}\end{array}$$

The following quantities: mass, momentum and energy are formally conserved [6] by the flow associated to (1):

$$\begin{array}{ccc}\hfill C\left(u\right)& =& {\int }_{\Omega }u(x,t)\mathrm{d}x,\hfill \\ \hfill M\left(u\right)& =& \frac{1}{2}{\int }_{\Omega }{u}^2(x,t)\mathrm{d}x,\hfill \\ \hfill E\left(u\right)& =& {\int }_{\Omega }\frac{c_1}{2}{\left({(-▵)}^{\frac{\alpha }{4}}u\right)}^2-\frac{c_2}{6}{u}^3\mathrm{d}x.\hfill \end{array}$$

We will discuss the case 1 < α ≤ 2. When $\alpha =2,$ the fKdV Eq. (1) reduces to the classical KdV equation.

In Eq. (1), the fractional Laplacian is defined as a pseudo-differential operator via the Fourier transform:

$$-{(-▵)}^{\frac{\alpha }{2}}u(x,t)=-\sum _{l\in Z}{\left|v_l\right|}^{\alpha }{\widehat{u}}_l{e}^{i{v}_l(x-a)},$$

where $v_l=\frac{2l\pi }{b-a}$ and the Fourier coefficient is defined by

$${\widehat{u}}_l=\frac{1}{b-a}{\int }_{\Omega }u(x,t)e^{-i{v}_l(x-a)}\mathrm{d}x.$$

We notice that most of the existing methods for solving KdV equations are nonlinear implicit. Since the KdV equations are nonlinear, we need to apply a nonlinear iteration technique. Conservative schemes require the nonlinear system to be solved to machine precision. However, the cost of solving the nonlinear system by iteration method is huge. And if the nonlinear scheme is not computed to reach high accuracy, the invariants of the equation may not be preserved. Motivated by the work [22], [23], we develop two linear-implicit schemes for the fKdV equation by applying the Fourier pseudospectral method in the space direction and the linear-implicit Crank–Nicolson scheme, the leap-frog scheme in the time direction. Firstly, those semi-discrete and fully-discrete schemes of Crank–Nicolson scheme, the leap-frog scheme are proved to preserve the momentum conservation law which implies a point-wise maximum bound on the numerical solution. Secondly, the two fully-discrete schemes are shown to be uniquely solvable. Thirdly, we prove that the fractional Laplacian discretized by Fourier pseudospectral approach can converge to machine precision regardless of its order. In addition, we prove that the two schemes are unconditionally convergent with the order of $O(N^{-r}+{\tau }^2)$.

The rest of this article is organized as follows. In Section 2, the definition and Lemmas are given and a semi-discrete scheme of the fKdV equation is introduced which not only preserves momentum conservation law but also admits a bounded solution. Besides, the semi-discrete scheme has spectral accuracy. We construct two linear-implicit schemes in Section 3. The momentum is conserved and there exists a unique solution for each scheme. To show the advantages and to verify the effectiveness of the proposed schemes, we perform some experiments in Section 4. Finally, some conclusions are made in Section 5.