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Conservative Fourier spectral scheme for the coupled Schrödinger–Boussinesq equations

Advances in Difference Equations20182018:405

https://doi.org/10.1186/s13662-018-1784-7

  • Received: 1 April 2018
  • Accepted: 31 August 2018
  • Published:

Abstract

In the paper, the conservative Fourier spectral scheme is presented for the coupled Schrödinger–Boussinesq equations. We apply the Fourier collocation scheme to spatial derivatives and the Crank–Nicolson scheme to the system in time direction, respectively. We find that the scheme can preserve mass and energy conservation laws. Moreover, the existence, uniqueness, stability and convergence of the scheme are discussed, and it is shown that the scheme is of the accuracy \(O(\tau^{2}+J^{-r})\). The numerical experiments are given to show that verify the correctness of theoretical results and the efficiency of the scheme.

Keywords

  • Schrödinger–Boussinesq equations
  • Conservative Fourier spectral method
  • Conservation laws
  • Convergence

1 Introduction

The coupled Schrödinger–Boussinesq (CSB) equations were given in the form [13]
$$\begin{aligned} &iu_{t}+u_{xx}-uv=0, \end{aligned}$$
(1)
$$\begin{aligned} &v_{tt}=v_{xx}-\alpha v_{xxxx}+f(v)_{xx}+ \omega \vert u \vert ^{2}_{xx}, \end{aligned}$$
(2)
where \(\alpha,\omega\) are constants, \(f(x)\) is a sufficiently smooth real function, \(u(x,t)\) represents the complex Schrödinger field, and \(v(x,t)\) represents the real Boussinesq field. Many researchers have devoted much energy to the study of the CSB system for a long time. As a result, many important properties such that existence and uniqueness of the solutions [4], exact solitary wave solutions [5, 6] and global attractors [7, 8] of some CSB systems have been discovered.

The nonlinear CSB equations are composed of a Schrödinger equation and a Boussinesq equation. Many scholars have given much numerical schemes to solve the Schrödinger equation. Numerical contributions for the good Boussinesq and nonlinear CSB equations have been topics of concern during the past decades. In [9], a quadratic B-spline finite element scheme was presented, and the corresponding results of the error estimates were obtained. In [10], a conservative multisymplectic scheme was presented and one simulated solitary waves for long times. In [11], Zhang obtained an optimal error estimate for an implicit conservative difference scheme with order \(O(\tau^{2} + h^{2})\). In [12], two conserved compact finite difference schemes were given, and the conservation property, and the existence, convergence, and stability of the difference schemes were theoretically analyzed.

As a class of high accuracy methods, spectral methods are often chosen to solve differential equations. In [13], the authors have given a Fourier pseudospectral method to solve the good Boussinesq equation with second-order temporal accuracy, and they have discussed the nonlinear stability and convergence of the scheme. In [14], the authors showed a second-order operator splitting numerical scheme and a Fourier pseudospectral scheme to solve the good Boussinesq equation, and they gave a stability and convergence analysis. In [15], the authors derived the fourth-order average vector field method and the Fourier pseudospectral method for the good Boussinesq equation. However, to the best of our knowledge, there was little attention paid to the spectral method for the CSB system. In [16], a time-splitting Fourier spectral method was presented, but there was no theoretical analysis available for the approximation error. In the paper, we construct a conservative Fourier spectral scheme to solve nonlinear CSB equations, and we give a rigorous numerical analysis of the scheme.

Let \(v_{t}=\phi_{xx}\), we consider the following initial-boundary value problem:
$$\begin{aligned} &iu_{t}+u_{xx}-uv=0,\quad x\in \Omega, 0< t\leq T, \end{aligned}$$
(3)
$$\begin{aligned} &v_{t}=\phi_{xx},\quad x\in\Omega, 0< t\leq T, \end{aligned}$$
(4)
$$\begin{aligned} &\phi_{t}=v-\alpha v_{xx}+f(v)+\omega \vert u \vert ^{2},\quad x \in\Omega, 0< t\leq T, \end{aligned}$$
(5)
$$\begin{aligned} &u(x,0)=u_{0}(x),\qquad v(x,0)=v_{0}(x),\qquad \phi(x,0)= \phi_{0}(x),\quad x\in\Omega, \end{aligned}$$
(6)
$$\begin{aligned} & u(a,t)=u(b,t),\qquad v(a,t)=v(b,t),\qquad \phi(a,t)=\phi(b,t), \end{aligned}$$
(7)
where \(\Omega=(a,b)\). The system (3)–(7) has the following mass and energy conservation laws:
$$\begin{aligned} &\operatorname{Mass}: \quad M(t)= \Vert u \Vert ^{2}=M(0), \\ &\operatorname{Energy}:\quad E(t)= \Vert v \Vert ^{2}+ \Vert \phi_{x} \Vert ^{2}+2\omega \Vert u_{x} \Vert ^{2}+\alpha \Vert v_{x} \Vert ^{2}+2 \bigl\langle F(v),1 \bigr\rangle +2\omega \bigl\langle \bigl\vert u^{2} \bigr\vert ,v \bigr\rangle =E(0), \end{aligned}$$
where \(F(v)>0\) is a primitive function of \(f(v)\).

It is well known that the conservative schemes perform better than the nonconservative ones. Thus, it is of interest to investigate conservative schemes for the CSB system. The Fourier pseudospectral method has attracted much attention in recent years due to its high accuracy and efficiency. In this paper, the Fourier pseudospectral method [1720] is utilized to discretize the CSB equations in space direction. In [14, 15], the authors employed the fourth-order average vector field method and the second-order operator splitting scheme, respectively, to solve the Boussinesq equation. However, the two method cannot satisfy the energy conservation law. Here, we apply the Crank–Nicolson method to solve the CSB equations in time direction. We find that the scheme can preserve mass and energy conservation laws simultaneously. Moreover, the existence, uniqueness and boundedness of the scheme are proved, and the convergence order is of \(O(\tau^{2}+J^{-r})\). Finally, we give numerical experiments to show the efficiency of the conservative scheme.

The remainder of this article is structured as follows. In Sect. 2, we give some useful lemmas and a conservative scheme, and we prove that the scheme preserve mass and energy conservation laws. Moreover, the existence, uniqueness, boundedness and convergence of the scheme are proved. In Sect. 3, we give the iterative algorithm of the scheme. In Sect. 4, numerical experiments are given, and the results verify the efficiency of the conservative schemes. Finally, a conclusion and some discussions are given in Sect. 5.

2 Conservative Fourier spectral scheme for the CSB system

2.1 Some useful lemmas

Let \(\tau=T/N, h =(b-a)/J\), and define
$$\begin{aligned} \Omega_{h}=\{x_{j}|0\leq j\leq J-1\},\qquad \Omega_{\tau}=\{t_{n}|0\leq n\leq N-1\}. \end{aligned}$$
Suppose \(w=\{w_{j}^{n}; j=0, 1, 2,\ldots, J, n=0, 1, 2,\ldots ,N\}\) be a discrete function, and define operators
$$\begin{aligned} & \bigl(w_{j}^{n} \bigr)_{x}= \frac{w_{j+1}^{n}-w_{j}^{n}}{h},\qquad w_{j}^{n+\frac{1}{2}}=\frac {w_{j}^{n+1}+w_{j}^{n}}{2},\qquad \bigl(w_{j}^{n} \bigr)_{t}=\frac{w_{j}^{n+1}-w_{j}^{n}}{\tau}. \end{aligned}$$
Let \(U_{j}^{n}, V_{j}^{n}, \Phi_{j}^{n}\) denote the numerical approximations to \(u(x_{j}, t_{n}), v(x_{j},t_{n}), \phi(x_{j}, t_{n})\), respectively. Denote
$$\begin{aligned} & \langle U,V\rangle=h\sum_{j=0}^{J-1} U_{j}\overline{V}_{j}, \qquad \Vert U \Vert ^{2}= \langle U,U \rangle, \qquad \vert U \vert ^{2}_{h,1}=\langle U_{x},U_{x} \rangle, \\ & \Vert U \Vert _{l_{h}^{p}}^{p}=h\sum _{j=0}^{J-1} \vert U_{j} \vert ^{p}, \quad 1\leq p< +\infty, \\ & \Vert U \Vert _{l_{h}^{\infty}}=\sup_{j\in Z} \vert U_{j} \vert . \end{aligned}$$
For \(\forall r> 0\), let \(H^{r}(R)= W^{r,2}(R)\) be Sobolev space. Define \(H_{p}^{r}(\Omega)\) as a subspace composed by periodic functions with period \(L=b-a\) on \(H^{r}(R)\), and
$$H_{p}^{r}(\Omega)= \bigl\{ u|u\in H^{r}(R), u(x+a)=u(x+b) \bigr\} . $$
Let equivalent norm and semi-norm of \(H_{p}^{r}(\Omega)\) be
$$\begin{aligned} & \Vert u \Vert _{r}= \Biggl[\sum _{l=-\infty}^{\infty} \bigl(1+ \vert l \vert ^{2} \bigr)^{r} \vert \hat{u}_{l} \vert ^{2} \Biggr]^{1/2}, \\ &\vert u \vert _{r}= \Biggl[\sum_{l=-\infty}^{\infty} \vert l \vert ^{2r} \vert \hat{u}_{l} \vert ^{2} \Biggr]^{1/2}, \end{aligned}$$
where
$$u(x)=\sum_{l=-\infty}^{\infty}\hat{u}_{l}e^{il\mu(x-a)}, \qquad \hat{u}_{l}=\frac {1}{b-a} \int_{\Omega}u(x)e^{-il\mu(x-a)}\,dx,\quad \mu= \frac{2\pi}{L}. $$
Denote the orthogonal projector \(P_{J}:L^{2}(\Omega)\rightarrow V_{J}\), where
$$\begin{aligned} V_{J}= \biggl\{ u(x)=\sum_{|k|\leq J/2} \hat{u}_{k}e^{ik\mu(x-a)} \biggr\} . \end{aligned}$$
We have following conclusions:
$$\begin{aligned} &P_{J}\partial_{x}u=\partial_{x} P_{J} u,\quad P_{J}u=u, \forall u\in V_{J}. \end{aligned}$$
Denote the interpolation operator \(I_{J}:L^{2}(\Omega)\rightarrow V_{J}''\) by
$$\begin{aligned} I_{J} u(x,t)=\sum_{j=0}^{J-1}u_{j}g_{j}(x), \end{aligned}$$
where
$$\begin{aligned} &V''_{J}= \biggl\{ u(x)=\sum _{|l|\leq J/2}\tilde{u}_{l}e^{il\mu(x-a)}, \tilde {u}_{J/2}=\tilde{u}_{-J/2} \biggr\} , \\ & \tilde{u}_{l}=\frac{1}{Jc_{l}}\sum_{j=0}^{J-1}u(x_{j})e^{-ik(x_{j}-a)}, \\ &g_{j}(x)=\frac{1}{J}\sum _{l=-\frac{J}{2}}^{\frac{J}{2}-1} \frac {1}{c_{l}}e^{il\mu(x-x_{j})},\quad c_{l}=1\ \biggl( \vert l \vert \neq \frac{J}{2} \biggr), c_{\frac{J}{2}}=c_{-\frac{J}{2}}=2. \end{aligned}$$
(8)
We have the following conclusions:
$$\begin{aligned} &I_{J}\partial_{x}u\neq\partial_{x} I_{J} u, \quad I_{J} u=u, \forall u\in V_{J}''. \end{aligned}$$
The values for the derivatives \(I_{J}u(x,t)\) at the collocation points \(x_{j}\) are obtained by [21]
$$\begin{aligned} &\frac{\partial^{k}I_{J}u(x_{i},t)}{\partial x^{k}}=\sum_{j=0}^{J-1}u_{j} \frac {d^{k}g_{j}(x_{i})}{dx^{k}}=(D_{k}u)_{i}, \\ &(D_{k})_{j,n}=\frac{d^{k}g_{n}(x_{j})}{dx^{k}}, \end{aligned}$$
(9)
where \(D_{k}\) represents Fourier spectral differential matrix.

Lemma 2.1

([21])

Let \(r>0, u\in H_{p}^{r}(\Omega)\),
$$\begin{aligned} & \Vert P_{J}u-u \Vert _{l}\leq CJ^{l-r} \vert u \vert _{r},\quad 0\leq l\leq r, \\ & \Vert P_{J}u \Vert _{l}\leq C \Vert u \Vert _{l}. \end{aligned}$$

Lemma 2.2

([21])

Let \(r>\frac{1}{2}, u\in H_{p}^{r}(\Omega)\),
$$\begin{aligned} & \Vert I_{J}u-u \Vert _{l}\leq CJ^{l-r} \vert u \vert _{r}, \quad 0\leq l\leq r, \\ & \Vert I_{J}u \Vert _{l}\leq C \Vert u \Vert _{l}. \end{aligned}$$

Lemma 2.3

([21])

Assume \(u^{*}=P_{J-2}u, u\in H_{p}^{r}(\Omega), r>\frac{1}{2}\), then \(\| u^{*}-u\|\leq CJ^{-r}|u|_{r}\).

Lemma 2.4

([22])

For any discrete function \(U\in w \), we can obtain
$$ \Vert U \Vert _{l^{p}}\leq C \bigl( \vert U \vert _{h,1}^{\alpha} \Vert U \Vert ^{1-\alpha}+ \Vert U \Vert \bigr), $$
(10)
where \(\alpha=\frac{1}{2}-\frac{1}{p}, p\in[2,+\infty)\), and C is a constant independent of h.
Here, we define a new seim-norm \(|U|_{h}=\sqrt{\langle-D_{2}U,U\rangle}\). Noting that \(|U|_{h,1}=\sqrt{\langle-AU, U\rangle}\), where
A = 1 h 2 [ 2 1 0 0 1 1 2 1 0 0 0 1 2 1 0 0 0 1 2 1 1 0 0 1 2 ] .

Lemma 2.5

([23])

For a real circulant matrix \(A_{1}=C(a_{0},a_{1},\ldots,a_{n-1})\), all eigenvalues of \(A_{1}\) are given by
$$f(\varepsilon_{k}),\quad k=0, 1, 2, \ldots, n-1, $$
where \(f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n-1}x^{n-1}\), and \(\varepsilon _{k}=e^{i\frac{2k\pi}{n}}\).

Lemma 2.6

([23])

For matrices \(A, D_{2}\), there exist relations
$$A=F^{T}_{1}\Lambda_{1}F_{1},\qquad D_{2}=F^{T}_{2}\Lambda_{2}F_{2}, $$
where
$$\begin{aligned} &\Lambda_{1}=\operatorname{diag}(\lambda_{A,0}, \lambda_{A,1},\ldots, \lambda _{A,J-1}), \\ &\Lambda_{2}=\operatorname{diag}(\lambda_{D_{2},0}, \lambda_{D_{2},1}, \ldots, \lambda_{D_{2},J-1}) \end{aligned}$$
and
$$0\leq-\frac{4}{\pi^{2}}\lambda_{D_{2},j}\leq-\lambda_{A,j}\leq- \lambda_{D_{2},j}. $$

Lemma 2.7

([12])

For any grid function \(U\in V_{J}''\), the following inequalities hold:
$$\vert U \vert _{h,1}\leq \vert U \vert _{h}\leq \frac{\pi}{2} \vert U \vert _{h,1}. $$

Lemma 2.8

For any grid functions \(U^{n}\in w\), we can obtain
$$\begin{aligned} &\operatorname{Im} \bigl\langle D_{2} U^{n+\frac{1}{2}},U^{n+\frac{1}{2}} \bigr\rangle =0, \end{aligned}$$
(11)
$$\begin{aligned} &\operatorname{Re} \bigl\langle D_{2} U^{n+\frac{1}{2}},U^{n}_{t} \bigr\rangle =-\frac{1}{2\tau } \bigl( \bigl\vert U^{n+1} \bigr\vert ^{2}_{h}- \bigl\vert U^{n} \bigr\vert ^{2}_{h} \bigr), \end{aligned}$$
(12)
where \(\operatorname{Im}(s), \operatorname{Re}(s)\) are for the imaginary part and the real part of a complex number s, respectively.

Lemma 2.9

([12])

Suppose that \(g(x)\in C^{2}[d_{1}, d_{2}]\) and \(a_{1}, a_{2}, b_{1}, b_{2}\in[d_{1}, d_{2}]\), there exist constants \(\theta\in(-1, 1)\) and \(\eta\in[d_{1}, d_{2}]\) such that
$$\begin{aligned} \frac{g(a_{2})-g(a_{1})}{a_{2}-a_{1}}-\frac{g(b_{2})-g(b_{1})}{b_{2}-b_{1}}&= g' \biggl( \frac {1-\theta}{2}a_{1}+\frac{1+\theta}{2}a_{2} \biggr)-g' \biggl(\frac{1-\theta}{2}b_{1}+ \frac {1+\theta}{2}b_{2} \biggr) \\ &= g''(\eta) \biggl(\frac{1-\theta}{2}(a_{1}-b_{1})+ \frac{1+\theta}{2}(a_{2}-b_{2}) \biggr). \end{aligned}$$

2.2 Conservative Fourier spectral scheme

In the paper, we give following conservative scheme for the CSB system:
$$\begin{aligned} &i \bigl(U_{j}^{n} \bigr)_{t}+ \bigl(D_{2}U^{n+\frac{1}{2}} \bigr)_{j}-U_{j}^{n+\frac{1}{2}}V_{j}^{n+\frac {1}{2}}=0, \quad 0\leq j\leq J-1, 0\leq n\leq N-1, \end{aligned}$$
(13)
$$\begin{aligned} & \bigl(V_{j}^{n} \bigr)_{t}= \bigl(D_{2}\Phi^{n+\frac{1}{2}} \bigr)_{j},\quad 0\leq j\leq J-1, 0 \leq n\leq N-1, \end{aligned}$$
(14)
$$\begin{aligned} \begin{aligned} & \bigl(\Phi_{j}^{n} \bigr)_{t}=V_{j}^{n+\frac{1}{2}}+ \frac {F(V_{j}^{n+1})-F(V_{j}^{n})}{V_{j}^{n+1}-V_{j}^{n}}-\alpha \bigl(D_{2}V^{n+\frac {1}{2}} \bigr)_{j}+\frac{\omega}{2} \bigl( \bigl\vert U_{j}^{n+1} \bigr\vert ^{2}+ \bigl\vert U_{j}^{n} \bigr\vert ^{2} \bigr), \\ &\quad 0\leq j\leq J-1, 1\leq n\leq N-1, \end{aligned} \end{aligned}$$
(15)
$$\begin{aligned} &U_{j}^{0}=u_{0}(x_{j}), \qquad V_{j}^{0}=v_{0}(x_{j}),\qquad \Phi_{j}^{0}=\phi_{0}(x_{j}),\quad 0\leq j \leq J, \end{aligned}$$
(16)
$$\begin{aligned} &U_{0}^{n}=U_{J}^{n}, \qquad V_{0}^{n}=V_{J}^{n},\qquad \Phi_{0}^{n}=\Phi_{J}^{n},\quad 0\leq n \leq N. \end{aligned}$$
(17)

Theorem 2.1

The scheme (13)(17) is conservative in the sense
$$\begin{aligned} &\operatorname{Mass}:\quad M^{n}=M^{n-1}= \cdots=M^{0}, \\ &\operatorname{Energy}:\quad E^{n}=E^{n-1}= \cdots=E^{0}, \end{aligned}$$
where
$$\begin{aligned} &M^{n}= \bigl\Vert U^{n} \bigr\Vert ^{2}, \\ &E^{n}= \bigl\Vert V^{n} \bigr\Vert - \bigl\langle D_{2} \Phi^{n},\Phi^{n} \bigr\rangle -2\omega \bigl\langle D_{2}U^{n},U^{n} \bigr\rangle -\alpha \bigl\langle D_{2} V^{n},V^{n} \bigr\rangle +2 \bigl\langle F \bigl(V^{n} \bigr),I \bigr\rangle +2\omega \bigl\langle V^{n}, \bigl\vert U^{n} \bigr\vert ^{2} \bigr\rangle . \end{aligned}$$

Proof

Computing the inner product of Eq. (13) with \(2 U^{n+\frac{1}{2}}\), we can obtain
$$\begin{aligned} \bigl\langle iU^{n}_{t}+D_{2}U^{n+\frac{1}{2}}+ U^{n+\frac{1}{2}}V^{n+\frac {1}{2}},2 U^{n+\frac{1}{2}} \bigr\rangle =0. \end{aligned}$$
By Lemma 2.8 and taking the imaginary part, we have
$$\bigl\Vert U^{n+1} \bigr\Vert ^{2}= \bigl\Vert U^{n} \bigr\Vert ^{2}. $$
This means that \(M^{n}=M^{n-1}=\cdots=M^{0}\).
Computing the inner product of Eq. (13) with \(2\tau U^{n}_{t}\), we can obtain
$$\begin{aligned} \bigl\langle iU^{n}_{t}+D_{2}U^{n+\frac{1}{2}}+ U^{n+\frac{1}{2}}V^{n+\frac {1}{2}},2\tau U^{n}_{t} \bigr\rangle =0. \end{aligned}$$
(18)
Taking the real part yields
$$\begin{aligned} \bigl\langle D_{2}U^{n+1},U^{n+1} \bigr\rangle - \bigl\langle D_{2}U^{n},U^{n} \bigr\rangle = \frac {1}{2} \bigl\langle V^{n+1}+V^{n}, \bigl\vert U^{n+1} \bigr\vert ^{2} \bigr\rangle - \frac{1}{2} \bigl\langle V^{n+1}+V^{n}, \bigl\vert U^{n} \bigr\vert ^{2} \bigr\rangle . \end{aligned}$$
(19)
Making the inner product of (14) and (15) with \(2\tau\Phi^{n}_{t}, 2\tau V^{n}_{t}\), respectively, then we have
$$\begin{aligned} 2\tau \bigl\langle V^{n}_{t}, \Phi^{n}_{t} \bigr\rangle ={}& \bigl\langle D_{2} \Phi^{n+1},\Phi ^{n+1} \bigr\rangle - \bigl\langle D_{2} \Phi^{n},\Phi^{n} \bigr\rangle , \\ 2\tau \bigl\langle \Phi^{n}_{t},V^{n}_{t} \bigr\rangle ={}& \biggl\langle V^{n+\frac {1}{2}}+\frac{F(V^{n+1})-F(V^{n})}{V^{n+1}-V^{n}}-\alpha \bigl(D_{2}V^{n+\frac {1}{2}} \bigr)+\frac{\omega}{2} \bigl( \bigl\vert U^{n+1} \bigr\vert ^{2}+ \bigl\vert U^{n} \bigr\vert ^{2} \bigr),2\tau V^{n}_{t} \biggr\rangle \\ ={}& \bigl\Vert V^{n+1} \bigr\Vert ^{2}+\omega \bigl\langle V^{n+1}, \bigl\vert U^{n+1} \bigr\vert ^{2} \bigr\rangle -\alpha \bigl\langle D_{2}V^{n+1},V^{n+1} \bigr\rangle \\ &{}+2 \bigl\langle F \bigl(V^{n+1} \bigr),I \bigr\rangle -\omega \bigl\langle V^{n}, \bigl\vert U^{n+1} \bigr\vert ^{2} \bigr\rangle \\ &{}- \bigl\Vert V^{n} \bigr\Vert ^{2}-\omega \bigl\langle V^{n}, \bigl\vert U^{n} \bigr\vert ^{2} \bigr\rangle +\alpha \bigl\langle D_{2}V^{n},V^{n} \bigr\rangle +2 \bigl\langle F \bigl(V^{n} \bigr),I \bigr\rangle -\omega \bigl\langle V^{n+1}, \bigl\vert U^{n} \bigr\vert ^{2} \bigr\rangle . \end{aligned}$$
It follows from the above equations and (19) that
$$\begin{aligned} & \bigl\Vert V^{n+1} \bigr\Vert - \bigl\langle D_{2} \Phi^{n+1}, \Phi^{n+1} \bigr\rangle -2\omega \bigl\langle D_{2}U^{n+1},U^{n+1} \bigr\rangle -\alpha \bigl\langle D_{2} V^{n+1},V^{n+1} \bigr\rangle \\ &\qquad{}+2 \bigl\langle F \bigl(V^{n+1} \bigr),I \bigr\rangle +2\omega \bigl\langle V^{n+1}, \bigl\vert U^{n+1} \bigr\vert ^{2} \bigr\rangle \\ &\quad = \bigl\Vert V^{n} \bigr\Vert - \bigl\langle D_{2} \Phi^{n}, \Phi^{n} \bigr\rangle -2\omega \bigl\langle D_{2}U^{n},U^{n} \bigr\rangle -\alpha \bigl\langle D_{2} V^{n},V^{n} \bigr\rangle +2 \bigl\langle F \bigl(V^{n} \bigr),I \bigr\rangle +2\omega \bigl\langle V^{n}, \bigl\vert U^{n} \bigr\vert ^{2} \bigr\rangle . \end{aligned}$$
This yields \(E^{n}=E^{n-1}=\cdots=E^{0}\). □

Theorem 2.2

The scheme (13)(17) is bounded in the discrete \(l_{h}^{\infty}\).

Proof

It follows from Theorem 2.1 that \(\|U^{n}\|=C_{1}\), and
$$\begin{aligned} E^{n}= {}& \bigl\Vert V^{n} \bigr\Vert - \bigl\langle D_{2} \Phi^{n},\Phi^{n} \bigr\rangle -2 \omega \bigl\langle D_{2}U^{n},U^{n} \bigr\rangle - \alpha \bigl\langle D_{2} V^{n},V^{n} \bigr\rangle \\ &{}+2 \bigl\langle F \bigl(V^{n} \bigr),I \bigr\rangle +2\omega \bigl\langle V^{n}, \bigl\vert U^{n} \bigr\vert ^{2} \bigr\rangle . \end{aligned}$$
(20)
According to the Young inequality, we can obtain
$$ \bigl\langle V^{n}, \bigl\vert U^{n} \bigr\vert ^{2} \bigr\rangle =h\sum_{j=0}^{J-1} \bigl\vert U_{j}^{n} \bigr\vert ^{2}V_{j}^{n} \leq h\sum_{j=0}^{J-1} \biggl( \bigl\vert U_{j}^{n} \bigr\vert ^{4}+ \frac{1}{4} \bigl\vert V_{j}^{n} \bigr\vert ^{2} \biggr) = \bigl\Vert U^{n} \bigr\Vert ^{4}_{l^{4}_{h}}+ \frac{1}{4} \bigl\Vert V^{n} \bigr\Vert ^{2}. $$
(21)
It follows from Lemma 2.4 that
$$\begin{aligned} \bigl\Vert U^{n} \bigr\Vert ^{4}_{l_{h}^{4}}\leq{}&C_{2}\bigl( \bigl\vert U^{n} \bigr\vert _{h,1}^{\frac{1}{4}} \bigl\Vert U^{n} \bigr\Vert ^{\frac {3}{4}}+ \bigl\Vert U^{n} \bigr\Vert \bigr)^{4}\leq8C_{2}\bigl( \vert U \vert _{h,1} \bigl\Vert U^{n} \bigr\Vert ^{3}+ \bigl\Vert U^{n} \bigr\Vert ^{4}\bigr) \\ \leq{}& 4C_{2}\biggl(\epsilon \bigl\vert U^{n} \bigr\vert _{h,1}^{2}+\frac{1}{\epsilon} \bigl\Vert U^{n} \bigr\Vert ^{6}+2 \bigl\Vert U^{n} \bigr\Vert ^{4}\biggr). \end{aligned}$$
(22)
By Lemma 2.7, there exist constants \(C_{3}, C_{4}, C_{5}\) such that
$$\begin{aligned} \begin{aligned}&\bigl\langle -D_{2} \Phi^{n}, \Phi^{n} \bigr\rangle =C_{3} \bigl\vert \Phi^{n} \bigr\vert _{h,1}^{2},\\ & 2\omega \bigl\langle -D_{2}U^{n},U^{n} \bigr\rangle =C_{4} \bigl\vert U^{n} \bigr\vert _{h,1}^{2}, \qquad\alpha \bigl\langle -D_{2} V^{n},V^{n} \bigr\rangle =C_{5} \bigl\vert V^{n} \bigr\vert _{h,1}^{2}. \end{aligned} \end{aligned}$$
(23)
Substituting Eqs. (21)–(23) into (20), we have
$$\begin{aligned} & \bigl\Vert V^{n} \bigr\Vert ^{2}+C_{3} \bigl\vert \Phi ^{n} \bigr\vert _{h,1}^{2}+C_{4} \bigl\vert U^{n} \bigr\vert _{h,1}^{2}+C_{5} \bigl\vert V^{n} \bigr\vert _{h,1}^{2} \\ &\quad\leq E^{n}+2\omega \bigl\vert \bigl\langle V^{n}, \bigl\vert U^{n} \bigr\vert ^{2} \bigr\rangle \bigr\vert \\ &\quad\leq E^{n}+2\omega \biggl( \bigl\Vert U^{n} \bigr\Vert ^{4}_{l^{4}_{h}}+ \frac{1}{2} \bigl\Vert V^{n} \bigr\Vert ^{2} \biggr) \\ &\quad \leq E^{n}+8\omega C_{2} \biggl(\epsilon \bigl\vert U^{n} \bigr\vert _{h,1}^{2}+ \frac{1}{\epsilon} \bigl\Vert U^{n} \bigr\Vert ^{6}+2 \bigl\Vert U^{n} \bigr\Vert ^{4} \biggr)+\frac{1}{4} \bigl\Vert V^{n} \bigr\Vert ^{2}. \end{aligned}$$
When \(\epsilon<\frac{C_{4}}{8\omega C_{2}} \), there exists a constant \(C_{6}\) such that
$$\bigl\Vert V^{n} \bigr\Vert \leq C_{6},\qquad \bigl\vert \Phi^{n} \bigr\vert _{h,1}^{2} \leq C_{6},\qquad \bigl\vert U^{n} \bigr\vert _{h,1}^{2} \leq C_{6},\qquad \bigl\vert V^{n} \bigr\vert _{h,1}^{2}\leq C_{6}. $$
By the Sobolev inequality, we have \(\|U^{n}\|_{l^{\infty}_{h}}\leq C_{7}, \| V^{n}\|_{l^{\infty}_{h}}\leq C_{7}, \|\Phi^{n}\|_{l^{\infty}_{h}}\leq C_{7}\). □

2.3 Existence and uniqueness

Theorem 2.3

The numerical solutions of the scheme (13)(17) exist.

Proof

Let \(U=(U_{0}, U_{1}, \ldots,U_{J-1})^{T}\), \(V=(V_{0}, V_{1}, \ldots, V_{J-1})^{T}\), \(\Phi=(\Phi_{0}, \Phi_{1}, \ldots, \Phi_{J-1})^{T}\) and \(X = (U^{T}, V^{T}, \Phi^{T} )^{T}\). Define a mapping \(T_{\lambda}\): \(R^{3J-3}\rightarrow R^{3J-3}\) with parameter \(\lambda\in(0,1)\),
$$\begin{aligned} &i\frac{U_{j}-u_{j}^{n}}{\tau}+\frac{\lambda}{2} \bigl(D_{2} \bigl(U+u^{n} \bigr) \bigr)_{j}+ \frac {\lambda}{4} \bigl(U_{j}+u_{j}^{n} \bigr) \bigl(V_{j}+v_{j}^{n} \bigr)=0,\quad 0\leq j\leq J-1, \end{aligned}$$
(24)
$$\begin{aligned} &\frac{V_{j}-v_{j}^{n}}{\tau}=\frac{\lambda}{2} \bigl(D_{2} \bigl(\Phi+ \phi^{n} \bigr) \bigr)_{j}, \quad 0\leq j\leq J-1, \end{aligned}$$
(25)
$$\begin{aligned} \begin{aligned}&\frac{\Phi_{j}-\phi_{j}^{n}}{\tau}=\frac{\lambda}{2} \bigl(V_{j}+v_{j}^{n} \bigr)+\lambda \frac{F(V_{j})- F(v_{j}^{n})}{V_{j}-v_{j}^{n}}-\lambda\alpha \bigl(D_{2} \bigl(V+v^{n} \bigr) \bigr)_{j}+\frac{\omega\lambda}{2} \bigl( \vert U_{j} \vert ^{2}+ \bigl\vert u_{j}^{n} \bigr\vert ^{2} \bigr), \\ &\quad 0\leq j\leq J-1. \end{aligned} \end{aligned}$$
(26)
It ie easy to see that the mapping \(T_{\lambda}(X)\) is continuous, and \(T_{0}(X)\) is a fixed point for any \(X\in R^{3J-3}\). Next, we prove that X is uniformly bounded. Similar to the proof of Theorem 2.1, we have
$$\begin{aligned} \Vert U \Vert ^{2}=C_{8}. \end{aligned}$$
Thus, U is uniformly bounded. Then we prove the uniformly bounded V and Φ. Computing the inner product of Eq. (25) and (26) with \(\alpha\tau (V-v^{n}),\tau(\Phi-\phi^{n})\), respectively, we obtain
$$\begin{aligned} & \biggl\langle \frac{V-v^{n}}{\tau},\alpha\tau \bigl(V-v^{n} \bigr) \biggr\rangle = \biggl\langle \frac {\lambda}{2} \bigl(D_{2} \bigl( \Phi+\phi^{n} \bigr) \bigr),\alpha\tau \bigl(V-v^{n} \bigr) \biggr\rangle , \end{aligned}$$
(27)
$$\begin{aligned} & \biggl\langle \frac{\Phi-\phi^{n}}{\tau},\tau \bigl(\Phi- \phi^{n} \bigr) \biggr\rangle = \biggl\langle \frac{\lambda}{2} \bigl(V+v^{n} \bigr)+ \lambda \frac{F(V)-F(v^{n})}{V-v^{n}}-\alpha \lambda \bigl(D_{2} \bigl(V+v^{n} \bigr) \bigr) \\ &\phantom{ \langle\frac{\Phi-\phi^{n}}{\tau},\tau (\Phi-\phi^{n} ) \rangle=}{}+\frac{\omega\lambda}{2} \bigl( \vert U \vert ^{2}+ \bigl\vert u^{n} \bigr\vert ^{2} \bigr), \tau \bigl(\Phi - \phi^{n} \bigr) \biggr\rangle . \end{aligned}$$
(28)
Addition (27) and (28) yields
$$\begin{aligned} & \bigl(\alpha \Vert V \Vert ^{2}+ \Vert \Phi \Vert ^{2} \bigr)- \bigl( \alpha \bigl\Vert v^{n} \bigr\Vert ^{2}+ \bigl\Vert \phi^{n} \bigr\Vert ^{2} \bigr) \\ &\quad=\frac {\lambda\tau}{2}h\sum_{j=0}^{J-1} \bigl(V_{j}+v_{j}^{n} \bigr)\cdot \bigl( \Phi_{j}+\phi_{j}^{n} \bigr) \\ &\qquad{}+ \frac{\lambda\omega\tau}{2}h\sum_{j=1}^{J-1} \bigl( \vert U_{j} \vert ^{2}+ \bigl\vert u_{j}^{n} \bigr\vert ^{2} \bigr)\cdot \bigl( \Phi_{j}+\phi_{j}^{n} \bigr)+ + \lambda\tau h\sum _{j=1}^{J-1} \biggl( \frac{F(V_{j})-F(v_{j}^{n})}{V_{j}-v_{j}^{n}} \biggr)\cdot \bigl(\Phi_{j}+\phi_{j}^{n} \bigr). \end{aligned}$$
Similar to the proof of [12], we get
$$\begin{aligned} &\alpha \Vert V \Vert ^{2}+ \Vert \Phi \Vert ^{2}=\lambda\tau C_{9} \bigl(\varepsilon \Vert U \Vert _{h,1}+C(\varepsilon) \Vert U \Vert ^{2} \bigr)+C_{9}\leq\tau rC_{9} \Vert U \Vert ^{2}+C_{9} \biggl(r=\frac {\tau}{h^{2}} \biggr). \end{aligned}$$
Thus, \(\|V\|\) and \(\|\Phi\|\) are uniformly bounded. According to Lemma 4.1 of [12], we see that the numerical solutions of the scheme exist. This completes the proof. □

Theorem 2.4

The numerical solution of the scheme (13)(17) is unique.

Proof

Assume that \((U^{n},V^{n}, \Phi^{n})\) and \((\widetilde{U}^{n},\widetilde {V}^{n},\widetilde{\Phi}^{n})\) satisfy scheme (13)–(17). Let \(P^{n}=U^{n}-\widetilde{U}^{n},Q^{n}=V^{n}-\widetilde{V}^{n}, S^{n}=\Phi^{n}-\widetilde {\Phi^{n}}\), we obtain
$$\begin{aligned} &i\frac{P^{n+1}-P^{n}}{\tau}+D_{2}P^{n+\frac{1}{2}}-F_{1}=0, \end{aligned}$$
(29)
$$\begin{aligned} &\frac{Q^{n+1}-Q^{n}}{\tau}=D_{2}S^{n+\frac{1}{2}}, \end{aligned}$$
(30)
$$\begin{aligned} &\frac{S^{n+1}-S^{n}}{\tau}=Q^{n+\frac{1}{2}}-\alpha D_{2}Q^{n+\frac{1}{2}}+F_{2}, \end{aligned}$$
(31)
where
$$\begin{aligned} F_{1}={}&U^{n+\frac{1}{2}}V^{n+\frac{1}{2}}- \widetilde{U}^{n+\frac {1}{2}}\widetilde{V}^{n+\frac{1}{2}}, \\ F_{2}={}&\frac{F(V^{n+1}-F(V^{n}))}{V^{n+1}-V^{n}}-\frac{F(\widetilde {V}^{n+1}-F(\widetilde{V}^{n}))}{\widetilde{V}^{n+1}-\widetilde{V}^{n}} \\ &{}+\frac{\omega}{2} \bigl( \bigl\vert U^{n+1} \bigr\vert ^{2}+ \bigl\vert U^{n} \bigr\vert ^{2} \bigr)- \frac{\omega}{2} \bigl( \bigl\vert \widetilde {U}^{n+1} \bigr\vert ^{2}+ \bigl\vert \widetilde{U}^{n} \bigr\vert ^{2} \bigr). \end{aligned}$$
Computing the inner product of (29) with \(2P^{n+\frac{1}{2}}\) and taking the imaginary part, we have
$$ \frac{ \Vert P^{n+1} \Vert ^{2}- \Vert P^{n} \Vert ^{2}}{\tau}-\operatorname{Im} \bigl\langle F_{1},2P^{n+\frac {1}{2}} \bigr\rangle =0. $$
(32)
Noting that
$$\begin{aligned} (F_{1})_{j}&=U_{j}^{n+\frac{1}{2}}V_{j}^{n+\frac{1}{2}}- \widetilde {U}_{j}^{n+\frac{1}{2}}\widetilde{V}_{j}^{n+\frac{1}{2}} \\ &= \bigl(U_{j}^{n+\frac{1}{2}}-\widetilde{U}_{j}^{n+\frac{1}{2}} \bigr)V_{j}^{n+\frac {1}{2}}+\widetilde{U}_{j}^{n+\frac{1}{2}} \bigl(V_{j}^{n+\frac{1}{2}}-\widetilde {V}_{j}^{n+\frac{1}{2}} \bigr), \end{aligned}$$
and according to Theorem 2.2, we get
$$\Vert F_{1} \Vert ^{2}\leq C_{10} \bigl( \bigl\Vert P^{n+1} \bigr\Vert ^{2}+ \bigl\Vert P^{n} \bigr\Vert ^{2}+ \bigl\Vert Q^{n+1} \bigr\Vert ^{2}+ \bigl\Vert Q^{n} \bigr\Vert ^{2} \bigr). $$
It follows from Eq. (32) that there exists a constant \(C_{11}\) such that
$$ \bigl\Vert P^{n+1} \bigr\Vert ^{2}- \bigl\Vert P^{n} \bigr\Vert ^{2}\leq C_{11}\tau \bigl( \bigl\Vert P^{n+1} \bigr\Vert ^{2}+ \bigl\Vert P^{n} \bigr\Vert ^{2}+ \bigl\Vert Q^{n+1} \bigr\Vert ^{2}+ \bigl\Vert Q^{n} \bigr\Vert ^{2} \bigr). $$
(33)
Computing the inner product of (30) and (31) with \(2\alpha Q^{n+\frac {1}{2}}, 2S^{n+\frac{1}{2}}\), we have
$$\begin{aligned} & \biggl\langle \frac{Q^{n+1}-Q^{n}}{\tau},2\alpha Q^{n+\frac{1}{2}} \biggr\rangle = \bigl\langle D_{2}S^{n+\frac{1}{2}}, 2\alpha Q^{n+\frac{1}{2}} \bigr\rangle , \end{aligned}$$
(34)
$$\begin{aligned} & \biggl\langle \frac{S^{n+1}-S^{n}}{\tau},2S^{n+\frac{1}{2}} \biggr\rangle = \bigl\langle Q^{n+\frac{1}{2}}-\alpha D_{2}Q^{n+\frac{1}{2}}+F_{2},2S^{n+\frac {1}{2}} \bigr\rangle . \end{aligned}$$
(35)
According to Lemma 2.9, and adding Eqs. (34) and (35), we get
$$\begin{aligned} & \bigl\Vert Q^{n+1} \bigr\Vert ^{2}- \bigl\Vert Q^{n} \bigr\Vert ^{2}+ \bigl\Vert S^{n+1} \bigr\Vert ^{2}- \bigl\Vert S^{n} \bigr\Vert ^{2} \\ &\quad\leq C_{12}\tau \bigl( \bigl\Vert P^{n+1} \bigr\Vert ^{2}+ \bigl\Vert P^{n} \bigr\Vert ^{2}+ \bigl\Vert Q^{n+1} \bigr\Vert ^{2}+ \bigl\Vert Q^{n} \bigr\Vert ^{2}+ \bigl\Vert S^{n+1} \bigr\Vert ^{2}+ \bigl\Vert S^{n} \bigr\Vert ^{2} \bigr). \end{aligned}$$
(36)
Noting that (28) holds, we have
$$\begin{aligned} & \bigl\Vert P^{n+1} \bigr\Vert ^{2}- \bigl\Vert P^{n} \bigr\Vert ^{2}+ \bigl\Vert Q^{n+1} \bigr\Vert ^{2}- \bigl\Vert Q^{n} \bigr\Vert ^{2}+ \bigl\Vert S^{n+1} \bigr\Vert ^{2}- \bigl\Vert S^{n} \bigr\Vert ^{2} \\ &\quad\leq C_{13}\tau \bigl( \bigl\Vert P^{n+1} \bigr\Vert ^{2}+ \bigl\Vert P^{n} \bigr\Vert ^{2}+ \bigl\Vert Q^{n+1} \bigr\Vert ^{2}+ \bigl\Vert Q^{n} \bigr\Vert ^{2}+ \bigl\Vert S^{n+1} \bigr\Vert ^{2}+ \bigl\Vert S^{n} \bigr\Vert ^{2} \bigr). \end{aligned}$$
(37)
Let \(B^{n}=\|P^{n}\|^{2}+\|Q^{n}\|^{2}+\|S^{n}\|^{2}\), then
$$\begin{aligned} B^{n+1}-B^{n}\leq C_{13}\tau \bigl(B^{n+1}+B^{n} \bigr). \end{aligned}$$
It follows from Gronwall’s inequality [24] that
$$\max_{1\leq n\leq N}B^{n}\leq B^{0}e^{4C_{13}T}. $$
Noting that \(P^{0}=Q^{0}=S^{0}=0\), then we get \(P^{n}=Q^{n}=S^{n}=0\). This completes the proof of the uniqueness for \(U^{n}, V^{n}, \Phi^{n}\). □

2.4 Convergence and error estimates

Theorem 2.5

Suppose that \(u_{0}(x),v_{0}(x),\phi_{0}(x)\in H^{r}(R),s\geq1\). Then the solution \(U^{n}, V^{n}, \Phi^{n}\) of the scheme (13)(17) converges to the true solution \(u, v, \phi\) with order \(O(\tau ^{2}+J^{-r})\) by the \(\|\cdot\|_{1}\) norm.

Proof

Define
$$\begin{aligned} &P_{J-2}(iu_{t}+u_{xx}-uv)=0, \end{aligned}$$
(38)
$$\begin{aligned} &P_{J-2}v_{t}=P_{J-2} \phi_{xx}, \end{aligned}$$
(39)
$$\begin{aligned} &P_{J-2} \bigl(\phi_{t}-v+\alpha v_{xx}-f(v)- \omega \vert u \vert ^{2} \bigr)=0. \end{aligned}$$
(40)
Let \(u^{*}=P_{J-2}u, v^{*}=P_{J-2}v, \phi^{*}=P_{J-2}\phi\), we have
$$\begin{aligned} &iu^{*}_{t}+u^{*}_{xx}-P_{J-2}(uv)=0, \end{aligned}$$
(41)
$$\begin{aligned} &v_{t}^{*}=\phi_{xx}^{*}, \end{aligned}$$
(42)
$$\begin{aligned} &\phi_{t}^{*}-v^{*}+\alpha v_{xx}^{*}-P_{J-2} \bigl(f(v)-\omega \vert u \vert ^{2} \bigr)=0. \end{aligned}$$
(43)
Define
$$\begin{aligned} &\xi_{j}^{n} =i \bigl(u_{j}^{*n} \bigr)_{t}+ \bigl(D_{2}u^{*n+\frac{1}{2}} \bigr)_{j}-P_{J-2} \bigl(u_{j}^{n+\frac {1}{2}}v_{j}^{n+\frac{1}{2}} \bigr), \\ &\eta_{j}^{n}= \bigl(v_{j}^{*n} \bigr)_{t}- \bigl(D_{2}\phi^{*n+\frac{1}{2}} \bigr)_{j}, \\ &\rho_{j}^{n}= \bigl(\phi_{j}^{*n} \bigr)_{t}-v^{*n+\frac{1}{2}}+\alpha \bigl(D_{2}v^{*n+\frac {1}{2}} \bigr)_{j}-P_{J-2} \biggl(\frac {F(v_{j}^{n+1})-F(v_{j}^{n})}{v_{j}^{n+1}-v_{j}^{n}} \biggr)- \frac{\omega }{2} \bigl( \bigl\vert u^{n+1} \bigr\vert ^{2}+ \bigl\vert u^{n} \bigr\vert ^{2} \bigr). \end{aligned}$$
Due to \(u^{*}, v^{*}, \phi^{*}\in V_{N}'', u_{xx}^{*}(x_{j},t_{n})=(D_{2}u^{*n})_{j}, v_{xx}^{*}(x_{j},t_{n})=(D_{2}v^{*n})_{j}, \phi _{xx}^{*}(x_{j},t_{n})=(D_{2}\phi^{*n})_{j}\), we have
$$\begin{aligned} &u_{xx}^{*}(x_{j},t_{n})= \bigl(D_{2}u^{*n+\frac{1}{2}} \bigr)_{j}+C_{14} \tau^{2}, \\ &\phi_{xx}^{*}(x_{j},t_{n})= \bigl(D_{2} \phi^{*n+\frac{1}{2}} \bigr)_{j}+C_{14}\tau^{2}, \\ &v_{xx}^{*}(x_{j},t_{n})= \bigl(D_{2}v^{*n+\frac{1}{2}} \bigr)_{j}+C_{14}\tau^{2}. \end{aligned}$$
Using the Taylor expansion, we have
$$\begin{aligned} \bigl\vert \xi_{j}^{n} \bigr\vert \leq C_{15} \tau^{2}, \qquad \bigl\vert \eta_{j}^{n} \bigr\vert \leq C_{15} \tau^{2},\qquad \bigl\vert \rho _{j}^{n} \bigr\vert \leq C_{15} \tau^{2}. \end{aligned}$$
(44)
Define \(e_{1}^{n}=(u^{*})^{n}-U^{n}, e_{2}^{n}=(v^{*})^{n}-V^{n}, e_{3}^{n}=(\phi^{*})^{n}-\Phi ^{n}\) we have
$$\begin{aligned} &\xi^{n}=i \bigl(e_{1}^{n} \bigr)_{t}+D_{2}e_{1}^{n+\frac{1}{2}}-F^{n+\frac{1}{2}}, \quad n=0, 1, 2, \ldots, \end{aligned}$$
(45)
$$\begin{aligned} &\eta^{n}= \bigl(e_{2}^{n} \bigr)_{t}-D_{2}e_{3}^{n+\frac{1}{2}},\quad n=0, 1, 2, \ldots, \end{aligned}$$
(46)
$$\begin{aligned} &\rho^{n}= \bigl(e_{3}^{n} \bigr)_{t}-e_{2}^{n+\frac{1}{2}}+ \alpha D_{2}e_{2}^{n+\frac {1}{2}}-G^{n+\frac{1}{2}}, \quad n=0, 1, 2, \ldots, \end{aligned}$$
(47)
$$\begin{aligned} &e_{1}^{0}=u^{*0}-U^{0}, \qquad e_{2}^{0}=v^{*0}-V^{0},\qquad e_{3}^{0}=\phi^{*^{0}}-\Phi^{0}, \end{aligned}$$
(48)
where
$$\begin{aligned} & F^{n+\frac{1}{2}}=P_{J-2} \bigl( u^{n+\frac{1}{2}} \phi^{n+\frac {1}{2}} \bigr)-U^{n+\frac{1}{2}}\Phi^{n+\frac{1}{2}}, \\ &G^{n+\frac{1}{2}}=P_{J-2} \biggl(\frac {F(v_{j}^{n+1})-F(v_{j}^{n})}{v_{j}^{n+1}-v_{j}^{n}}+\frac{\omega }{2} \bigl( \bigl\vert u^{n+1} \bigr\vert ^{2}+ \bigl\vert u^{n} \bigr\vert ^{2} \bigr) \biggr) \\ &\phantom{G^{n+\frac{1}{2}}=}{}- \biggl( \frac {F(V_{j}^{n+1})-F(V_{j}^{n})}{V_{j}^{n+1}-V_{j}^{n}}+\frac{\omega }{2} \bigl( \bigl\vert U_{j}^{n+1} \bigr\vert ^{2}+ \bigl\vert U_{j}^{n} \bigr\vert ^{2} \bigr) \biggr). \end{aligned}$$
Let
$$\begin{aligned} & F^{n+\frac{1}{2}}=(F_{1})^{n+\frac{1}{2}}+(F_{2})^{n+\frac {1}{2}}+(F_{3})^{n+\frac{1}{2}}, \\ & G^{n+\frac{1}{2}}=(G_{1})^{n+\frac{1}{2}}+(G_{2})^{n+\frac {1}{2}}+(G_{3})^{n+\frac{1}{2}}, \end{aligned}$$
where
$$\begin{aligned} &(F_{1})^{n+\frac{1}{2}}=P_{J-2} \bigl(u^{n+\frac{1}{2}}v^{n+\frac {1}{2}} \bigr)-u^{n+\frac{1}{2}}v^{n+\frac{1}{2}}, \\ &(F_{2})^{n+\frac{1}{2}}=u^{n+\frac{1}{2}}v^{n+\frac{1}{2}}-u^{*n+\frac {1}{2}}v^{*n+\frac{1}{2}}, \\ &(F_{3})^{n+\frac{1}{2}}=u^{*n+\frac{1}{2}}v^{*n+\frac{1}{2}}-U^{n+\frac {1}{2}}V^{n+\frac{1}{2}}, \\ &(G_{1})^{n+\frac{1}{2}}= P_{J-2} \biggl(\frac {F(v_{j}^{n+1})-F(v_{j}^{n})}{v_{j}^{n+1}-v_{j}^{n}} \\ &\phantom{(G_{1})^{n+\frac{1}{2}}=}{}+ \frac{\omega }{2} \bigl( \bigl\vert u^{n+1} \bigr\vert ^{2}+ \bigl\vert u^{n} \bigr\vert ^{2} \bigr) \biggr)- \biggl(\frac {F(v^{n+1})-F(v^{n})}{v^{n+1}-v^{n}}+\frac{\omega }{2} \bigl( \bigl\vert u^{n+1} \bigr\vert ^{2}+ \bigl\vert u^{n} \bigr\vert ^{2} \bigr) \biggr), \\ &(G_{2})^{n+\frac{1}{2}}=\frac {F(v_{j}^{n+1})-F(v_{j}^{n})}{v_{j}^{n+1}-v_{j}^{n}}+\frac{\omega }{2} \bigl( \bigl\vert u^{n+1} \bigr\vert ^{2}+ \bigl\vert u^{n} \bigr\vert ^{2} \bigr)) \\ &\phantom{(G_{2})^{n+\frac{1}{2}}=}{}- \biggl(\frac {F(v^{*n+1})-F(v^{*n})}{v^{*n+1}-v^{*n}}+ \frac{\omega }{2} \bigl( \bigl\vert u^{*n+1} \bigr\vert ^{2}+ \bigl\vert u^{*n} \bigr\vert ^{2} \bigr) \biggr), \\ &(G_{3})^{n+\frac{1}{2}}= \biggl(\frac {F(v^{*n+1})-F(v^{*n})}{v^{*n+1}-v^{*n}}+\frac{\omega }{2} \bigl( \bigl\vert u^{*n+1} \bigr\vert ^{2}+ \bigl\vert u^{*n} \bigr\vert ^{2} \bigr) \biggr) \\ &\phantom{(G_{3})^{n+\frac{1}{2}}= }{}- \biggl( \frac {F(V_{j}^{n+1})-F(V_{j}^{n})}{V_{j}^{n+1}-V_{j}^{n}}+\frac{\omega }{2} \bigl( \bigl\vert U_{j}^{n+1} \bigr\vert ^{2}+ \bigl\vert U_{j}^{n} \bigr\vert ^{2} \bigr) \biggr). \end{aligned}$$
It follows from Lemmas 2.1, 2.3 and 2.9 that
$$\begin{aligned} \bigl\Vert (F_{1})^{n+\frac{1}{2}} \bigr\Vert \leq{}& C_{16}J^{-r}, \qquad\bigl\Vert (G_{1})^{n} \bigr\Vert \leq C_{16}J^{-r}, \\ \bigl\Vert (F_{2})^{n+\frac{1}{2}} \bigr\Vert ={}& \bigl\Vert u^{n+\frac{1}{2}}v^{n+\frac {1}{2}}-u^{*n+\frac{1}{2}}v^{*n+\frac{1}{2}} \bigr\Vert \\ ={}& \bigl\Vert u^{n+\frac{1}{2}} \bigl(v^{n+\frac{1}{2}}-v^{*n+\frac{1}{2}} \bigr)+ \bigl(u^{n+\frac{1}{2}}-u^{*n+\frac{1}{2}} \bigr) \bigl(v^{*n+\frac{1}{2}} \bigr) \bigr\Vert \leq C_{17}J^{-r}, \\ \bigl\Vert (G_{2})^{n+\frac{1}{2}} \bigr\Vert ={}& \biggl\| \frac {F(v_{j}^{n+1})-F(v_{j}^{n})}{v_{j}^{n+1}-v_{j}^{n}}+\frac{\omega }{2} \bigl( \bigl\vert u^{n+1} \bigr\vert ^{2}+ \bigl\vert u^{n} \bigr\vert ^{2} \bigr)) \\ &{}- \biggl(\frac {F(v^{*n+1})-F(v^{*n})}{v^{*n+1}-v^{*n}}+\frac{\omega }{2} \bigl( \bigl\vert u^{*n+1} \bigr\vert ^{2}+ \bigl\vert u^{*n} \bigr\vert ^{2} \bigr) \biggr) \biggr\| \leq C_{18}J^{-r}, \\ \bigl\Vert (F_{3})^{n+\frac{1}{2}} \bigr\Vert ={}&C_{19} \bigl( \bigl\Vert e_{1}^{n+\frac{1}{2}} \bigr\Vert + \bigl\Vert e_{2}^{n+\frac{1}{2}} \bigr\Vert \bigr), \\ \bigl\Vert (G_{3})^{n+\frac{1}{2}} \bigr\Vert ={}&C_{19} \bigl( \bigl\Vert e_{1}^{n+\frac{1}{2}} \bigr\Vert + \bigl\Vert e_{2}^{n+\frac{1}{2}} \bigr\Vert \bigr). \end{aligned}$$
Computing the inner product of Eq. (45) with \(2\tau e_{1}^{n+\frac {1}{2}}\), then taking the imaginary part, we obtain
$$\begin{aligned} \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}- \bigl\Vert e_{1}^{n} \bigr\Vert ^{2} \bigr)- \operatorname{Im} \bigl\langle F_{1}^{n+\frac {1}{2}}+F_{2}^{n+\frac{1}{2}}+F_{3}^{n+\frac{1}{2}},2 \tau e^{n+\frac {1}{2}} \bigr\rangle = \bigl\langle \xi^{n}, 2\tau e_{1}^{n+\frac{1}{2}} \bigr\rangle . \end{aligned}$$
(49)
Using the Cauchy–Schwartz inequality, we obtain
$$\begin{aligned} & \bigl\langle F_{1}^{n+\frac{1}{2}}+F_{2}^{n+\frac{1}{2}},2 \tau e_{1}^{n+\frac {1}{2}} \bigr\rangle \leq \tau \bigl( \bigl\Vert F_{1}^{n+\frac{1}{2}} \bigr\Vert ^{2}+ \bigl\Vert F_{2}^{n+\frac {1}{2}} \bigr\Vert ^{2} \bigr)+ \frac{\tau}{2} \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{1}^{n} \bigr\Vert ^{2} \bigr), \\ & \bigl\vert \operatorname{Im} \bigl\langle F_{3}^{n+\frac{1}{2}},2 \tau e_{1}^{n+\frac{1}{2}} \bigr\rangle \bigr\vert \leq C_{20}\tau \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{1}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n} \bigr\Vert ^{2} \bigr), \\ & \bigl\langle \xi^{n}, 2\tau e_{1}^{n+\frac{1}{2}} \bigr\rangle \leq\tau \bigl\Vert \xi^{n} \bigr\Vert ^{2}+ \frac{\tau}{2} \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{1}^{n} \bigr\Vert ^{2} \bigr). \end{aligned}$$
It is easy to see that
$$\begin{aligned} & \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}- \bigl\Vert e_{1}^{n} \bigr\Vert ^{2} \bigr) \\ &\quad\leq C_{21}\tau \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{1}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n} \bigr\Vert ^{2}+ \bigl\Vert F_{1}^{n+\frac{1}{2}} \bigr\Vert ^{2}+ \bigl\Vert F_{2}^{n+\frac {1}{2}} \bigr\Vert ^{2}+ \bigl\Vert \xi^{n} \bigr\Vert ^{2} \bigr). \end{aligned}$$
(50)
Computing the inner product of Eqs. (46) and (47) with \(2\alpha\tau e_{2}^{n+\frac{1}{2}},2\tau e_{3}^{n+\frac{1}{2}}\), respectively, we obtain
$$\begin{aligned} & \bigl\langle \eta^{n},2\alpha\tau e_{2}^{n+\frac{1}{2}} \bigr\rangle = \bigl\langle \bigl(e_{2}^{n} \bigr)_{t}-D_{2}e_{3}^{n+\frac{1}{2}},2\alpha\tau e_{2}^{n+\frac {1}{2}} \bigr\rangle , \end{aligned}$$
(51)
$$\begin{aligned} & \bigl\langle \rho^{n},2\tau e_{3}^{n+\frac{1}{2}} \bigr\rangle = \bigl\langle \bigl(e_{3}^{n} \bigr)_{t}-e_{2}^{n+\frac{1}{2}}+ \alpha D_{2}e_{2}^{n+\frac{1}{2}}-G^{n+\frac {1}{2}},2 \tau e_{3}^{n+\frac{1}{2}} \bigr\rangle . \end{aligned}$$
(52)
Noting that
$$\begin{aligned} & \bigl\langle \eta^{n}, 2\alpha\tau e_{2}^{n+\frac{1}{2}} \bigr\rangle \leq\alpha\tau \bigl\Vert \eta^{n} \bigr\Vert ^{2}+\frac{\alpha\tau}{2} \bigl( \bigl\Vert e_{2}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n} \bigr\Vert ^{2} \bigr), \\ & \bigl\langle \rho^{n},2\tau e_{3}^{n+\frac{1}{2}} \bigr\rangle \leq\tau \bigl\Vert \rho^{n} \bigr\Vert ^{2}+ \frac{\tau}{2} \bigl( \bigl\Vert e_{3}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n} \bigr\Vert ^{2} \bigr), \\ & \bigl\langle G_{1}^{n+\frac{1}{2}}+G_{2}^{n+\frac{1}{2}},2 \tau e_{3}^{n+\frac {1}{2}} \bigr\rangle \leq\tau \bigl( \bigl\Vert G_{1}^{n+\frac{1}{2}} \bigr\Vert ^{2}+ \bigl\Vert G_{2}^{n+\frac{1}{2}} \bigr\Vert ^{2} \bigr)+ \frac{\tau}{2} \bigl( \bigl\Vert e_{3}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n} \bigr\Vert ^{2} \bigr), \\ & \bigl\vert \bigl\langle G_{3}^{n+\frac{1}{2}},2\tau e_{3}^{n+\frac{1}{2}} \bigr\rangle \bigr\vert \leq C_{22} \tau \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{1}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n} \bigr\Vert ^{2} \bigr), \end{aligned}$$
and adding (51)–(52), we get
$$\begin{aligned} & \bigl( \bigl\Vert e_{2}^{n+1} \bigr\Vert ^{2}- \bigl\Vert e_{2}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n+1} \bigr\Vert ^{2}- \bigl\Vert e_{3}^{n} \bigr\Vert ^{2} \bigr) \\ &\quad\leq C_{23}\tau \bigl( \bigl\Vert G_{1}^{n+\frac{1}{2}} \bigr\Vert ^{2}+ \bigl\Vert G_{2}^{n+\frac{1}{2}} \bigr\Vert ^{2}+ \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{1}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n} \bigr\Vert ^{2} \\ &\qquad{}+ \bigl\Vert e_{3}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n} \bigr\Vert ^{2}+ \bigl\Vert \eta^{n} \bigr\Vert ^{2}+ \bigl\Vert \rho^{n} \bigr\Vert ^{2} \bigr). \end{aligned}$$
Add the equations and (50), we have
$$\begin{aligned} & \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}- \bigl\Vert e_{1}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n+1} \bigr\Vert ^{2}- \bigl\Vert e_{2}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n+1} \bigr\Vert ^{2}- \bigl\Vert e_{3}^{n} \bigr\Vert ^{2} \bigr) \\ &\quad\leq C_{22}\tau \bigl( \bigl\Vert e_{1}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{1}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{2}^{n} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n+1} \bigr\Vert ^{2}+ \bigl\Vert e_{3}^{n} \bigr\Vert ^{2} \\ &\qquad{}+ \bigl\Vert \eta^{n} \bigr\Vert ^{2}+ \bigl\Vert \rho^{n} \bigr\Vert ^{2}+ \bigl\Vert F_{1}^{n+\frac{1}{2}} \bigr\Vert ^{2}+ \bigl\Vert F_{2}^{n+\frac{1}{2}} \bigr\Vert ^{2}+ \bigl\Vert G_{1}^{n+\frac {1}{2}} \bigr\Vert ^{2}+ \bigl\Vert G_{2}^{n+\frac{1}{2}} \bigr\Vert ^{2} \bigr). \end{aligned}$$
Let \(B^{n}=\|e_{1}^{n}\|^{2}+\|e_{2}^{n}\|^{2}+\|e_{3}^{n}\|^{2}\), we can obtain
$$\begin{aligned} B^{n+1}-B^{n}\leq{}& C_{24}\tau \bigl( \bigl\Vert \xi^{n} \bigr\Vert ^{2}+ \bigl\Vert \eta^{n} \bigr\Vert ^{2}+ \bigl\Vert F_{1}^{n} \bigr\Vert ^{2}+ \bigl\Vert F_{2}^{n} \bigr\Vert ^{2}+ \bigl\Vert G_{1}^{n} \bigr\Vert ^{2}+ \bigl\Vert G_{2}^{n} \bigr\Vert ^{2} \bigr) \\ &{}+C_{22}\tau \bigl( B^{n+1}+B^{n} \bigr). \end{aligned}$$
It follows from Gronwall’s inequality [24] that
$$\begin{aligned} \max_{1\leq n\leq N}B^{n}\leq \bigl(B^{0}+C_{25}T \bigl(J^{-2r}+\tau^{4} \bigr) \bigr)e^{4C_{24}T}. \end{aligned}$$
Noting that \(B^{0}\leq C_{26} J^{-r}\), we can obtain
$$\bigl\Vert e_{1}^{n} \bigr\Vert \leq C_{27} \bigl(J^{-r}+ \tau^{2} \bigr), \qquad \bigl\Vert e_{2}^{n} \bigr\Vert \leq C_{27} \bigl(J^{-r}+\tau^{2} \bigr), \qquad \bigl\Vert e_{3}^{n} \bigr\Vert \leq C_{27} \bigl(J^{-r}+\tau^{2} \bigr). $$
Finally, we prove \(|e_{1}^{n}|_{h}=O(J^{-r}+\tau^{2}), |e_{2}^{n}|_{h}=O(J^{-r}+\tau^{2}), |e_{3}^{n}|_{h}=O(J^{-r}+\tau^{2})\). Computing the inner product of Eqs. (46) and (47) with \(e_{3}^{n+1}-e_{3}^{n}, e_{2}^{n+1}-e_{2}^{n}\), respectively, we obtain
$$\begin{aligned} & \bigl\langle \eta^{n},e_{3}^{n+1}-e_{3}^{n} \bigr\rangle = \bigl\langle \bigl(e_{2}^{n} \bigr)_{t}-D_{2}e_{3}^{n+\frac{1}{2}},e_{3}^{n+1}-e_{3}^{n} \bigr\rangle , \end{aligned}$$
(53)
$$\begin{aligned} & \bigl\langle \rho^{n},e_{2}^{n+1}-e_{2}^{n} \bigr\rangle = \bigl\langle \bigl(e_{3}^{n} \bigr)_{t}-e_{2}^{n+\frac {1}{2}}+ \alpha D_{2}e_{2}^{n+\frac{1}{2}}-G^{n+\frac {1}{2}},e_{2}^{n+1}-e_{2}^{n} \bigr\rangle . \end{aligned}$$
(54)
It follows from (46) that
$$\begin{aligned} \bigl\langle -G^{n+\frac{1}{2}},e_{2}^{n+1}-e_{2}^{n} \bigr\rangle = \bigl\langle -G^{n+\frac {1}{2}},\tau\eta^{n}+\tau D_{2}e_{3}^{n+\frac{1}{2}} \bigr\rangle =\tau \bigl[ \bigl\langle -G^{n+\frac{1}{2}}, \eta^{n} \bigr\rangle + \bigl\langle -G^{n+\frac {1}{2}},D_{2}e_{3}^{n+\frac{1}{2}} \bigr\rangle \bigr]. \end{aligned}$$
Noting that
$$\begin{aligned} & \bigl\vert \bigl\langle -G^{n+\frac{1}{2}}, D_{2}e_{3}^{n+\frac{1}{2}} \bigr\rangle \bigr\vert \leq C_{28} \bigl( \bigl\vert G^{n+\frac{1}{2}} \bigr\vert _{h}+ \bigl\vert e_{3}^{n+1} \bigr\vert _{h}+ \bigl\vert e_{3}^{n} \bigr\vert _{h} \bigr), \\ & \bigl\vert G^{n+\frac{1}{2}} \bigr\vert _{h}\leq C_{28} \bigl( \bigl\vert e_{1}^{n+1} \bigr\vert _{h}+ \bigl\vert e_{1}^{n} \bigr\vert _{h}+ \bigl\vert e_{2}^{n+1} \bigr\vert _{h}+ \bigl\vert e_{2}^{n} \bigr\vert _{h}+O \bigl(\tau^{2}+J^{-r} \bigr) \bigr), \end{aligned}$$
we get
$$\begin{aligned} & \bigl\vert e_{2}^{n+1} \bigr\vert _{h}^{2}- \bigl\vert e_{2}^{n} \bigr\vert _{h}^{2}+ \bigl\vert e_{3}^{n+1} \bigr\vert _{h}^{2}- \bigl\vert e_{3}^{n} \bigr\vert _{h}^{2} \\ &\quad\leq C_{29} \bigl( \bigl\vert e_{1}^{n+1} \bigr\vert _{h}^{2}+ \bigl\vert e_{1}^{n} \bigr\vert _{h}^{2}+ \bigl\vert e_{2}^{n+1} \bigr\vert _{h}^{2}+ \bigl\vert e_{2}^{n} \bigr\vert _{h}^{2}+ \bigl\vert e_{3}^{n+1} \bigr\vert _{h}^{2}+ \bigl\vert e_{3}^{n} \bigr\vert _{h}^{2} \bigr) \\ &\qquad{}+O \bigl(\tau^{2}+J^{-r} \bigr)^{2}+ \bigl\vert \bigl\langle \eta^{n},e_{3}^{n+1}-e_{3}^{n} \bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \rho^{n},e_{2}^{n+1}-e_{2}^{n} \bigr\rangle \bigr\vert . \end{aligned}$$
(55)
Computing the inner product of Eqs. (45) with \(e_{1}^{n+1}-e_{1}^{n}\), we obtain
$$\begin{aligned} \bigl\langle \xi^{n},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle = \bigl\langle i \bigl(e_{1}^{n} \bigr)_{t}+D_{2}e_{1}^{n+\frac {1}{2}}-F^{n+\frac{1}{2}},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle . \end{aligned}$$
(56)
Taking the real part of Eq. (56), we get
$$\begin{aligned} \operatorname{Re} \bigl\langle -D_{2}e_{1}^{n+\frac{1}{2}},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle =-\operatorname{Re} \bigl\langle \xi ^{n},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle -\operatorname{Re} \bigl\langle F^{n+\frac {1}{2}},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle . \end{aligned}$$
It follows from Lemma 2.8 and Eq. (45) that
$$\begin{aligned} &\operatorname{Re} \bigl\langle -D_{2}e_{1}^{n+\frac{1}{2}},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle =\frac {1}{2} \bigl( \bigl\vert e_{1}^{n+1} \bigr\vert _{h}- \bigl\vert e_{1}^{n} \bigr\vert _{h} \bigr), \\ &\operatorname{Re} \bigl\langle F^{n+\frac{1}{2}},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle =\tau \operatorname{Re} \bigl\langle F^{n+\frac{1}{2}},iD_{2}e_{1}^{n+\frac{1}{2}}-iF^{n+\frac{1}{2}}-i \xi ^{n} \bigr\rangle \\ &\phantom{\operatorname{Re} \langle F^{n+\frac{1}{2}},e_{1}^{n+1}-e_{1}^{n} \rangle} =\tau \operatorname{Im} \bigl\langle F^{n+\frac{1}{2}},D_{2}e_{1}^{n+\frac{1}{2}}- \xi^{n} \bigr\rangle . \end{aligned}$$
Then we get
$$\begin{aligned} & \bigl\vert e_{1}^{n+1} \bigr\vert _{h}^{2}- \bigl\vert e_{1}^{n} \bigr\vert _{h}^{2} \\ &\quad=C_{30} \bigl( \bigl\vert e_{1}^{n+1} \bigr\vert _{h}^{2}+ \bigl\vert e_{1}^{n} \bigr\vert _{h}^{2}+ \bigl\vert e_{2}^{n+1} \bigr\vert _{h}^{2}+ \bigl\vert e_{2}^{n} \bigr\vert _{h}^{2} \bigr)+O \bigl(\tau ^{2}+J^{-r} \bigr)^{2}+ \bigl\vert \bigl\langle \xi^{n},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle \bigr\vert . \end{aligned}$$
(57)
Adding Eqs. (55)–(57), we get
$$\begin{aligned} & \bigl\vert e_{1}^{n+1} \bigr\vert _{h}^{2}- \bigl\vert e_{1}^{n} \bigr\vert _{h}^{2}+ \bigl\vert e_{2}^{n+1} \bigr\vert _{h}^{2}- \bigl\vert e_{2}^{n} \bigr\vert _{h}^{2}+ \bigl\vert e_{3}^{n+1} \bigr\vert _{h}^{2}- \bigl\vert e_{3}^{n} \bigr\vert _{h}^{2} \\ &\quad\leq C_{31} \bigl( \bigl\vert e_{1}^{n+1} \bigr\vert _{h}^{2}+ \bigl\vert e_{1}^{n} \bigr\vert _{h}^{2}+ \bigl\vert e_{2}^{n+1} \bigr\vert _{h}^{2}+ \bigl\vert e_{2}^{n} \bigr\vert _{h}^{2}+ \bigl\vert e_{3}^{n+1} \bigr\vert _{h}^{2}+ \bigl\vert e_{3}^{n} \bigr\vert _{h}^{2} \bigr) \\ &\qquad{}+O \bigl(\tau^{2}+J^{-r} \bigr)^{2}+ \bigl\vert \bigl\langle \xi^{n},e_{1}^{n+1}-e_{1}^{n} \bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \eta ^{n},e_{3}^{n+1}-e_{3}^{n} \bigr\rangle \bigr\vert + \bigl\vert \bigl\langle \rho^{n},e_{2}^{n+1}-e_{2}^{n} \bigr\rangle \bigr\vert . \end{aligned}$$
According to Gronwall’s inequality [24], we get \(|e_{1}^{n}|_{h}=O(J^{-r}+\tau^{2}), |e_{2}^{n}|_{h}=O(J^{-r}+\tau^{2}), |e_{3}^{n}|_{h}=O(J^{-r}+\tau^{2})\). □

3 Iterative algorithm

In order to derive the algorithms conveniently, we also give some notations:
$$\begin{aligned} \begin{aligned} &U_{j}^{n}=\sum _{l=-\frac{J}{2}}^{\frac{J}{2}-1} \widehat{U}_{l}^{n}e^{i\mu _{l}(x_{j}-a)}, \quad j=0,1,\ldots,J-1, \\ & \widehat{U}_{j}^{n}=\frac{1}{J}\sum _{j=0}^{J-1}U_{j}^{n}e^{i\mu_{l}(x_{j}-a)}, \quad l= \frac{J}{2},\ldots,\frac{J}{2}-1, \end{aligned} \end{aligned}$$
(58)
$$\begin{aligned} \begin{aligned} &V_{j}^{n}=\sum _{l=-\frac{J}{2}}^{\frac{J}{2}-1} \widehat{V}_{l}^{n}e^{i\mu _{l}(x_{j}-a)}, \quad j=0,1,\ldots,J-1, \\ & \widehat{V}_{j}^{n}=\frac{1}{J}\sum _{j=0}^{J-1}V_{j}^{n}e^{i\mu_{l}(x_{j}-a)}, \quad l= \frac{J}{2},\ldots,\frac{J}{2}-1, \end{aligned} \end{aligned}$$
(59)
$$\begin{aligned} \begin{aligned} &\Phi_{j}^{n}=\sum _{l=-\frac{J}{2}}^{\frac{J}{2}-1} \widehat{\Phi}_{l}^{n}e^{i\mu _{l}(x_{j}-a)}, \quad j=0,1,\ldots,J-1, \\ &\widehat{\Phi}_{j}^{n}=\frac{1}{J}\sum _{j=0}^{J-1}\Phi_{j}^{n}e^{i\mu_{l}(x_{j}-a)}, \quad l= \frac{J}{2},\ldots,\frac{J}{2}-1. \end{aligned} \end{aligned}$$
(60)
Substituting (58)–(60) into (13)–(15), we can obtain
$$\begin{aligned} &i\frac{\widehat{U}_{l}^{n+1}-\widehat{U}_{l}^{n}}{\tau}-\frac{\mu _{l}^{2}}{2} \bigl(\widehat{U}_{l}^{n+1}+ \widehat{U}_{l}^{n} \bigr)-\frac{1}{4}\widehat { \bigl(U_{j}^{n+1}+U_{j}^{n} \bigr) \bigl(V_{j}^{n+1}+V_{j}^{n} \bigr)}=0, \\ &\frac{\widehat{V}_{l}^{n+1}-\widehat{V}_{l}^{n}}{\tau}=-\frac{\mu _{l}^{2}}{2} \bigl(\widehat{\Phi}_{l}^{n+1}+ \widehat{\Phi}_{l}^{n} \bigr), \\ &\frac{\widehat{\Phi}_{l}^{n+1}-\widehat{\Phi}_{l}^{n}}{\tau}=\frac {1}{2} \bigl(\widehat{V}_{l}^{n+1}+ \widehat{V}_{l}^{n} \bigr)+\frac{\alpha\mu _{l}^{2}}{2} \bigl( \widehat{V}_{l}^{n+1}+\widehat{V}_{l}^{n} \bigr)-\mu_{l}^{2}\widehat{\frac {F(V_{j}^{n+1})-F(V_{j}^{n})}{V_{j}^{n+1}-V_{j}^{n}}} \\ &\phantom{\frac{\widehat{\Phi}_{l}^{n+1}-\widehat{\Phi}_{l}^{n}}{\tau}=}{}+ \mu_{l}^{2}\frac{\omega}{2}\widehat{ \bigl\vert U_{j}^{n+1} \bigr\vert ^{2}+ \bigl\vert U_{j}^{n} \bigr\vert ^{2}}. \end{aligned}$$
By direct calculation, we get
$$\begin{aligned} &\widehat{U}_{l}^{n+1}=\frac{i+\frac{\mu_{l}^{2}\tau}{2}}{i-\frac{\mu _{l}^{2}\tau}{2}} \widehat{U}_{l}^{n}+\frac{1}{4}\widehat { \bigl(U_{j}^{n+1}+U_{j}^{n} \bigr) \bigl(V_{j}^{n+1}+V_{j}^{n} \bigr)}, \\ & \widehat{V}_{l}^{n+1}=\frac{1}{\frac{\tau}{2}+\frac{\alpha\mu_{l}^{2}\tau }{2}+\frac{2}{\mu_{l}^{2}\tau}} \biggl[ \biggl( \frac{2}{\mu_{l}^{2}\tau}-\frac{\tau}{2}-\frac {\alpha\mu_{l}^{2}\tau}{2} \biggr) \widehat{V}_{l}^{n}-2\widehat{\Phi}_{l}^{n} \\ &\phantom{\widehat{V}_{l}^{n+1}=}+ \mu_{l}^{2}\tau\widehat{ \frac{F(V_{j}^{n+1})-F(V_{j}^{n})}{V_{j}^{n+1}-V_{j}^{n}}}- \frac{\mu_{l}^{2}\tau\omega}{2}\widehat{ \bigl\vert U_{j}^{n+1} \bigr\vert ^{2}+ \bigl\vert U_{j}^{n} \bigr\vert ^{2}} \biggr], \\ &\widehat{\Phi}_{l}^{n+1}=\frac{2}{\mu_{l}^{2}\tau}\widehat {V}_{l}^{n}-\widehat{\Phi}_{l}^{n}- \frac{2}{\mu_{l}^{2}\tau}\widehat{V}_{l}^{n+1}. \end{aligned}$$
Let \(U_{j}^{n+1,0}=U_{j}^{n},V_{j}^{n+1,0}=V_{j}^{n},\Phi_{j}^{n+1,0}=\Phi _{j}^{n}\). We use the following iterative method to solve the algebraic systems:
$$\begin{aligned} &\widehat{U}_{l}^{n+1,s+1}=\frac{l+\frac{\mu_{l}^{2}\tau}{2}}{i-\frac{\mu _{l}^{2}\tau}{2}} \widehat{U}_{j}^{n}+\frac{1}{4}\widehat { \bigl(U_{j}^{n+1,s}+U_{j}^{n} \bigr) \bigl(V_{j}^{n+1,s}+V_{j}^{n} \bigr)}, \\ & \widehat{V}_{l}^{n+1,s+1}=\frac{1}{\frac{\tau}{2}+\frac{\alpha\mu _{l}^{2}\tau}{2}+\frac{2}{\mu_{l}^{2}\tau}} \biggl[ \biggl( \frac{2}{\mu_{l}^{2}\tau}-\frac{\tau }{2}-\frac{\alpha\mu_{l}^{2}\tau}{2} \biggr) \widehat{V}_{l}^{n}-2\widehat{\Phi}_{l}^{n} \\ &\phantom{\widehat{V}_{l}^{n+1,s+1}=}{}+ \mu_{l}^{2}\tau\widehat{ \frac{F(V_{j}^{n+1,s})-F(V_{j}^{n})}{V_{j}^{n+1,s}-V_{j}^{n}}}- \frac{\mu_{l}^{2}\tau\omega}{2}\widehat{ \bigl\vert U_{j}^{n+1,s+1} \bigr\vert ^{2}+ \bigl\vert U_{j}^{n} \bigr\vert ^{2}} \biggr], \\ &\widehat{\Phi}_{l}^{n+1,s+1}=\frac{2}{\mu_{l}^{2}\tau}\widehat {V}_{l}^{n}-\widehat{\Phi}_{l}^{n}- \frac{2}{\mu_{l}^{2}\tau}\widehat{V}_{l}^{n+1,s+1}. \end{aligned}$$

4 Numerical example

Taking \(f(v)=\frac{1}{2}v^{2}, \alpha=1, \omega=\frac{1}{2}\), then we consider the following initial condition:
$$\begin{aligned} &u(x,0)=u_{0}(x)=\frac{9}{10}\operatorname{sec}h^{2} \biggl(\frac{\sqrt{15}}{10}x \biggr)\exp \biggl(i\frac {\sqrt{10}}{10}x \biggr), \\ &v(x,0)=v_{0}(x)=-\frac{9}{10}\operatorname{sec}h^{2} \biggl( \frac{\sqrt{15}}{10}x \biggr), \\ &v_{t}(x,0)=v_{1}(x)=-\frac{9\sqrt{6}}{50} \operatorname{sec}h^{2} \biggl(\frac{\sqrt {15}}{10}x \biggr)\tanh \biggl( \frac{\sqrt{15}}{10}x \biggr). \end{aligned}$$
The forms of the exact solutions are available as
$$\begin{aligned} &u(x,t)=\frac{9}{10}\operatorname{sec}h^{2} \biggl( \frac{\sqrt{15}}{10} \biggl(x-\frac{2\sqrt {10}}{10}t \biggr) \biggr)\exp \biggl(i \biggl( \frac{\sqrt{10}}{10}x+\frac{1}{2}t \biggr) \biggr), \\ &v(x,t)=-\frac{9}{10}\operatorname{sec}h^{2} \biggl( \frac{\sqrt{15}}{10} \biggl(x- \frac{2\sqrt {10}}{10}t \biggr) \biggr), \\ &v_{t}(x,t)=-\frac{9\sqrt{6}}{50}\operatorname{sec}h^{2} \biggl( \frac{\sqrt{15}}{10} \biggl(x-\frac {2\sqrt{10}}{10}t \biggr) \biggr)\tanh \biggl( \frac{\sqrt{15}}{10} \biggl(x-\frac{2\sqrt{10}}{10}t \biggr) \biggr). \end{aligned}$$
Firstly, the numerical accuracy of the scheme (13)–(17) is examined. Table 1 shows the time errors and the convergence orders in \(l_{h}^{2}\) norms and \(l_{h}^{\infty}\) norms of the scheme (13)–(17), respectively. The data in Table 1 indicate that the scheme (13)–(17) are of second order in time, and confirm the theoretical accuracy in Theorem 3.5. Then we fix the time step \(\tau=0.001\) to test the space accuracy of the scheme (13)–(17). The results are listed in Table 2. From Table 2, it is found that the errors decrease as fast as the number of grid points J increases.
Table 1

Errors and orders in time for \(J=1024, t=1\)

  

τ

0.05

0.025

0.0125

0.00625

u

\(l_{h}^{2}\)

error

8.750 × 10−5

2.188 × 10−5

5.470 × 10−6

1.367 × 10−6

order

*

1.99

2.00

2.00

\(l_{h}^{\infty}\)

error

4.179 × 10−5

1.060 × 10−5

2.692 × 10−6

6.765 × 10−7

order

*

1.97

1.97

2.00

v

\(l_{h}^{2}\)

error

1.150 × 10−4

2.880 × 10−5

7.211 × 10−6

1.803 × 10−6

order

*

1.99

1.99

1.99

\(l_{h}^{\infty}\)

error

7.487 × 10−5

1.92 × 10−5

4.862 × 10−6

1.223 × 10−6

order

*

1.96

1.98

1.99

Table 2

Errors and orders in space for \(\tau =0.001, t=1\)

  

J

64

128

256

512

u

\(l_{h}^{2}\)

error

5.076 × 10−5

3.501 × 10−8

3.5015 × 10−8

3.5015 × 10−8

\(l_{h}^{\infty}\)

error

1.982 × 10−5

1.425 × 10−8

1.739 × 10−8

1.739 × 10−8

v

\(l_{h}^{2}\)

error

1.654 × 10−5

4.615 × 10−8

4.615 × 10−8

4.615 × 10−8

\(l_{h}^{\infty}\)

error

9.350 × 10−5

3.146 × 10−8

3.146 × 10−8

3.146 × 10−8

Secondly, we test the numerical performance for the long time computation with \(x \in[-15, 15], t \in[0, 100], \tau=0.01, J=256 \). Figure 1 show the errors of mass \(M^{n}\) and energy \(E^{n}\) at different time for the scheme (13)–(17). From Fig. 1, we find that the scheme (13)–(17) preserves the mass and energy conservation very well.
Figure 1
Figure 1

Errors of mass \(M^{n}\) (left column) and energy \(E^{n}\) (right column) of the numerical solutions

Finally, the numerical solutions for the system (13)–(17) are depicted. We simulate the solitary wave solutions with \(x \in[-15, 15], t \in [0, 10], \tau=0.01, J=256\). Figures 24 show the wave forms of the numerical solutions.
Figure 2
Figure 2

The wave forms of the numerical solutions for \(t=3\)

Figure 3
Figure 3

The wave forms of the numerical solutions for \(t=6\)

Figure 4
Figure 4

The wave forms of the numerical solutions

5 Conclusion

In the paper, we introduce a conservative Fourier spectral scheme to solve the CSB system. We give the iterative algorithm of the scheme and prove that the scheme preserves the mass and energy conservation laws. The convergence of the scheme is discussed, and it is shown that the scheme is of the accuracy \(O(\tau^{2}+J^{-r})\). Numerical tests are presented to demonstrate the obtained theoretical results and the method availability.

Declarations

Funding

This work is supported by Pu’er University innovation team (CXTD003).

Authors’ contributions

All authors read and approved the final manuscript.

Competing interests

The author declares to have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistical, Pu’er University, Yunnan, China

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