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
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 \(|e_{1}^{n}|_{h}=O(J^{-r}+\tau^{2}), |e_{2}^{n}|_{h}=O(J^{-r}+\tau^{2}), |e_{3}^{n}|_{h}=O(J^{-r}+\tau^{2})\). □