TT-M Finite Element Algorithm for the Coupled Schrödinger–Boussinesq Equations

Abstract

In this article, the coupled Schrödinger–Boussinesq equations are solved numerically using the finite element method combined with the time two-mesh (TT-M) fast algorithm. The spatial direction is discretized by the standard Galerkin finite element method, the temporal direction is approximated by the TT-M Crank–Nicolson scheme, and then the numerical scheme of TT-M finite element (FE) system is formulated. The method includes three main steps: for the first step, the nonlinear system is solved on the coarse time mesh; for the second step, by an interpolation formula, the numerical solutions at the fine time mesh point are computed based on the numerical solutions on the coarse mesh system; for the last step, the linearized temporal fine mesh system is constructed based on Taylor’s formula for two variables, and then the TT-M FE solutions can be obtained. Furthermore, theory analyses on the TT-M system including the stability and error estimations are conducted. Finally, a large number of numerical examples are provided to verify the accuracy of the algorithm, the correctness of theoretical results, and the computational efficiency with a comparison to the numerical results calculated by using the standard FE method.

1. Introduction

The coupled Schrödinger–Boussinesq equations have often been proposed to describe many complex physical processes, such as the dynamic behaviors of the Langmuir soliton formation and the interaction of long-wave with short-wave packets in diatomic lattice systems [1] and nonlinear dispersive media [2,3], where the Schrödinger equation describes the short-wave term and the Boussinesq equation describes the long-wave term. Many theoretical studies have been conducted on this important equation. Guo and Du studied the attractor and its regularity of the damped Schrödinger–Boussinesq equations [4]. The new exact traveling wave solutions of the coupled Schrödinger–Boussinesq equations were constructed, which are expressed by the hyperbolic functions, the trigonometric functions and the rational functions in [5]. Guo studied the existence and uniqueness of global solutions in [6]. Li and Chen proved the existence of global attractors of dissipative Schrödinger–Boussinesq equations [7]. Many other scholars have done a significant amount of work to study the analytical solution of the Schrödinger–Boussinesq equations [8,9,10,11,12,13]. In recent years, many scholars have studied the numerical methods of the Schrödinger–Boussinesq equations. In [3], Liao et al. constructed cubic orthogonal spline collocation schemes to solve Schrödinger–Boussinesq equations, gave conservation laws for schemes, and proved the convergence and stability of the nonlinear scheme by discrete energy methods. In [14], Liao et al. considered the time-splitting Fourier spectral discretization method for solving Schrödinger–Boussinesq equations. In [15], Zhang and Bai developed a conservative difference method for solving Schrödinger–Boussinesq equations, gave the calculation procedure for the numerical solution, and analyzed the related properties. Liao and Zhang [16] developed a conservative compact difference scheme for the N-coupled nonlinear Schrödinger–Boussinesq equations and proved the scheme is unconditionally convergent in the maximum norm. Deng and Wu [17] studied a finite difference method for one-dimensional and two-dimensional nonlinear coupled Schrödinger–Boussinesq equations. In [18], Zheng and Xiang studied the finite element algorithm of coupled Schrödinger–Boussinesq equations and its error theory. Bai and Zhang proposed a quadratic B-spline finite-element method for the coupled Schrödinger–Boussinesq equations and gave an error analysis based on the semi-discrete finite element scheme of the equations in [19]. Other numerical methods to solve the Schrödinger–Boussinesq equations can be found in [20,21,22]. It can be seen from a large number of studies that although there are many numerical methods for solving the coupled Schrödinger–Boussinesq equations, there are few studies on finite element algorithms with fast computing techniques.

In 2018, Liu and his collaborators proposed the fast time two-mesh (TT-M) finite element method in [23] for numerically solving the fractional water wave model. They provided rigorous theory analyses including stability and error estimations, verified the effectiveness of the method by a large number of calculation data, and illustrated the calculation efficiency by making a comparison with the standard nonlinear finite element algorithm. It is easy to find that the TT-M algorithm can save computing time and improve the calculation efficiency to a great extent, and it can be combined with many other algorithms, as summarized in [23]. In view of the advantages of its fast calculation and maintaining calculation accuracy, this method has been developed rapidly. Subsequently, in 2019, Yin et al. developed the TT-M finite element algorithm for a space fractional Allen–Cahn model in [24] and discussed the problem of parameter selection in detail. In [25], Liu et al. used the TT-M finite element method to numerically solve the two-dimensional Gray–Scott model with space fractional derivatives. In [26], Wen et al. used the TT-M algorithm combined with the $H^1$-Galerkin mixed finite element method to numerically solve the nonlinear distributed order diffusion model and verified the computational efficiency of the algorithm and the correctness of the theoretical results by numerical examples with smooth and non-smooth solutions. In [27], Wang et al. used the TT-M fast algorithm based on the finite element method to overcome the time-consuming problem caused by the nonlinear term of the strongly nonlinear Allen–Cahn equation. Qiu et al. [28] studied the difference scheme based on the TT-M algorithm to numerically solve the fractional moving/non-moving transport model. Recently, Niu et al. developed a TT-M fourth-order compact difference scheme in [29] and numerically solved the nonlinear distributed fractional Sobolev equation model.

Here, we will develop a TT-M finite element fast algorithm for the initial and boundary value problems of the coupled Schrödinger–Boussinesq equations

$$i\epsilon E_t+\gamma E_{xx}=\lambda NE,(x,t)\in (a,b)\times (0,T],$$

(1)

$$N_{tt}-N_{xx}+\alpha N_{xxxx}-\theta {\left(N^2\right)}_{xx}={\omega \left(\right|E|}^2{)}_{xx},(x,t)\in (a,b)\times (0,T],$$

(2)

$$E(x,0)=E_0\left(x\right),N(x,0)=N_0\left(x\right),N_t(x,0)=N_1\left(x\right),x\in [a,b],$$

(3)

$$E(a,t)=E(b,t)=0,N(a,t)=N(b,t)=0,t\in [0,T],$$

(4)

where E is the short-wave amplitude, and N is the long-wave amplitude. E and N are complex functions and real functions, respectively. $\epsilon ={(m_e/m_i)}^{1/2}$, $m_e$ and $m_i$ are the electronic quality and the ion quality. Parameters $\alpha$, $\theta$, $\gamma$, $\lambda$ and $\omega$ are five positive coefficients. $E_0\left(x\right)$, $N_0\left(x\right)$ and $N_1\left(x\right)$ are given initial functions.

Now, by introducing the following unknown function $\Phi (x,t)$ found in many references,

$${\Phi }_x(x,t)={\int }_a^x{N}_t(y,t)dy,$$

(5)

one can write the original problem (1)–(4) as follows

$$i\epsilon E_t+\gamma E_{xx}=\lambda NE,(x,t)\in (a,b)\times (0,T],$$

(6)

$$N_t={\Phi }_{xx},(x,t)\in (a,b)\times (0,T],$$

(7)

$${\Phi }_t-N+\alpha N_{xx}-\theta N^2=\omega {\left|E\right|}^2,(x,t)\in (a,b)\times (0,T],$$

(8)

$$E(x,0)=E_0\left(x\right),N(x,0)=N_0\left(x\right),\Phi (x,0)={\Phi }_0\left(x\right),x\in [a,b],$$

(9)

$$E(a,t)=E(b,t)=0,N(a,t)=N(b,t)=0,\Phi (a,t)=\Phi (b,t)=0,t\in [0,T],$$

(10)

where $\Phi (x,0)={\Phi }_0\left(x\right)$ is yielded by (5).

In this article, we use the finite element method combined with the TT-M fast algorithm to solve the coupled Schrödinger–Boussinesq equations. The TT-M Crank–Nicolson fast algorithm is used to discretize the time direction and speed up the calculation, and the Galerkin finite element method is used to approximate the space direction. The main work of this article is as follows:

• Detailed proofs of stability and error estimates for TT-M FE system are done;
• Numerical examples are given to verify the correctness of the theory results, to reflect the dynamic behavior of solutions and to illustrate the computational efficiency compared with the standard nonliear finite element method.

The remaining structure of this article is as follows: in Section 2, we derive the weak formulation, finite element scheme and the time two-mesh numerical scheme. In Section 3, we give the stability analysis of the TT-M FE system. In Section 4, we derive the optimal error results based on the TT-M system. In Section 5, we carry out the numerical computing by taking some numerical examples. Finally, in Section 6, we provide some discussions for our algorithm and the future studies.

2. Weak Formulation and TT-M Finite Element Scheme

2.1. Weak Formulation

Let $\overline{\Omega }=[a,b]$. We denote the complex Sobolev space on $\Omega$ by the notation $H^r(\Omega )=W^{r,2}(\Omega )$,

$$H_0^1(\Omega )=\{u\in H^1(\Omega )|u\left(a\right)=u\left(b\right)=0\}.$$

(11)

We denote the inner product in space $L^2(\Omega )$ by $(·,·)$, the norm in space $L^2(\Omega )$ by $\parallel ·\parallel$ and the norm in space $H^r(\Omega )$ by ${\parallel ·\parallel }_r$,

$$\begin{array}{cc}\hfill & (u,v)={\int }_{\Omega }u\overline{v}dx,\parallel v\parallel ={(v,\overline{v})}^{1/2}={\left({\int }_{\Omega }v\overline{v}dx\right)}^{1/2},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & {\parallel v\parallel }_r={\parallel v\parallel }_{H^r}={\left(\sum _{\left|\alpha \right|\le r}{\parallel D^{\alpha }v\parallel }^2\right)}^{1/2},\hfill \end{array}$$

(13)

where $\overline{v}$ represents the complex conjugate of v.

Now, the weak formulation for (6)–(10) is to find N, $\Phi$, E$\in H_0^1(\Omega )$, for $\forall g\in H_0^1(\Omega )$ such that

$$i\epsilon (E_t,g)-\gamma (E_x,g_x)=\lambda (NE,g),$$

(14)

$$(N_t,g)=-({\Phi }_x,g_x),$$

(15)

$$({\Phi }_t,g)-(N,g)-\alpha (N_x,g_x)-\theta (N^2,g)={\omega \left(\right|E|}^2,g).$$

(16)

We use the Crank–Nicolson method and rewrite (14)–(16) as follows:

$$i\epsilon (E_t^{n+\frac{1}{2}},g)-\gamma (E_x^{n+\frac{1}{2}},g_x)=\lambda (N^{n+\frac{1}{2}}{E}^{n+\frac{1}{2}},g)+(e_{1\left(\tau \right)}^{n+\frac{1}{2}},g),$$

(17)

$$(N_t^{n+\frac{1}{2}},g)=-({\Phi }_x^{n+\frac{1}{2}},g_x)+(e_{2\left(\tau \right)}^{n+\frac{1}{2}},g),$$

(18)

$$({\Phi }_t^{n+\frac{1}{2}},g)-(N^{n+\frac{1}{2}},g)-\alpha (N_x^{n+\frac{1}{2}},g_x)-\theta ({\left(N^{n+\frac{1}{2}}\right)}^2,g)=\omega \left(\right|E^{n+\frac{1}{2}}{|}^2,g)+(e_{3\left(\tau \right)}^{n+\frac{1}{2}},g),$$

(19)

where the truncation errors $e_{1\left(\tau \right)}^{n+\frac{1}{2}}$, $e_{2\left(\tau \right)}^{n+\frac{1}{2}}$ and $e_{3\left(\tau \right)}^{n+\frac{1}{2}}$ are as follows:

$$e_{1\left(\tau \right)}^{n+\frac{1}{2}}=O\left({\tau }^2\right),e_{2\left(\tau \right)}^{n+\frac{1}{2}}=O\left({\tau }^2\right),e_{3\left(\tau \right)}^{n+\frac{1}{2}}=O\left({\tau }^2\right).$$

(20)

2.2. TT-M Finite Element Scheme

In order to give the fully discrete scheme of the TT-M FE method, we divide the time interval $[0,T]$ into a uniform partition with the nodes $t_n=nM\tau$$(n=0,1,\cdots ,P)$, satisfying $0=t_0<t_1<\cdots <t_P=T$ with a positive integer $2\le M\le \frac{1}{{\tau }_c}$, where $\tau =T/\left(PM\right)$ and ${\tau }_c=M\tau$ are the fine time step size and the coarse time step size, respectively. For convenience, we define $E_t^{n+\frac{1}{2}}=\frac{E^{n+1}-E^n}{{\Delta }_t}$ and $E^{n+\frac{1}{2}}=\frac{E^{n+1}+E^n}{2}$.

The standard nonlinear fully discrete FE scheme corresponding to the weak formulation (17)–(19) is as follows:

$$i\epsilon (E_{ht}^{n+\frac{1}{2}},g)-\gamma (E_{hx}^{n+\frac{1}{2}},g_x)=\lambda (N_h^{n+\frac{1}{2}}{E}_h^{n+\frac{1}{2}},g),$$

(21)

$$(N_{ht}^{n+\frac{1}{2}},g)=-({\Phi }_{hx}^{n+\frac{1}{2}},g_x),$$

(22)

$$({\Phi }_{ht}^{n+\frac{1}{2}},g)-(N_h^{n+\frac{1}{2}},g)-\alpha (N_{hx}^{n+\frac{1}{2}},g_x)-\theta ({\left(N_h^{n+\frac{1}{2}}\right)}^2,g)=\omega \left(\right|E_h^{n+\frac{1}{2}}{|}^2,g).$$

(23)

If the numerical calculation is carried out directly through the nonlinear fully discrete FE system (21)–(23), it will take a lot of calculation time. Therefore, we need to develop fast computing methods to improve the computing efficiency. Here, we consider the TT-M fast algorithm based on the FE method, which includes three main calculating steps (see [23]):

Step 1: The coarse time mesh system based on the coarse time step ${\tau }_c$ is solved directly by the iterative algorithm.

$$i\epsilon (E_{ct}^{n+\frac{1}{2}},g)-\gamma (E_{cx}^{n+\frac{1}{2}},g_x)=\lambda (N_c^{n+\frac{1}{2}}{E}_c^{n+\frac{1}{2}},g),$$

(24)

$$(N_{ct}^{n+\frac{1}{2}},g)=-({\Phi }_{cx}^{n+\frac{1}{2}},g_x),$$

(25)

$$({\Phi }_{ct}^{n+\frac{1}{2}},g)-(N_c^{n+\frac{1}{2}},g)-\alpha (N_{cx}^{n+\frac{1}{2}},g_x)-\theta ({\left(N_c^{n+\frac{1}{2}}\right)}^2,g)=\omega \left(\right|E_c^{n+\frac{1}{2}}{|}^2,g).$$

(26)

Step 2: Now, we need to expand the nodes and the corresponding point values. We let m be the index of the fine time mesh, $m=0,1,\cdots ,M-1,M,\cdots ,MP$. Taking $n=\lceil \frac{m}{M}\rceil$ to be the smallest integer that is equal to or greater than $\frac{m}{M}$, we use an interpolation formula and obtain

$$E_I^m={\lambda }_m{E}_c^{n-1}+(1-{\lambda }_m)E_c^n,$$

(27)

where ${\lambda }_m=n-\frac{m}{M}\in [0,1)$. We obtain all the interpolated values $E_I^m$$(E_I^0=E_c^0)$ at time $t_m$$(m=1,\cdots ,M-1,M,\cdots ,PM)$ between $E_c^{n-1}$ and $E_c^n$$(n=1,2,\cdots ,P)$. Similarly, we can define $N_I^m$ as

$$N_I^m={\lambda }_m{N}_c^{n-1}+(1-{\lambda }_m)N_c^n.$$

(28)

Step 3: Based on all the coarse numerical solutions obtained in Step 2, Taylor’s formula for two variables is performed to obtain a linearized system on a fine time step $\tau$,

$$\begin{array}{cc}\hfill i\epsilon (E_{ft}^{m+\frac{1}{2}},g)-\gamma (E_{fx}^{m+\frac{1}{2}},g_x)=& \lambda (f(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})+f_{N_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(N_f^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})\hfill \\ & +f_{E_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(E_f^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}),g),\hfill \end{array}$$

(29)

$$(N_{ft}^{m+\frac{1}{2}},g)=-({\Phi }_{fx}^{m+\frac{1}{2}},g_x),$$

(30)

$$\begin{array}{cc}\hfill ({\Phi }_{ft}^{m+\frac{1}{2}},g)-(N_f^{m+\frac{1}{2}},g)-\alpha (N_{fx}^{m+\frac{1}{2}},g_x)=& \theta (y\left(N_I^{m+\frac{1}{2}}\right)+y_{N_I}\left(N_I^{m+\frac{1}{2}}\right)(N_f^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}}),g)\hfill \\ & +\omega (q\left(E_I^{m+\frac{1}{2}}\right)+q_{E_I}\left(E_I^{m+\frac{1}{2}}\right)(E_f^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}),g),\hfill \end{array}$$

(31)

where

$$f(N_f^{m+\frac{1}{2}},E_f^{m+\frac{1}{2}})=N_f^{m+\frac{1}{2}}{E}_f^{m+\frac{1}{2}},y\left(N_f^{m+\frac{1}{2}}\right)={\left(N_f^{m+\frac{1}{2}}\right)}^2,q\left(E_f^{m+\frac{1}{2}}\right)={\left|E_f^{m+\frac{1}{2}}\right|}^2.$$

The TT-M numerical solutions are obtained by directly solving (29)–(31).

We will discuss the stability and error estimation of the TT-M FE system. Without losing generality, in the following analysis process, we take the coefficient parameter in the model (1) as $\epsilon =\alpha =\theta =1$, $\gamma =\frac{3}{2}$, $\lambda =\frac{1}{2}$ and $\omega =\frac{1}{4}$.

Remark 1.

Compared with semi-implicit and semi-explicit schemes [17,30,31], the purpose of the TT-M FE algorithm is to solve the nonlinear system directly. Moreover, with a comparison to the standard nonlinear Galerkin FE method, our algorithm can save on computing time, which can be seen by observing the numerical results.

3. Stability Analysis

In this section, we give the stability analysis of the TT-M FE scheme.

Theorem 1.

The following stability for the fully discrete scheme (24)–(26) on the coarse time mesh holds:

$$\parallel E_c^n{\parallel }^2+\parallel {\Phi }_c^n{\parallel }^2+\parallel N_c^n{\parallel }^2\le C(\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^0{\parallel }^2+\parallel N_c^0{\parallel }^2).$$

(32)

Proof. 

Choosing $g=E_c^{n+\frac{1}{2}}$ in (24) and taking the imaginary part and setting $g=N_c^{n+\frac{1}{2}}$ in (25) and $g={\Phi }_c^{n+\frac{1}{2}}$ in (26), we combine the three resulting equations to arrive at

$$\begin{array}{cc}\hfill & \frac{1}{2\tau }(\parallel E_c^{n+1}{\parallel }^2-\parallel E_c^n{\parallel }^2+\parallel {\Phi }_c^{n+1}{\parallel }^2-\parallel {\Phi }_c^n{\parallel }^2+\parallel N_c^{n+1}{\parallel }^2-\parallel N_c^n{\parallel }^2)-(\frac{N_c^{n+1}+N_c^n}{2},\frac{{\Phi }_c^{n+1}+{\Phi }_c^n}{2})\hfill \\ \hfill =& ({\left(N_c^{n+\frac{1}{2}}\right)}^2,\frac{{\Phi }_c^{n+1}+{\Phi }_c^n}{2})+\frac{1}{4}\left(\right|E_c^{n+\frac{1}{2}}{|}^2,\frac{{\Phi }_c^{n+1}+{\Phi }_c^n}{2}).\hfill \end{array}$$

(33)

Summing (33) from 0 to n and using the Hölder inequality and Young inequality, we have

$$\begin{array}{cc}\hfill & \frac{1}{2\tau }(\parallel E_c^{n+1}{\parallel }^2-\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^{n+1}{\parallel }^2-\parallel {\Phi }_c^0{\parallel }^2+\parallel N_c^{n+1}{\parallel }^2-\parallel N_c^0{\parallel }^2)\hfill \\ \hfill \le & \sum _{k=0}^n(\parallel N_c^{k+\frac{1}{2}}{\parallel }_{\infty }\parallel N_c^{k+\frac{1}{2}}\parallel +\frac{1}{4}\parallel E_c^{k+\frac{1}{2}}{\parallel }_{\infty }\parallel E_c^{k+\frac{1}{2}}\parallel )\parallel \frac{{\Phi }_c^{k+1}+{\Phi }_c^k}{2}\parallel +C\sum _{k=0}^n(\parallel N_c^{k+\frac{1}{2}}{\parallel }^2+\parallel {\Phi }_c^{k+\frac{1}{2}}{\parallel }^2).\hfill \end{array}$$

(34)

Using the Gronwall inequality with a small enough $\tau$, we obtain

$$\parallel E_c^{n+1}{\parallel }^2+\parallel {\Phi }_c^{n+1}{\parallel }^2+\parallel N_c^{n+1}{\parallel }^2\le C(\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^0{\parallel }^2+\parallel N_c^0{\parallel }^2).$$

(35)

□

Remark 2.

Actually, we easily obtain the following two conservation relationships by referring to [18]:

$$\parallel E_c^n\parallel =\parallel E_c^0\parallel ,\parallel N_c^n\parallel =\parallel N_c^0\parallel .$$

(36)

Theorem 2.

The stability holds for the TT-M FE system (29)–(31)

$$\begin{array}{cc}\hfill & \parallel E_f^{m+1}{\parallel }^2+\parallel {\Phi }_f^{m+1}{\parallel }^2+{\parallel N_f^{m+1}\parallel }^2\hfill \\ \hfill \le & C(\parallel E_f^0{\parallel }^2+\parallel {\Phi }_f^0{\parallel }^2+\parallel N_f^0{\parallel }^2+\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^0{\parallel }^2+\parallel N_c^0{\parallel }^2).\hfill \end{array}$$

(37)

Proof. 

Setting $g=E_f^{m+\frac{1}{2}}$ in (29) and taking the imaginary part, and using the Hölder inequality as well as the Young inequality, we have

$$\begin{array}{cc}\hfill \frac{1}{2\tau }(\parallel E_f^{m+1}{\parallel }^2-\parallel E_f^m{\parallel }^2)=& Im(N_f^{m+\frac{1}{2}}{E}_I^{m+\frac{1}{2}}+N_I^{m+\frac{1}{2}}{E}_f^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}}{E}_I^{m+\frac{1}{2}},E_f^{m+\frac{1}{2}})\hfill \\ \hfill \le & C(\parallel N_I^{m+\frac{1}{2}}{\parallel }^2+\parallel E_I^{m+\frac{1}{2}}{\parallel }^2+\parallel N_f^{m+\frac{1}{2}}{\parallel }^2+\parallel E_f^{m+\frac{1}{2}}{\parallel }^2).\hfill \end{array}$$

(38)

Summing (38) from 0 to $m-1$, we get

$$\parallel E_f^m{\parallel }^2-\parallel E_f^0{\parallel }^2\le C\tau \sum _{k=0}^{m-1}(\parallel N_I^k{\parallel }^2+\parallel E_I^k{\parallel }^2+\parallel N_f^k{\parallel }^2+\parallel E_f^k{\parallel }^2).$$

(39)

Using (27) and (28), the triangle inequality and Theorem 1, we have

$$\begin{array}{cc}\hfill \parallel E_I^m\parallel =& \parallel {\lambda }_m{E}_c^{n-1}+(1-{\lambda }_m)E_c^n\parallel \hfill \\ \hfill \le & {\lambda }_m\parallel E_c^{n-1}\parallel +(1-{\lambda }_m)\parallel E_c^n\parallel \le C(\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^0{\parallel }^2+\parallel N_c^0{\parallel }^2),\hfill \end{array}$$

(40)

and

$$\begin{array}{cc}\hfill \parallel N_I^m\parallel =& \parallel {\lambda }_m{N}_c^{n-1}+(1-{\lambda }_m)N_c^n\parallel \hfill \\ \hfill \le & {\lambda }_m\parallel N_c^{n-1}\parallel +(1-{\lambda }_m)\parallel N_c^n\parallel \le C(\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^0{\parallel }^2+\parallel N_c^0{\parallel }^2).\hfill \end{array}$$

(41)

Combining (39)–(41) with the Gronwall inequality, we arrive at

$$\parallel E_f^m{\parallel }^2\le C(\parallel N_c^0{\parallel }^2+\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^0{\parallel }^2+\parallel N_f^0{\parallel }^2+\parallel E_f^0{\parallel }^2).$$

(42)

Setting $g=N_f^{m+\frac{1}{2}}$ in (30) and $g={\Phi }_f^{m+\frac{1}{2}}$ in (31) and combining two resulting equations, we arrive at

$$\begin{array}{cc}\hfill & (\frac{{\Phi }_f^{m+1}-{\Phi }_f^m}{\tau },\frac{{\Phi }_f^{m+1}+{\Phi }_f^m}{2})+(\frac{N_f^{m+1}-N_f^m}{\tau },\frac{N_f^{m+1}+N_f^m}{2})-(\frac{N_f^{m+1}+N_f^m}{2},\frac{{\Phi }_f^{m+1}+{\Phi }_f^m}{2})\hfill \\ \hfill =& ({\left(N_I^{m+\frac{1}{2}}\right)}^2+2N_I^{m+\frac{1}{2}}(N_f^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}}),\frac{{\Phi }_f^{m+1}+{\Phi }_f^m}{2})\hfill \\ & +\frac{1}{4}\left(\right|E_I^{m+\frac{1}{2}}{|}^2+2|E_I^{m+\frac{1}{2}}|(E_f^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}),\frac{{\Phi }_f^{m+1}+{\Phi }_f^m}{2}).\hfill \end{array}$$

(43)

Summing (43) from 0 to $m-1$ and using the Hölder inequality as well as the Young inequality, we have

$$\begin{array}{cc}\hfill & \frac{1}{2\tau }(\parallel {\Phi }_f^{m+1}{\parallel }^2-\parallel {\Phi }_f^0{\parallel }^2)+\frac{1}{2\tau }(\parallel N_f^{m+1}{\parallel }^2-\parallel N_f^0{\parallel }^2)\hfill \\ \hfill \le & \sum _{k=0}^{m-1}(\parallel N_I^{k+\frac{1}{2}}{\parallel }_{\infty }\parallel N_I^{k+\frac{1}{2}}\parallel +2\parallel N_I^{k+\frac{1}{2}}{\parallel }_{\infty }\parallel N_f^{k+\frac{1}{2}}-N_I^{k+\frac{1}{2}}\parallel \hfill \\ & +\frac{1}{4}\parallel E_I^{k+\frac{1}{2}}{\parallel }_{\infty }\parallel E_I^{k+\frac{1}{2}}\parallel +\frac{1}{2}\parallel E_I^{k+\frac{1}{2}}{\parallel }_{\infty }\parallel E_f^{k+\frac{1}{2}}-E_I^{k+\frac{1}{2}}\parallel )\parallel \frac{{\Phi }_f^{k+1}+{\Phi }_f^k}{2}\parallel \hfill \\ & +C\sum _{k=0}^{m-1}(\parallel N_f^{k+\frac{1}{2}}{\parallel }^2+\parallel {\Phi }_f^{k+\frac{1}{2}}{\parallel }^2).\hfill \end{array}$$

(44)

Combining (44) with (42) and using the Young inequality, we have

$$\begin{array}{cc}\hfill & \parallel E_f^{m+1}{\parallel }^2+\parallel {\Phi }_f^{m+1}{\parallel }^2+{\parallel N_f^{m+1}\parallel }^2\hfill \\ \hfill \le & C(\parallel N_c^0{\parallel }^2+\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^0{\parallel }^2+\parallel N_f^0{\parallel }^2+\parallel E_f^0{\parallel }^2+\parallel {\Phi }_f^0{\parallel }^2)\hfill \\ & +C\tau \sum _{k=0}^m\parallel N_I^{k+\frac{1}{2}}{\parallel }_{\infty }^2\parallel N_I^{k+\frac{1}{2}}{\parallel }^2+C\tau \sum _{k=0}^m\parallel N_I^{k+\frac{1}{2}}{\parallel }_{\infty }^2{\parallel N_f^{k+\frac{1}{2}}-N_I^{k+\frac{1}{2}}\parallel }^2\hfill \\ & +C\tau \sum _{k=0}^m\parallel E_I^{k+\frac{1}{2}}{\parallel }_{\infty }^2\parallel E_I^{k+\frac{1}{2}}{\parallel }^2+C\tau \sum _{k=0}^m\parallel E_I^{k+\frac{1}{2}}{\parallel }_{\infty }^2{\parallel E_f^{k+\frac{1}{2}}-E_I^{k+\frac{1}{2}}\parallel }^2\hfill \\ & +C\epsilon \tau \sum _{k=0}^m(\parallel N_f^{k+\frac{1}{2}}{\parallel }^2+\parallel {\Phi }_f^{k+\frac{1}{2}}{\parallel }^2)\hfill \\ \hfill \le & C(\parallel N_c^0{\parallel }^2+\parallel E_c^0{\parallel }^2+\parallel {\Phi }_c^0{\parallel }^2+\parallel N_f^0{\parallel }^2+\parallel E_f^0{\parallel }^2+\parallel {\Phi }_f^0{\parallel }^2)\hfill \\ & +C\tau \sum _{k=0}^m(\parallel N_I^{k+\frac{1}{2}}{\parallel }^2+\parallel E_I^{k+\frac{1}{2}}{\parallel }^2+\parallel N_f^{k+\frac{1}{2}}{\parallel }^2+\parallel E_f^{k+\frac{1}{2}}{\parallel }^2+\parallel {\Phi }_f^{k+\frac{1}{2}}{\parallel }^2).\hfill \end{array}$$

(45)

Using the Gronwall inequality with a small enough $\tau$, we obtain the conclusion of theorem. □

4. Error Analysis

In this section, we will present an error analysis on the TT-M FE scheme. We introduce the auxiliary projection $Q_h$ on $S_h(\Omega )$ defined by: for $v\in H_0^1(\Omega )$,

$$({(Q_{h}v-v)}_x,w_{hx})=0,\forall w_h\in S_h,$$

(46)

where $Q_h$ satisfies the following inequalities.

Lemma 1

(See [32]). Assuming that a function v satisfies the condition

$$v\in H_0^1(\Omega )\cap H^{r+1}(\Omega ),$$

then the following inequalities hold:

$$\parallel Q_{h}v-v\parallel +\parallel {(Q_{h}v-v)}_t\parallel +h\parallel Q_h{v-v\parallel }_1⩽C{h}^{r+1}{(\parallel v\parallel }_{r+1}+\parallel v_t{\parallel }_{r+1}),$$

(47)

where C is the constant independent of h.

Theorem 3.

Let $E^n$, $N^n$, ${\Phi }^n$ be the solution of system (6)–(8), and suppose $E_c^n$, $N_c^n$, ${\Phi }_c^n$ and $E_f^m$, $N_f^m$, ${\Phi }_f^m$ are the solutions ofthe FE system (24)–(26) and the TT-M FE system (29)–(31), respectively. Then, there exists a constant $C>0$ that depends only on $E^n$, $N^n$, ${\Phi }^n$, such that

$$\parallel E_c^n-E^n\parallel +\parallel N_c^n-N^n\parallel +\parallel {\Phi }_c^n-{\Phi }^n\parallel ⩽C({\tau }_c^2+h^{r+1}),$$

(48)

and

$$\parallel E_f^m-E^m\parallel +\parallel N_f^m-N^m\parallel +\parallel {\Phi }_f^m-{\Phi }^m\parallel ⩽C({\tau }_c^4+{\tau }^2+h^{r+1}).$$

(49)

Proof. 

Here, we will prove the theorem by two main steps: (i) estimates on the coarse time mesh; (ii) estimates on the fine time mesh.

(i) Firstly, we derive error results based on the coarse time mesh. To facilitate the derivation, we split the errors as the following:

$$E_c^{n+\frac{1}{2}}-E^{n+\frac{1}{2}}=E_c^{n+\frac{1}{2}}-Q_h{E}^{n+\frac{1}{2}}+Q_h{E}^{n+\frac{1}{2}}-E^{n+\frac{1}{2}}={\mu }_1^{n+\frac{1}{2}}+{\sigma }_1^{n+\frac{1}{2}},$$

$$N_c^{n+\frac{1}{2}}-N^{n+\frac{1}{2}}=N_c^{n+\frac{1}{2}}-Q_h{N}^{n+\frac{1}{2}}+Q_h{N}^{n+\frac{1}{2}}-N^{n+\frac{1}{2}}={\mu }_2^{n+\frac{1}{2}}+{\sigma }_2^{n+\frac{1}{2}},$$

$${\Phi }_c^{n+\frac{1}{2}}-{\Phi }^{n+\frac{1}{2}}={\Phi }_c^{n+\frac{1}{2}}-Q_h{\Phi }^{n+\frac{1}{2}}+Q_h{\Phi }^{n+\frac{1}{2}}-{\Phi }^{n+\frac{1}{2}}={\mu }_3^{n+\frac{1}{2}}+{\sigma }_3^{n+\frac{1}{2}}.$$

Combining (17)–(19) with (24)–(26), we easily write the error equations as

$$\begin{array}{cc}\hfill i({\mu }_{1t}^{n+\frac{1}{2}},g)+i({\sigma }_{1t}^{n+\frac{1}{2}},g)-\frac{3}{2}({\mu }_{1x}^{n+\frac{1}{2}},g_x)=& \frac{1}{2}(N_c^{n+\frac{1}{2}}{E}_c^{n+\frac{1}{2}}-N^{n+\frac{1}{2}}{E}^{n+\frac{1}{2}},g)+(e_{1\left({\tau }_c\right)}^{n+\frac{1}{2}},g),\hfill \end{array}$$

(50)

$$({\mu }_{2t}^{n+\frac{1}{2}},g)+({\sigma }_{2t}^{n+\frac{1}{2}},g)=-({\mu }_{3x}^{n+\frac{1}{2}},g_x)+(e_{2\left({\tau }_c\right)}^{n+\frac{1}{2}},g),$$

(51)

$$\begin{array}{cc}\hfill ({\mu }_{3t}^{n+\frac{1}{2}},g)+({\sigma }_{3t}^{n+\frac{1}{2}},g)& -({\mu }_2^{n+\frac{1}{2}},g)-({\sigma }_2^{n+\frac{1}{2}},g)-({\mu }_{2x}^{n+\frac{1}{2}},g_x)\hfill \\ & =({\left(N_c^{n+\frac{1}{2}}\right)}^2-{\left(N^{n+\frac{1}{2}}\right)}^2,g)+\frac{1}{4}\left(\right|E_c^{n+\frac{1}{2}}{|}^2-|E^{n+\frac{1}{2}}{|}^2,g)+(e_{3\left({\tau }_c\right)}^{n+\frac{1}{2}},g).\hfill \end{array}$$

(52)

Choosing $g={\mu }_1^{n+\frac{1}{2}}$ in (50) and taking the imaginary part of the resulting equation, using the Hölder inequality and the boundedness of $\parallel E_c{\parallel }_{\infty }$, $\parallel N_c{\parallel }_{\infty }$, ${\parallel E\parallel }_{\infty }$ and ${\parallel N\parallel }_{\infty }$, we get

$$\begin{array}{cc}\hfill \parallel {\mu }_1^{n+1}{\parallel }^2-{\parallel {\mu }_1^n\parallel }^2=& -2\tau Re(\frac{{\sigma }_1^{n+1}-{\sigma }_1^n}{\tau },{\mu }_1^{n+\frac{1}{2}})+\tau Im(N_c^{n+\frac{1}{2}}{E}_c^{n+\frac{1}{2}}-N^{n+\frac{1}{2}}{E}^{n+\frac{1}{2}},{\mu }_1^{n+\frac{1}{2}})\hfill \\ & +2\tau Im(e_{1\left({\tau }_c\right)}^{n+\frac{1}{2}},{\mu }_1^{n+\frac{1}{2}})\hfill \\ \hfill ⩽& -2\tau (\parallel \frac{{\sigma }_1^{n+1}-{\sigma }_1^n}{\tau }{\parallel }^2+\parallel {\mu }_1^{n+1}{\parallel }^2+\parallel {\mu }_1^n{\parallel }^2)+C\tau (\parallel {\mu }_1^{n+1}{\parallel }^2+{\parallel {\mu }_1^n\parallel }^2\hfill \\ & +\parallel {\sigma }_1^{n+1}{\parallel }^2+\parallel {\sigma }_1^n{\parallel }^2+\parallel {\mu }_2^{n+1}{\parallel }^2+\parallel {\mu }_2^n{\parallel }^2+\parallel {\sigma }_2^{n+1}{\parallel }^2+{\parallel {\sigma }_2^n\parallel }^2\hfill \\ & +\parallel e_{1\left({\tau }_c\right)}^{n+1}{\parallel }^2+\parallel e_{1\left({\tau }_c\right)}^n{\parallel }^2).\hfill \end{array}$$

(53)

Setting $g={\mu }_2^{n+\frac{1}{2}}$ in (51) and taking $g={\mu }_3^{n+\frac{1}{2}}$ in (52), by the similar derivation to [18], we easily have

$$\begin{array}{cc}\hfill & \parallel {\mu }_1^{n+1}{\parallel }^2-\parallel {\mu }_1^n{\parallel }^2+\parallel {\mu }_2^{n+1}{\parallel }^2-\parallel {\mu }_2^n{\parallel }^2+\parallel {\mu }_3^{n+1}{\parallel }^2-{\parallel {\mu }_3^n\parallel }^2\hfill \\ \hfill ⩽& C\tau (\parallel {\mu }_1^{n+1}{\parallel }^2+\parallel {\mu }_1^n{\parallel }^2+\parallel {\mu }_2^{n+1}{\parallel }^2+\parallel {\mu }_2^n{\parallel }^2+\parallel {\mu }_3^{n+1}{\parallel }^2+\parallel {\mu }_3^n{\parallel }^2\hfill \\ & +\parallel {\sigma }_1^{n+1}{\parallel }^2+\parallel {\sigma }_1^n{\parallel }^2+\parallel {\sigma }_2^{n+1}{\parallel }^2+{\parallel {\sigma }_2^n\parallel }^2\hfill \\ & +\parallel \frac{{\sigma }_1^{n+1}-{\sigma }_1^n}{\tau }{\parallel }^2+\parallel \frac{{\sigma }_2^{n+1}-{\sigma }_2^n}{\tau }{\parallel }^2+{\parallel \frac{{\sigma }_3^{n+1}-{\sigma }_3^n}{\tau }\parallel }^2\hfill \\ & +\parallel e_{1\left({\tau }_c\right)}^{n+1}{\parallel }^2+\parallel e_{1\left({\tau }_c\right)}^n{\parallel }^2+\parallel e_{2\left({\tau }_c\right)}^{n+1}{\parallel }^2+\parallel e_{2\left({\tau }_c\right)}^n{\parallel }^2+\parallel e_{3\left({\tau }_c\right)}^{n+1}{\parallel }^2+\parallel e_{3\left({\tau }_c\right)}^n{\parallel }^2).\hfill \end{array}$$

(54)

Summing (54) from 0 to n and using the Gronwall inequality, Lemma 1 and the triangle inequality, we have

$$\parallel E_c^{n+1}-E^{n+1}\parallel +\parallel N_c^{n+1}-N^{n+1}\parallel +\parallel {\Phi }_c^{n+1}-{\Phi }^{n+1}\parallel ⩽C({\tau }_c^2+h^{r+1}).$$

(55)

Taking the similar techniques to (40) and (41), we easily obtain

$$\parallel E_I^{m+1}-E^{m+1}\parallel +\parallel N_I^{m+1}-N^{m+1}\parallel +\parallel {\Phi }_I^{m+1}-{\Phi }^{m+1}\parallel ⩽C({\tau }_c^2+h^{r+1}),$$

(56)

which will be used in the following analysis.

(ii) Secondly, we give a detailed analysis on the fine time mesh. We rewrite the errors as

$$E_f^{m+\frac{1}{2}}-E^{m+\frac{1}{2}}=E_f^{m+\frac{1}{2}}-Q_h{E}^{m+\frac{1}{2}}+Q_h{E}^{m+\frac{1}{2}}-E^{m+\frac{1}{2}}={\xi }_1^{m+\frac{1}{2}}+{\rho }_1^{m+\frac{1}{2}},$$

$$N_f^{m+\frac{1}{2}}-N^{m+\frac{1}{2}}=N_f^{m+\frac{1}{2}}-Q_h{N}^{m+\frac{1}{2}}+Q_h{N}^{m+\frac{1}{2}}-N^{m+\frac{1}{2}}={\xi }_2^{m+\frac{1}{2}}+{\rho }_2^{m+\frac{1}{2}},$$

$${\Phi }_f^{m+\frac{1}{2}}-{\Phi }^{m+\frac{1}{2}}={\Phi }_f^{m+\frac{1}{2}}-Q_h{\Phi }^{m+\frac{1}{2}}+Q_h{\Phi }^{m+\frac{1}{2}}-{\Phi }^{m+\frac{1}{2}}={\xi }_3^{m+\frac{1}{2}}+{\rho }_3^{m+\frac{1}{2}}.$$

Combining (17)–(19) with (29)–(31), we can obtain the following error equations:

$$\begin{array}{cc}\hfill & i({\xi }_{1t}^{m+\frac{1}{2}},g)+i({\rho }_{1t}^{m+\frac{1}{2}},g)-\frac{3}{2}({\xi }_{1x}^{m+\frac{1}{2}},g_x)\hfill \\ \hfill =& \frac{1}{2}(f(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})+f_{N_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(N_f^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})\hfill \\ & +f_{E_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(E_f^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})-f(N^{m+\frac{1}{2}},E^{m+\frac{1}{2}}),g)+(e_{1\left(\tau \right)}^{m+\frac{1}{2}},g),\hfill \end{array}$$

(57)

$$({\xi }_{2t}^{m+\frac{1}{2}},g)+({\rho }_{2t}^{m+\frac{1}{2}},g)=-({\xi }_{3x}^{m+\frac{1}{2}},g_x)+(e_{2\left(\tau \right)}^{m+\frac{1}{2}},g),$$

(58)

and

$$\begin{array}{cc}\hfill & ({\xi }_{3t}^{m+\frac{1}{2}},g)+({\rho }_{3t}^{m+\frac{1}{2}},g)-({\xi }_2^{m+\frac{1}{2}},g)-({\rho }_2^{m+\frac{1}{2}},g)-({\xi }_{2x}^{m+\frac{1}{2}},g_x)\hfill \\ \hfill =& (y\left(N_I^{m+\frac{1}{2}}\right)+y_{N_I}\left(N_I^{m+\frac{1}{2}}\right)(N_f^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})-y\left(N^{m+\frac{1}{2}}\right),g)\hfill \\ & +\frac{1}{4}(q\left(E_I^{m+\frac{1}{2}}\right)+q_{E_I}\left(E_I^{m+\frac{1}{2}}\right)(E_f^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})-q\left(E^{m+\frac{1}{2}}\right),g)+(e_{3\left(\tau \right)}^{m+\frac{1}{2}},g).\hfill \end{array}$$

(59)

By Taylor’s formula for two variables, we can derive

$$\begin{array}{cc}\hfill & f(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})+f_{N_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(N_f^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})\hfill \\ & +f_{E_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(E_f^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})-f(N^{m+\frac{1}{2}},E^{m+\frac{1}{2}})\hfill \\ \hfill =& f_{N_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})({\xi }_2^{m+\frac{1}{2}}+{\rho }_2^{m+\frac{1}{2}})+f_{E_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})({\xi }_1^{m+\frac{1}{2}}+{\rho }_1^{m+\frac{1}{2}})\hfill \\ & -\frac{1}{2}\{f_{N_I{N}_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}}){(N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})}^2+f_{E_I{E}_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}}){(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})}^2\hfill \\ & +2f_{N_I{E}_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})\}\hfill \\ \hfill =& N_I^{m+\frac{1}{2}}({\xi }_1^{n+\frac{1}{2}}+{\rho }_1^{n+\frac{1}{2}})+E_I^{m+\frac{1}{2}}({\xi }_2^{n+\frac{1}{2}}+{\rho }_2^{n+\frac{1}{2}})-(N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}).\hfill \end{array}$$

(60)

In (57), we set $g={\xi }_1^{m+\frac{1}{2}}$ and take the imaginary part to get

$$\begin{array}{cc}\hfill \parallel {\xi }_1^{m+1}{\parallel }^2-{\parallel {\xi }_1^m\parallel }^2=& -2\tau Re\left(\frac{{\rho }_1^{m+1}-{\rho }_1^m}{\tau },{\xi }_1^{m+\frac{1}{2}}\right)+\tau Im(f(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})\hfill \\ & +f_{N_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(N_f^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})\hfill \\ & +f_{E_I}(N_I^{m+\frac{1}{2}},E_I^{m+\frac{1}{2}})(E_f^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})\hfill \\ & -f(N^{m+\frac{1}{2}},E^{m+\frac{1}{2}}),{\xi }_1^{m+\frac{1}{2}})+2\tau Im(e_{1\left(\tau \right)}^{m+\frac{1}{2}},{\xi }_1^{m+\frac{1}{2}})\hfill \\ \hfill ⩽& -2\tau \left({∥\frac{{\rho }_1^{m+1}-{\rho }_1^m}{\tau }∥}^2+\parallel {\xi }_1^{m+1}{\parallel }^2+{\parallel {\xi }_1^m\parallel }^2\right)\hfill \\ & +C\tau (\parallel {\xi }_1^{m+1}{\parallel }^2+\parallel {\xi }_1^m{\parallel }^2+\parallel {\rho }_1^{m+1}{\parallel }^2+\parallel {\rho }_1^m{\parallel }^2\hfill \\ & +\parallel {\xi }_2^{m+1}{\parallel }^2+\parallel {\xi }_2^m{\parallel }^2+\parallel {\rho }_2^{m+1}{\parallel }^2+{\parallel {\rho }_2^m\parallel }^2\hfill \\ & +\parallel (N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}){\parallel }^2+\parallel e_{1\left(\tau \right)}^{m+1}{\parallel }^2+\parallel e_{1\left(\tau \right)}^m{\parallel }^2).\hfill \end{array}$$

(61)

Setting $g={\xi }_2^{m+\frac{1}{2}}$ in (58), we get

$$\begin{array}{cc}\hfill & \frac{1}{2\tau }{\int }_a^b({\left({\xi }_2^{m+1}\right)}^2-{\left({\xi }_2^m\right)}^2)dx+\frac{1}{2}{\int }_a^b\left(\frac{{\rho }_2^{m+1}-{\rho }_2^m}{\tau }\right)({\xi }_2^{m+1}+{\xi }_2^m)dx\hfill \\ \hfill =& -\frac{1}{4}{\int }_a^b({\xi }_{3x}^{m+1}+{\xi }_{3x}^m)({\xi }_{2x}^{m+1}+{\xi }_{2x}^m)dx+\frac{1}{4}{\int }_a^b(e_{2\left(\tau \right)}^{m+1}+e_{2\left(\tau \right)}^m)({\xi }_2^{m+1}+{\xi }_2^m)dx.\hfill \end{array}$$

(62)

Take $g={\xi }_3^{m+\frac{1}{2}}$ in (59), we arrive at

$$\begin{array}{cc}\hfill & \frac{1}{2\tau }{\int }_a^b({\left({\xi }_3^{m+1}\right)}^2-{\left({\xi }_3^m\right)}^2)dx+\frac{1}{2}{\int }_a^b\left(\frac{{\rho }_3^{m+1}-{\rho }_3^m}{\tau }\right)({\xi }_3^{m+1}+{\xi }_3^m)dx\hfill \\ & -\frac{1}{4}{\int }_a^b({\xi }_2^{m+1}+{\xi }_2^m)({\xi }_3^{m+1}+{\xi }_3^m)dx\hfill \\ & -\frac{1}{4}{\int }_a^b({\rho }_2^{m+1}+{\rho }_2^m)({\xi }_3^{m+1}+{\xi }_3^m)dx-\frac{1}{4}{\int }_a^b({\xi }_{2x}^{m+1}+{\xi }_{2x}^m)({\xi }_{3x}^{m+1}+{\xi }_{3x}^m)dx\hfill \\ \hfill =& \frac{1}{8}{\int }_a^b\left[\right(2|E_I^{m+\frac{1}{2}}|({\xi }_1^{m+\frac{1}{2}}+{\rho }_1^{m+\frac{1}{2}})+{(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})}^2]({\xi }_3^{m+1}+{\xi }_3^m)dx\hfill \\ & +\frac{1}{2}{\int }_a^b[(2N_I^{m+\frac{1}{2}}({\xi }_2^{m+\frac{1}{2}}+{\rho }_2^{m+\frac{1}{2}})+{(N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})}^2]({\xi }_3^{m+1}+{\xi }_3^m)dx\hfill \\ & +\frac{1}{4}{\int }_a^b(e_{3\left(\tau \right)}^{m+1}+e_{3\left(\tau \right)}^m)({\xi }_3^{m+1}+{\xi }_3^m)dx.\hfill \end{array}$$

(63)

Combining (62) with (63), we obtain

$$\frac{1}{2\tau }{\int }_a^b({\left({\xi }_2^{m+1}\right)}^2-{\left({\xi }_2^m\right)}^2+{\left({\xi }_3^{m+1}\right)}^2-{\left({\xi }_3^m\right)}^2)dx=R3+R4,$$

(64)

where $R3$ and $R4$ are given as follows:

$$\begin{array}{cc}\hfill R3=-& \frac{1}{2}{\int }_a^b\left(\frac{{\rho }_2^{m+1}-{\rho }_2^m}{\tau }\right)({\xi }_2^{m+1}+{\xi }_2^m)dx-\frac{1}{4}{\int }_a^b({\xi }_{3x}^{m+1}+{\xi }_{3x}^m)({\xi }_{2x}^{m+1}+{\xi }_{2x}^m)dx\hfill \\ \hfill +& \frac{1}{4}{\int }_a^b(e_{2\left(\tau \right)}^{m+1}+e_{2\left(\tau \right)}^m)({\xi }_2^{m+1}+{\xi }_2^m)dx,\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill R4=-& \frac{1}{2}{\int }_a^b\left(\frac{{\rho }_3^{m+1}-{\rho }_3^m}{\tau }\right)({\xi }_3^{m+1}+{\xi }_3^m)dx+\frac{1}{4}{\int }_a^b({\xi }_2^{m+1}+{\xi }_2^m)({\xi }_3^{m+1}+{\xi }_3^m)dx\hfill \\ \hfill +& \frac{1}{4}{\int }_a^b({\rho }_2^{m+1}+{\rho }_2^m)({\xi }_3^{m+1}+{\xi }_3^m)dx+\frac{1}{4}{\int }_a^b({\xi }_{2x}^{m+1}+{\xi }_{2x}^m)({\xi }_{3x}^{m+1}+{\xi }_{3x}^m)dx\hfill \\ \hfill +& \frac{1}{8}{\int }_a^b\left[\right(2|E_I^{m+\frac{1}{2}}|({\xi }_1^{m+\frac{1}{2}}+{\rho }_1^{m+\frac{1}{2}})+{(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})}^2]({\xi }_3^{m+1}+{\xi }_3^m)dx\hfill \\ \hfill +& \frac{1}{2}{\int }_a^b[(2N_I^{m+\frac{1}{2}}({\xi }_2^{m+\frac{1}{2}}+{\rho }_2^{m+\frac{1}{2}})+{(N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})}^2]({\xi }_3^{m+1}+{\xi }_3^m)dx\hfill \\ \hfill +& \frac{1}{4}{\int }_a^b(e_{3\left(\tau \right)}^{m+1}+e_{3\left(\tau \right)}^m)({\xi }_3^{m+1}+{\xi }_3^m)dx.\hfill \end{array}$$

(66)

Adding (65) and (66) together, and using the Cauchy–Schwarz inequality as well as the Young inequality, we get

$$\begin{array}{cc}\hfill R3+R4⩽& \frac{1}{2}{\int }_a^b\left[{\left(\frac{{\rho }_2^{m+1}-{\rho }_2^m}{\tau }\right)}^2+{\left({\xi }_2^{m+1}\right)}^2+{\left({\xi }_2^m\right)}^2\right]dx\hfill \\ & +C{\int }_a^b[{\left({\xi }_2^{m+1}\right)}^2+{\left({\xi }_2^m\right)}^2+{\left(e_{2\left(\tau \right)}^{m+1}\right)}^2+{\left(e_{2\left(\tau \right)}^m\right)}^2]dx\hfill \\ & +\frac{1}{2}{\int }_a^b\left[{\left(\frac{{\rho }_3^{m+1}-{\rho }_3^m}{\tau }\right)}^2+{\left({\xi }_3^{m+1}\right)}^2+{\left({\xi }_3^m\right)}^2\right]dx\hfill \\ & +C{\int }_a^b[{\left({\xi }_3^{m+1}\right)}^2+{\left({\xi }_3^m\right)}^2+{\left({\xi }_2^{m+1}\right)}^2+{\left({\xi }_2^m\right)}^2+{\left({\xi }_1^{m+1}\right)}^2+{\left({\xi }_1^m\right)}^2\hfill \\ & +{\left({\rho }_1^{m+1}\right)}^2+{\left({\rho }_1^m\right)}^2+{\left({\rho }_2^{m+1}\right)}^2+{\left({\rho }_2^m\right)}^2+{\left({(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})}^2\right)}^2\hfill \\ & +{\left({(N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})}^2\right)}^2+{\left(e_{3\left(\tau \right)}^{m+1}\right)}^2+{\left(e_{3\left(\tau \right)}^m\right)}^2]dx.\hfill \end{array}$$

(67)

Combining (61) with (67), we have

$$\begin{array}{cc}\hfill & \parallel {\xi }_1^{m+1}{\parallel }^2-\parallel {\xi }_1^m{\parallel }^2+\parallel {\xi }_2^{m+1}{\parallel }^2-\parallel {\xi }_2^m{\parallel }^2+\parallel {\xi }_3^{m+1}{\parallel }^2-{\parallel {\xi }_3^m\parallel }^2\hfill \\ \hfill ⩽& C\tau (\parallel {\xi }_1^{m+1}{\parallel }^2+\parallel {\xi }_1^m{\parallel }^2+\parallel {\xi }_2^{m+1}{\parallel }^2+\parallel {\xi }_2^m{\parallel }^2+\parallel {\xi }_3^{m+1}{\parallel }^2+\parallel {\xi }_3^m{\parallel }^2\hfill \\ & +\parallel {\rho }_1^{m+1}{\parallel }^2+\parallel {\rho }_1^m{\parallel }^2+\parallel {\rho }_2^{m+1}{\parallel }^2+{\parallel {\rho }_2^m\parallel }^2\hfill \\ & +{∥\frac{{\rho }_1^{m+1}-{\rho }_1^m}{\tau }∥}^2+{∥\frac{{\rho }_2^{m+1}-{\rho }_2^m}{\tau }∥}^2+{∥\frac{{\rho }_3^{m+1}-{\rho }_3^m}{\tau }∥}^2\hfill \\ & +\parallel {(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})}^2{\parallel }^2+\parallel {(N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})}^2{\parallel }^2+\parallel (N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}){\parallel }^2\hfill \\ & +\parallel e_{1\left(\tau \right)}^{m+1}{\parallel }^2+\parallel e_{1\left(\tau \right)}^m{\parallel }^2+\parallel e_{2\left(\tau \right)}^{m+1}{\parallel }^2+\parallel e_{2\left(\tau \right)}^m{\parallel }^2+\parallel e_{3\left(\tau \right)}^{m+1}{\parallel }^2+\parallel e_{3\left(\tau \right)}^m{\parallel }^2).\hfill \end{array}$$

(68)

Making use of (56), the Hölder inequality and the similar technique as shown in [33], we have

$$\begin{array}{cc}\hfill & \parallel {(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}})}^2{\parallel }^2+\parallel {(N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})}^2{\parallel }^2+\parallel (N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}})(E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}){\parallel }^2\hfill \\ \hfill ⩽& \parallel E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}{\parallel }_{\infty }^2\parallel E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}{\parallel }^2\hfill \\ & +\parallel N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}}{\parallel }_{\infty }^2(\parallel N^{m+\frac{1}{2}}-N_I^{m+\frac{1}{2}}{\parallel }^2+\parallel E^{m+\frac{1}{2}}-E_I^{m+\frac{1}{2}}{\parallel }^2)\hfill \\ \hfill ⩽& C({\tau }_c^4+h^{2r+2}).\hfill \end{array}$$

(69)

Combining (68) with (69), and summing the resulting equation from 0 to m, we arrive at

$$\begin{array}{cc}\hfill & \parallel {\xi }_1^{m+1}{\parallel }^2+\parallel {\xi }_2^{m+1}{\parallel }^2+{\parallel {\xi }_3^{m+1}\parallel }^2\hfill \\ \hfill ⩽& \parallel {\xi }_1^0{\parallel }^2+\parallel {\xi }_2^0{\parallel }^2+\parallel {\xi }_3^0{\parallel }^2+C\tau \sum _{k=0}^m(\parallel {\xi }_1^{k+1}{\parallel }^2+\parallel {\xi }_2^{k+1}{\parallel }^2+\parallel {\xi }_3^{k+1}{\parallel }^2+\parallel {\rho }_1^{k+1}{\parallel }^2+{\parallel {\rho }_2^{k+1}\parallel }^2\hfill \\ & +{∥\frac{{\rho }_1^{k+1}-{\rho }_1^k}{\tau }∥}^2+{∥\frac{{\rho }_2^{k+1}-{\rho }_2^k}{\tau }∥}^2+{∥\frac{{\rho }_3^{k+1}-{\rho }_3^k}{\tau }∥}^2+{\tau }_c^4+h^{2r+2}+\parallel e_{1\left(\tau \right)}^{k+1}{\parallel }^2+\parallel e_{2\left(\tau \right)}^{k+1}{\parallel }^2+\parallel e_{3\left(\tau \right)}^{k+1}{\parallel }^2).\hfill \end{array}$$

(70)

Using the Gronwall inequality and Lemma 1, we finish the proof of the theorem. □

5. Numerical Tests

We denote ${\Xi }_{{\mathfrak{Q}}_{\eta }}^n(\tau ,h)\triangleq \underset{n}{max}\parallel {\mathfrak{Q}}^n-{\mathfrak{Q}}_{\eta }^n\parallel$, where $\eta =h$ or f and define the convergence rate by the following formula:

$$\mathrm{Rate}=\left\{\begin{array}{cc}{\displaystyle {log}_2[{\Xi }_{{\mathfrak{Q}}_{\eta }}^n(2\tau ,h)/{\Xi }_{{\mathfrak{Q}}_{\eta }}^n(\tau ,h)],}\hfill & \mathrm{in}\mathrm{time},\hfill \\ {\displaystyle {log}_2[{\Xi }_{{\mathfrak{Q}}_{\eta }}^n(\tau ,2h)/{\Xi }_{{\mathfrak{Q}}_{\eta }}^n(\tau ,h)],}\hfill & \mathrm{in}\mathrm{space},\hfill \\ {\displaystyle {log}_2[{\Xi }_{{\mathfrak{Q}}_{\eta }}^n(2\tau ,2h)/{\Xi }_{{\mathfrak{Q}}_{\eta }}^n(\tau ,h)],}\hfill & \mathrm{in}\mathrm{both}\mathrm{space}\mathrm{and}\mathrm{time},\hfill \end{array}\right.$$

(71)

where $\mathfrak{Q}$ is taken as the function E, N or $\Phi$.

5.1. Example 1

In order to test the correctness of the convergence rates, we select the parameters $\epsilon =\gamma =\lambda =\alpha =\theta =\omega =1$ in (6)–(10). By adding the source terms to the system, we obtain the following initial boundary value problem:

$$\left\{\begin{array}{c}i{E}_t+E_{xx}-EN=a\left(x,t\right),(x,t)\in (0,\pi )\times (0,1],\hfill \\ N_t-{\Phi }_{xx}=b\left(x,t\right),(x,t)\in (0,\pi )\times (0,1],\hfill \\ {\Phi }_t-N+N_{xx}-N^2-{\left|E\right|}^2=c\left(x,t\right),(x,t)\in (0,\pi )\times (0,1],\hfill \\ E(x,0)=sin\left(2x\right)+isin\left(x\right),N(x,0)={sin}^2\left(x\right),\Phi (x,0)=sin\left(x\right),x\in [0,\pi ],\hfill \\ E(0,t)=E(\pi ,t)=0,N(0,t)=N(\pi ,t)=0,\Phi (0,t)=\Phi (\pi ,t)=0,t\in [0,1],\hfill \end{array}\right.$$

(72)

where $(0,\pi )$ and $(0,1]$ are the space domain and time interval, respectively. The source terms are given in advance, and the equations have the following exact solution:

$$E\left(x,t\right)={\left(t+1\right)}^{2}sin\left(2x\right)+i{e}^{-t}sin\left(x\right),$$

$$N\left(x,t\right)={\left(t+1\right)}^2{sin}^2\left(x\right),$$

$$\Phi \left(x,t\right)=\left(t+1\right)sin\left(x\right).$$

Taking a fixed time step $\tau =\frac{1}{3000}$ and varying spatial steps $h=\frac{\pi }{20},\frac{\pi }{40},\frac{\pi }{80},\frac{\pi }{160},\frac{\pi }{320}$, in

Table 1. Space convergence results for Example 1 with ${\tau }_C=4\tau$.

Standard FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{\pi }{20},\frac{1}{3000})$	1.5458 × ${10}^{-2}$		2.8961 × ${10}^{-2}$		6.7647 × ${10}^{-2}$		1.77
$(\frac{\pi }{40},\frac{1}{3000})$	3.8651 × ${10}^{-3}$	1.9997	7.3072 × ${10}^{-3}$	1.9867	1.7086 × ${10}^{-2}$	1.9852	3.25
$(\frac{\pi }{80},\frac{1}{3000})$	9.6635 × ${10}^{-4}$	1.9999	1.8335 × ${10}^{-3}$	1.9948	4.2815 × ${10}^{-3}$	1.9966	8.83
$(\frac{\pi }{160},\frac{1}{3000})$	2.4162 × ${10}^{-4}$	1.9998	4.5865 × ${10}^{-4}$	1.9991	1.0709 × ${10}^{-3}$	1.9993	53.00
$(\frac{\pi }{320},\frac{1}{3000})$	6.0442 × ${10}^{-5}$	1.9991	1.1459 × ${10}^{-4}$	2.0009	2.6761 × ${10}^{-4}$	2.0006	317.05
TT-M FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{\pi }{20},\frac{1}{3000})$	1.5458 × ${10}^{-2}$		2.8962 × ${10}^{-2}$		6.7647 × ${10}^{-2}$		1.77
$(\frac{\pi }{40},\frac{1}{3000})$	3.8651 × ${10}^{-3}$	1.9997	7.3073 × ${10}^{-3}$	1.9867	1.7086 × ${10}^{-2}$	1.9852	3.07
$(\frac{\pi }{80},\frac{1}{3000})$	9.6632 × ${10}^{-4}$	1.9999	1.8336 × ${10}^{-3}$	1.9947	4.2816 × ${10}^{-3}$	1.9966	8.76
$(\frac{\pi }{160},\frac{1}{3000})$	2.4160 × ${10}^{-4}$	1.9999	4.5878 × ${10}^{-4}$	1.9988	1.0710 × ${10}^{-3}$	1.9991	53.56
$(\frac{\pi }{320},\frac{1}{3000})$	6.0414 × ${10}^{-5}$	1.9996	1.1472 × ${10}^{-4}$	1.9997	2.6779 × ${10}^{-4}$	1.9998	282.96

and

Table 2. Space-time convergence results for Example 1 with $\tau =\frac{1}{\pi }h$ and ${\tau }_C=4\tau$.

Standard FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{\pi }{20},\frac{1}{20})$	1.5913 × ${10}^{-2}$		2.5619 × ${10}^{-2}$		6.2220 × ${10}^{-2}$		0.02
$(\frac{\pi }{40},\frac{1}{40})$	3.9807 × ${10}^{-3}$	1.9991	6.4235 × ${10}^{-3}$	1.9958	1.5737 × ${10}^{-2}$	1.9833	0.06
$(\frac{\pi }{80},\frac{1}{80})$	9.9505 × ${10}^{-4}$	2.0002	1.6056 × ${10}^{-3}$	2.0002	3.9462 × ${10}^{-3}$	1.9956	0.31
$(\frac{\pi }{160},\frac{1}{160})$	2.4882 × ${10}^{-4}$	1.9997	4.0175 × ${10}^{-4}$	1.9988	9.8720 × ${10}^{-4}$	1.9991	3.13
$(\frac{\pi }{320},\frac{1}{320})$	6.2204 × ${10}^{-5}$	2.0000	1.0039 × ${10}^{-4}$	2.0007	2.4666 × ${10}^{-4}$	2.0008	36.44
$(\frac{\pi }{640},\frac{1}{640})$	1.5551 × ${10}^{-5}$	2.0000	2.5105 × ${10}^{-5}$	1.9996	6.1688 × ${10}^{-5}$	1.9994	443.56
TT-M FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{\pi }{20},\frac{1}{20})$	1.5984 × ${10}^{-2}$		2.8722 × ${10}^{-2}$		6.7901 × ${10}^{-2}$		0.04
$(\frac{\pi }{40},\frac{1}{40})$	3.9951 × ${10}^{-3}$	2.0003	7.1917 × ${10}^{-3}$	1.9978	1.7111 × ${10}^{-2}$	1.9885	0.05
$(\frac{\pi }{80},\frac{1}{80})$	9.9906 × ${10}^{-4}$	1.9996	1.7973 × ${10}^{-3}$	2.0005	4.2867 × ${10}^{-3}$	1.9970	0.26
$(\frac{\pi }{160},\frac{1}{160})$	2.4977 × ${10}^{-4}$	2.0000	4.4967 × ${10}^{-4}$	1.9989	1.0721 × ${10}^{-3}$	1.9995	2.85
$(\frac{\pi }{320},\frac{1}{320})$	6.2444 × ${10}^{-5}$	2.0000	1.1240 × ${10}^{-4}$	2.0003	2.6806 × ${10}^{-4}$	1.9998	31.30
$(\frac{\pi }{640},\frac{1}{640})$	1.5611 × ${10}^{-5}$	2.0000	2.8100 × ${10}^{-5}$	1.9999	6.7016 × ${10}^{-5}$	2.0000	343.06

, we obtain the results of space convergence and space-time convergence. The results show that both the standard FE method and the fast TT-M FE method obtain the approximate second-order convergence, which is consistent with the theoretical results. At the same time, we can also find that the fast TT-M FE algorithm can save calculation time and maintain the same calculation accuracy as the standard FE method.

5.2. Example 2

In (6)–(10), choosing $\epsilon =1$, we obtain the following initial boundary value problem with the parameters

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}i{E}_t+\gamma E_{xx}-\lambda EN=0,\hfill & \left(x,t\right)\in \Omega \times J,\hfill \\ N_t-{\Phi }_{xx}=0,\hfill & \left(x,t\right)\in \Omega \times J,\hfill \\ {\Phi }_t-N+\alpha N_{xx}-\theta N^2-\omega {\left|E\right|}^2=0,\hfill & \left(x,t\right)\in \Omega \times J,\hfill \end{array}\right.\end{array}\end{array}$$

(73)

where $\Omega =(a,b)=\left(x_L,x_R\right)$ and $J=\left(0,T\right]$. Based on Refs. [6,8], the Schrödinger–Boussinesq system (73) has the following parametric analytical solution:

Case 1.If$\gamma \lambda d_1=2b_1\left(3\gamma \theta -\alpha \lambda \right)$and$3\alpha \lambda \ne \gamma \theta$and$4\alpha b_1\ne \gamma d_1$,

$$\left\{\begin{array}{c}E\left(x,t\right)=\pm \frac{6b_1}{\lambda }\sqrt{\frac{\gamma \theta -\alpha \lambda }{\gamma \omega }}sech\left(\mu \zeta \right)tanh\left(\mu \zeta \right)e^{i\left(\frac{m}{2\gamma }x+\delta t\right)},\hfill \\ N\left(x,t\right)=-\frac{6b_1}{\lambda }sec{h}^2\left(\mu \zeta \right),\hfill \\ \Phi \left(x,t\right)=\frac{12m{b}_1}{\mu \lambda }\left(\frac{x_R-x}{x_R-x_L}-\frac{1}{1+e^{2\mu \zeta }}\right).\hfill \end{array}\right.$$

(74)

Case 2.If$3\alpha \lambda =\gamma \theta$and$4\alpha b_1\ne \gamma d_1$,

$$\left\{\begin{array}{c}E\left(x,t\right)=\sqrt{\frac{6\alpha b_1}{{\gamma }^2\theta \omega }\left(\gamma d_1-4\alpha b_1\right)}sech\left(\mu \zeta \right)e^{i\left(\frac{m}{2\gamma }x+\delta t\right)},\hfill \\ N\left(x,t\right)=-\frac{2b_1}{\lambda }sec{h}^2\left(\mu \zeta \right),\hfill \\ \Phi \left(x,t\right)=\frac{4m{b}_1}{\mu \lambda }\left(\frac{x_R-x}{x_R-x_L}-\frac{1}{1+e^{2\mu \zeta }}\right).\hfill \end{array}\right.$$

(75)

Therefore, $b_1=\delta +\frac{m^2}{4\gamma }$, $d_1=1-m^2$, $\mu =\sqrt{\frac{b_1}{\gamma }}$, $\zeta =x-mt$ and m, $\delta$ are free parameters.

The following numerical experiments will be carried out based on two cases to verify the effectiveness and computational efficiency of the algorithm.

For Case 1, we choose

$$\gamma =1,\lambda =1,\alpha =1,\theta =\frac{4}{3},\omega =\frac{1}{18},m=\sqrt{\frac{1}{5}},\delta =\frac{1}{12},\Omega =\left[-40,40\right].$$

Taking the time interval $\overline{J}=\left[0,10\right]$, the fixed time step $\tau =\frac{{\tau }_C}{4}=1/300$, and changed space step length $h=1/2,1/4,1/8$, we obtain the convergence data of the standard nonlinear finite element method and the fast TT-M finite element method in

Table 3. Space convergence results for Example 2 in Case 1 with ${\tau }_C=4\tau$.

Standard FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{1}{2},\frac{1}{300})$	9.7274 × ${10}^{-2}$		5.1130 × ${10}^{-2}$		9.3026 × ${10}^{-2}$		62.39
$(\frac{1}{4},\frac{1}{300})$	2.4921 × ${10}^{-2}$	1.9647	1.3201 × ${10}^{-2}$	1.9535	2.3933 × ${10}^{-2}$	1.9586	385.42
$(\frac{1}{8},\frac{1}{300})$	6.2693 × ${10}^{-3}$	1.9910	3.3276 × ${10}^{-3}$	1.9881	6.0274 × ${10}^{-3}$	1.9894	3093.74
TT-M FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{1}{2},\frac{1}{300})$	9.7274 × ${10}^{-2}$		5.1130 × ${10}^{-2}$		9.3027 × ${10}^{-2}$		51.98
$(\frac{1}{4},\frac{1}{300})$	2.4921 × ${10}^{-2}$	1.9647	1.3201 × ${10}^{-2}$	1.9535	2.3934 × ${10}^{-2}$	1.9586	297.16
$(\frac{1}{8},\frac{1}{300})$	6.2698 × ${10}^{-3}$	1.9909	3.3279 × ${10}^{-3}$	1.9880	6.0284 × ${10}^{-3}$	1.9892	2521.57

, which implies that the fast TT-M FE method can maintain the same calculation accuracy as the standard finite element method and reduce the calculation time to a great extent. A similar case can also be seen in the space-time convergence calculation data in

Table 4. Space-time convergence results for Example 2 in Case 1 with $\tau =\frac{1}{2}h$ and ${\tau }_C=4\tau$.

Standard FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{1}{2},\frac{1}{4})$	9.6515 × ${10}^{-2}$		4.9930 × ${10}^{-2}$		9.0913 × ${10}^{-2}$		1.09
$(\frac{1}{4},\frac{1}{8})$	2.4741 × ${10}^{-2}$	1.9638	1.2908 × ${10}^{-2}$	1.9516	2.3431 × ${10}^{-2}$	1.9561	13.35
$(\frac{1}{8},\frac{1}{16})$	6.2254 × ${10}^{-3}$	1.9907	3.2549 × ${10}^{-3}$	1.9876	5.9038 × ${10}^{-3}$	1.9887	137.33
$(\frac{1}{16},\frac{1}{32})$	1.5589 × ${10}^{-3}$	1.9977	8.1550 × ${10}^{-4}$	1.9969	1.4789 × ${10}^{-3}$	1.9971	2000.16
TT-M FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{1}{2},\frac{1}{4})$	9.9179 × ${10}^{-2}$		5.1934 × ${10}^{-2}$		9.7061 × ${10}^{-2}$		1.05
$(\frac{1}{4},\frac{1}{8})$	2.5426 × ${10}^{-2}$	1.9637	1.3388 × ${10}^{-2}$	1.9558	2.4873 × ${10}^{-2}$	1.9643	9.97
$(\frac{1}{8},\frac{1}{16})$	6.3972 × ${10}^{-3}$	1.9908	3.3734 × ${10}^{-3}$	1.9886	6.2584 × ${10}^{-3}$	1.9907	112.46
$(\frac{1}{16},\frac{1}{32})$	1.6019 × ${10}^{-3}$	1.9977	8.4504 × ${10}^{-4}$	1.9971	1.5671 × ${10}^{-3}$	1.9977	1246.46

. By taking the time interval $\overline{J}=\left[0,30\right]$ and time step size $\tau =\frac{{\tau }_C}{4}=\frac{1}{100}$, we more vividly show the dynamic behavior of the numerical solution in Figure 1, Figure 2 and Figure 3, from which it is easy to see the performance behavior of the numerical images of the three functions at each fixed time point, and we can also observe the moving process, the behavior performance change of the numerical solution and the change of spatial position from $t=0$, $t=10$, $t=20$ to $t=30$.

For Case 2, we choose

$$\gamma =1,\lambda =1,\alpha =\frac{1}{2},\theta =\frac{3}{2},\omega =\frac{1}{12},m=\sqrt{\frac{1}{3}},\delta =\frac{1}{5},\overline{\Omega }=\left[-40,40\right].$$

When taking the time interval $\overline{J}=\left[0,10\right]$ and the same space-time step length as Case 1, we obtain the space and space-time convergence data in

Table 5. Space convergence results for Example 2 in Case 2 with ${\tau }_C=4\tau$.

Standard FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{1}{2},\frac{1}{300})$	8.1725 × ${10}^{-2}$		5.0737 × ${10}^{-2}$		6.2883 × ${10}^{-2}$		49.55
$(\frac{1}{4},\frac{1}{300})$	2.1093 × ${10}^{-2}$	1.9540	1.3275 × ${10}^{-2}$	1.9343	1.6315 × ${10}^{-2}$	1.9465	364.28
$(\frac{1}{8},\frac{1}{300})$	5.3172 × ${10}^{-3}$	1.9880	3.3580 × ${10}^{-3}$	1.9830	4.1181 × ${10}^{-3}$	1.9861	3037.39
TT-M FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{1}{2},\frac{1}{300})$	8.1726 × ${10}^{-2}$		5.0738 × ${10}^{-2}$		6.2884 × ${10}^{-2}$		53.02
$(\frac{1}{4},\frac{1}{300})$	2.1094 × ${10}^{-2}$	1.9540	1.3276 × ${10}^{-2}$	1.9343	1.6315 × ${10}^{-2}$	1.9465	284.42
$(\frac{1}{8},\frac{1}{300})$	5.3179 × ${10}^{-3}$	1.9879	3.3584 × ${10}^{-3}$	1.9829	4.1189 × ${10}^{-3}$	1.9859	2551.25

and

Table 6. Space-time convergence results for Example 2 in Case 2 with $\tau =\frac{1}{2}h$ and ${\tau }_C=4\tau$.

Standard FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathit{h}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{1}{2},\frac{1}{4})$	8.2622 × ${10}^{-2}$		4.8094 × ${10}^{-2}$		5.9645 × ${10}^{-2}$		1.14
$(\frac{1}{4},\frac{1}{8})$	2.1377 × ${10}^{-2}$	1.9505	1.2609 × ${10}^{-2}$	1.9314	1.5536 × ${10}^{-2}$	1.9408	12.62
$(\frac{1}{8},\frac{1}{16})$	5.3921 × ${10}^{-3}$	1.9871	3.1915 × ${10}^{-3}$	1.9821	3.9261 × ${10}^{-3}$	1.9845	125.44
$(\frac{1}{16},\frac{1}{32})$	1.3509 × ${10}^{-3}$	1.9970	8.0053 × ${10}^{-4}$	1.9952	9.8422 × ${10}^{-4}$	1.9961	1333.25
TT-M FE Algorithm
$(\mathit{h},\mathbf{\tau })$	${\Xi }_{{\mathit{E}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\mathit{N}}_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	${\Xi }_{{\Phi }_{\mathbf{f}}}^{\mathit{n}}(\mathbf{\tau },\mathit{h})$	Rate	CPU(s)
$(\frac{1}{2},\frac{1}{4})$	8.9536 × ${10}^{-2}$		5.1280 × ${10}^{-2}$		6.7564 × ${10}^{-2}$		0.99
$(\frac{1}{4},\frac{1}{8})$	2.2549 × ${10}^{-2}$	1.9894	1.3252 × ${10}^{-2}$	1.9521	1.6945 × ${10}^{-2}$	1.9954	9.36
$(\frac{1}{8},\frac{1}{16})$	5.6463 × ${10}^{-3}$	1.9977	3.3410 × ${10}^{-3}$	1.9879	4.2393 × ${10}^{-3}$	1.9989	100.56
$(\frac{1}{16},\frac{1}{32})$	1.4121 × ${10}^{-3}$	1.9995	8.3702 × ${10}^{-4}$	1.9970	1.0600 × ${10}^{-3}$	1.9998	1112.97

. Similarly, compared with the standard FE method, the TT-M FE algorithm can maintain the calculation accuracy and improve the calculation efficiency. In Figure 4, Figure 5 and Figure 6, we also show the performance behavior of the numerical solution when taking the time interval $\overline{J}=\left[0,30\right]$ and $\tau =\frac{{\tau }_C}{4}=\frac{1}{100}$.

In Figure 7, we depict the relationship between errors in the log scale based on the data computed in Table 3, Table 4, Table 5 and Table 6 and $1/h$, from which one can clearly see that as the parameter $1/h$ becomes larger, the errors becomes smaller. In Figure 8, where the error-axis is in the log scale, we show the comparison of performance of calculation time between the TT-M FE method and the standard FE method, which visually illustrates that the TT-M FE method has the advantage of saving computing time.

6. Conclusions

In this article, the coupled Schrödinger–Boussinesq equations are numerically solved by using the FE method combined with the TT-M fast algorithm so as to improve the calculation speed and reduce the calculation time. A theoretical analysis on the TT-M FE system is derived. In numerical experiments, the effectiveness and correctness of the algorithm are confirmed by a significant amount of calculation data, the dynamic behaviors of the numerical solutions are shown by several numerical images, and the advantage of the calculation speed of the algorithm is reflected by making a comparison with the standard nonlinear Galerkin FE method.

In the future, we will apply the TT-M FE method to two- or three-dimensional coupled Schrödinger–Boussinesq models, nonlinear fractional Schrödinger equations [34,35,36,37,38,39,40,41] and the coupled nonlinear Schrödinger equations [42,43,44], and also investigate the method’s conservation properties.