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
-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
where
E is the short-wave amplitude, and
N is the long-wave amplitude.
E and
N are complex functions and real functions, respectively.
,
and
are the electronic quality and the ion quality. Parameters
,
,
,
and
are five positive coefficients.
,
and
are given initial functions.
Now, by introducing the following unknown function
found in many references,
one can write the original problem (
1)–(
4) as follows
where
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.