A Time Two-mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

1. Introduction

The regularized long wave (RLW) equation [1,2] is a nonlinear partial differential equation that mainly describes the evolution of waves in shallow water channels and ion acoustic etc. It is a simplified version of the more complex Korteweg-de Vries (KdV) equation [3], which includes higher-order nonlinearities and dispersion effects. The symmetric regularized long wave (SRLW) equation [4] is a modified version of the RLW equation that includes a symmetry-breaking term. This term allows for the formation of asymmetric solutions, making the SRLW equation a more realistic model for waves in shallow water channels.

In this paper, the following initial boundary value problem of the SRLW equation is considered:

$$\left\{\begin{array}{cc}\hfill & u_t+{\rho }_x+u{u}_x-u_{xxt}=0,x_L\le x\le x_R,0<t\le T,\hfill \\ \hfill & {\rho }_t+u_x=0,x_L\le x\le x_R,0<t\le T,\hfill \\ \hfill & u\left(x_L,t\right)=u\left(x_R,t\right)=0,\rho \left(x_L,t\right)=\rho \left(x_R,t\right)=0,0<t\le T,\hfill \\ \hfill & u(x,0)=u_0\left(x\right),\rho (x,0)={\rho }_0\left(x\right),x_L\le x\le x_R.\hfill \end{array}\right.$$

(1)

The SRLW equation has attracted significant attention and has been extensively studied in the literatures. Numerous methods have been developed for obtaining numerical solutions to the SRLW equation, ranging from conservative finite difference schemes to mixed finite element methods. Wang et al. [5] proposed three conservative finite difference schemes that are all of second-order accuracy in both space and time. They also proved that the energy is preserved for all schemes while the mass is preserved only for the first scheme. Yimnet et al. [6] presented a novel finite difference method for the SRLW equation that utilizes a four-level average difference technique for solving the fluid velocity independently from the density. Hu et al. [7] developed a coupled conservative three-level implicit scheme that achieves a fourth–order rate of convergence. Li [8] considered a weighted and compact conservative difference scheme that is decoupled and linearized in practical computation, thus requiring only the solution of two tridiagonal systems of linear algebraic equations at each time step. Bai et al. [9] investigated a two-layer conservative finite difference scheme for the SRLW equation with homogeneous boundary conditions and analyzed the scheme’s convergence and stability using a discrete functional analysis method. Xu et al. [10] applied a mixed finite element method to solve the dissipative SRLW equations with damping term. He et al. [11] developed a fourth-order accurate compact difference scheme for the SRLW equation for a single nonlinear velocity form and conducted theoretical analysis using the discrete energy method.

From the view of numerical calculation, the time two-mesh (TT-M) method combined with finite element method or finite difference method can also solve plenty of nonlinear partial differential equation with better computational efficiency. For instance, Liu et al. [12] proposed the fast TT-M finite element method to solve the fractional water wave model, which has also been applied to other fractional models. Yin et al. [13] developed the TT-M finite element algorithm to solve a space fractional Allen–Cahn model and discussed in detail the problem of parameter selection. The TT-M finite element method was utilized by Liu et al. [14] to numerically solve the two-dimensional Gray–Scott model with space fractional derivatives. Wen et al. [15] used the TT-M algorithm in combination with the $H_1$-Galerkin mixed finite element method to numerically solve the nonlinear distributed order diffusion model. The computational efficiency of the algorithm was demonstrated, and the theoretical results were verified by numerical examples with both smooth and non-smooth solutions. Tian et al. [16] developed the finite element method combined with the TT-M technique to solve the coupled Schrödinger–Boussinesq equations. In recent years, there has been a amount of research on using the combined TT-M and finite difference methods to solve partial differential equations. Qiu and Xu et al. [17,18] developed and analyzed a TT-M algorithm based on finite difference (FD) methods for solving nonlinear fractional partial differential equations. Similarly, Niu et al. [19] used the TT-M technique to propose a fast high-order compact difference scheme for the nonlinear distributed order fractional Sobolev model appearing in porous media. He et al. [20] further extended the application of the TT-M method by studying a time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation and the scheme of second-order convergence rate in time as well as fourth-order in space. Despite the extensive research on the TT-M method in various fields, to the best of our knowledge, no study on the application of the TT-M method combined with finite difference to the SRLW equation has been discovered. Hence, investigations on the TT-M finite difference method’s performance when applied to the SRLW equation are still required.

The main contributions of this paper are as follows: (i) A novel TT-M finite difference numerical approach, incorporating a TT-M Crank-Nicolson algorithm for time discretization and finite difference method for space approximation, has been proposed to solve the SRLW equation. (ii) The TT-M finite difference method is used for the first time to solve partial differential equation with nonlinear term including derivatives. (iii) The detailed proofs of convergence analysis of the scheme are given, which are more complicated than existing methods. (iv) Numerical examples have been provided to demonstrate the computational speed and accuracy of the proposed method, which outperforms standard nonlinear finite difference method.

The remaining part of this article is organized as follows. In Section 2, some notations and useful lemmas are given. In Section 3, the TT-M finite difference numerical scheme is presented. In Section 4, the convergence of the scheme is analyzed. In Section 5, some numerical results are provided to test the theoretical results, computational efficiency of the scheme. Finally, in Section 6, we provide a brief conclusion.

2. Notations and Some Lemmas

As usual, the time interval $(0,T]$ and spatial interval $\left[x_L,x_R\right]$ are divided into N and J uniform partitions. The following notations will be used in this paper:

$$\begin{array}{cc}& {\left(u_j^n\right)}_x=\frac{u_{j+1}^n-u_j^n}{h},{\left(u_j^n\right)}_{\overline{x}}=\frac{u_j^n-u_{j-1}^n}{h},{\left(u_j^n\right)}_{\widehat{x}}=\frac{u_{j+1}^n-u_{j-1}^n}{2h},\hfill \\ & {\left(u_j^n\right)}_t=\frac{u_j^{n+1}-u_j^n}{\tau },u_j^{n+\frac{1}{2}}=\frac{1}{2}(u_j^{n+1}+u_j^n),{\left(u_j^n\right)}_{x\overline{x}}=\frac{u_{j+1}^n-2u_j^n+u_{j-1}^n}{h^2},\hfill \end{array}$$

where $\tau ,h$ denote the uniform time and spatial step length respectively, $x_j=x_L+jh,j=0,1,2,\cdots ,J$, $t_n=n\tau ,n=1,2,\cdots ,[T/\tau ]=N$, superscript n denotes a quantity associated with the time level $t_n$, subscript j denotes a quantity associated with space mesh point $x_j$. In this paper, M denotes general constant, which may have different value in different place.

Since $u\to 0$ for $x\to +\infty$ or $x\to -\infty$, we may assume $u_{-1}=u_{J+1}=0,1\le n\le N$ for simplicity, where $j=-1$ and $J+1$ are ghost points. Let $H_{h,0}$ denote the set of mesh functions $u^n$ defined on $I_h$ with boundary conditions $u_{-1}^n=u_0^n=u_J^n=u_{J+1}^n=0$. For any two mesh functions $u^n,w^n\in H_{h,0}$, we define the discrete inner product and norms as follows:

$$\begin{array}{cc}& (u^n,w^n)=h\sum _{j=1}^{J-1}{u}_j^n{w}_j^n,\parallel u^n\parallel =\sqrt{(u^n,u^n)},{\parallel u^n\parallel }_{\infty }=\underset{1\le j\le J-1}{max}\left|u_j^n\right|.\hfill \end{array}$$

Next, we presented some useful lemmas.

Lemma 1.

(See [11]). For any mesh functions $u^n,w^n\in H_{h,0}$, we have

$$\left(a\right)\left(u_x^n,w^n\right)=-\left(u^n,w_{\overline{x}}^n\right)=-\left(u^n,w_x^n\right),\left(b\right)\left(u_{x\overline{x}}^n,w^n\right)=-\left(u_x^n,w_x^n\right),\left(c\right)\left(u_{\widehat{x}}^n,w^n\right)=-\left(u^n,w_{\widehat{x}}^n\right).$$

Lemma 2.

(See [20,21]). Assume that a sequence of nonnegative real numbers ${\left\{a_j\right\}}_{j=0}^{\infty }$ satisfying

$$a_{n+1}\le \alpha +\beta \sum _{j=0}^n{a}_j\tau ,n\ge 0,$$

then there has the inequality $a_{n+1}\le \left(\alpha +\tau \beta a_0\right)e^{\beta (n+1)\tau }$, where $\alpha \ge 0,\beta$ and τ are positive constants.

Lemma 3.

(See [9,21]). For any discrete mesh function $u^n\in H_{h,0}$, there exists constants $C_1$ and $C_2$, such that

$${∥u^n∥}_{\infty }\le C_1∥u^n∥+C_2∥u_x^n∥.$$

3. The TT-M Finite Difference Scheme

In this paper, we studied a TT-M finite difference fast numerical method for the SRLW equation (1). In order to give the TT-M finite difference scheme, firstly, the time interval $(0,T]$ is partitioned uniformly into P coarse time intervals and then each coarse time interval is divided into $s\left(2\le s\in Z^{+}\right)$ fine time intervals. The coarse time mesh with the nodes $t_{ks}=k{\tau }_C(k=1,\cdots ,P)$ satisfying $0=t_0<t_s<t_{2s}<\cdots <t_{Ps}=T$ and the fine time mesh with the nodes $t_n=n{\tau }_F(n=1,2,\cdots ,Ps=N)$ satisfying $0=t_0<t_1<t_2<\cdots <t_{Ps}=T$, where ${\tau }_C=s{\tau }_F$ and ${\tau }_F$ are the coarse time and the fine time step size, respectively.

Secondly, the truncation errors of the problem (1) is considered, let $v_j^n=u\left(x_j,t_n\right),{\phi }_j^n=\rho \left(x_j,t_n\right)$ be the exact solutions of $u\left(x,t\right)$ and $\rho \left(x,t\right)$ in term of the point $\left(x_j,t_n\right)$, then we have

$$\begin{array}{cc}\hfill & E{r}_j^n={\left(v_j^n\right)}_t+{\left({\phi }_j^{n+1}\right)}_{\widehat{x}}-{\left(v_j^n\right)}_{x\overline{x}t}+\frac{1}{3}\left\{v_j^{n+\frac{1}{2}}{\left(v_j^{n+\frac{1}{2}}\right)}_{\widehat{x}}+{\left[{\left(v_j^{n+\frac{1}{2}}\right)}^2\right]}_{\widehat{x}}\right\},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & E{s}_j^n={\left({\phi }_j^n\right)}_t+{\left(v_j^n\right)}_{\widehat{x}},\hfill \end{array}$$

(3)

$$\begin{array}{cc}& v_0^n=v_J^n=0,{\phi }_0^n={\phi }_J^n=0,\hfill \\ & v_j^0=v_0\left(x_L+jh\right),{\phi }_j^0={\phi }_0\left(x_L+jh\right).\hfill \end{array}$$

By Taylor series expansion, we have

$$\begin{array}{cc}& E{r}_j^n={(u_t+{\rho }_x-u_{xxt}+u{u}_x)}_{(x_j,t_n)}=O(h^2+\tau ),\hfill \\ & E{s}_j^n={({\rho }_t+u_x)}_{(x_j,t_n)}=O(h^2+\tau ).\hfill \end{array}$$

Next, based on equations (2) and (3), a TT-M finite difference scheme for problem (1) is constructed with three steps.

Step 1: On the coarse time mesh, let $u_{C,j}^{ks}=u\left(x_j,t_{ks}\right),{\rho }_{C,j}^{ks}=\rho \left(x_j,t_{ks}\right)$ be the numerical solutions of of $u\left(x,t\right)$ and $\rho \left(x,t\right)$ in term of the point $\left(x_j,t_{ks}\right)$, then coarse time nonlinear finite difference scheme is given as

$${\left(u_{C,j}^{ks}\right)}_t+{\left({\rho }_{C,j}^{(k+1)s}\right)}_{\widehat{x}}-{\left(u_{C,j}^{ks}\right)}_{x\overline{x}t}+\frac{1}{3}\left\{u_{C,j}^{ks+\frac{1}{2}}{\left(u_{C,j}^{ks+\frac{1}{2}}\right)}_{\widehat{x}}+{\left[{\left(u_{C,j}^{ks+\frac{1}{2}}\right)}^2\right]}_{\widehat{x}}\right\}=0,$$

(4)

$${\left({\rho }_{C,j}^{ks}\right)}_t+{\left(u_{C,j}^{ks}\right)}_{\widehat{x}}=0,$$

(5)

$$\begin{array}{cc}{u}_{C,0}^{ks}=u_{C,J}^{ks}=0,{\rho }_{C,0}^{ks}={\rho }_{C,J}^{ks}=0,\hfill & k=0,1,\cdots ,P,\hfill \\ u_{C,j}^0=u_0\left(x_L+jh\right),{\rho }_{C,j}^0={\rho }_0\left(x_L+jh\right),\hfill & j=1,2,\cdots ,J-1,\hfill \end{array}$$

where $u_{C,j}^{ks+\frac{1}{2}}=\frac{1}{2}(u_{C,j}^{(k+1)s}+u_{C,j}^{ks}).$

Step 2: Based on the solutions $u_C^{ks},{\rho }_C^{ks}$ at time levels $t_{ks}$ obtained from step 1, we apply the Lagrange’s linear interpolation formula to compute $u_C^{ks-l},{\rho }_C^{ks-l}$ at time levels $t_{ks-l}(l=1,2,\cdots ,s-1$ and $k=1,2,\cdots ,P,ks-l=n)$, we have

$$u_C^{ks-l}=\frac{t_{ks-l}-t_{ks}}{t_{(k-1)s}-t_{ks}}{u}_C^{(k-1)s}+\frac{t_{ks-l}-t_{(k-1)s}}{t_{ks}-t_{(k-1)s}}{u}_C^{ks}=\frac{l}{s}{u}_C^{(k-1)s}+(1-\frac{l}{s})u_C^{ks},$$

(6)

$${\rho }_C^{ks-l}=\frac{t_{ks-1}-t_{ks}}{t_{(k-1)s}-t_{ks}}{\rho }_C^{(k-1)s}+\frac{t_{ks-1}-t_{(k-1)s}}{t_{ks}-t_{(k-1)s}}{\rho }_C^{ks}=\frac{l}{s}{\rho }_C^{(k-1)s}+(1-\frac{l}{s}){\rho }_C^{ks}.$$

(7)

Remark 1.

The equation (7) is only employed for theoretical analysis of the scheme. In numerical simulation, the coarse numerical solutions ${\rho }_C^{ks-l}$ are no need to compute since it does not used in step 3.

Step 3: Based on all the coarse numerical solutions $u_{C,j}^n(n=0,1,2,\cdots ,Ps=N,j=1,2,\cdots ,J-1)$ obtained in the first two steps, Taylor’s formula is used to construct a linearized system on the fine time mesh, which is expressed as follows. Let $u_{F,j}^n=u\left(x_j,t_n\right),{\rho }_{F,j}^n=\rho \left(x_j,t_n\right)$ be the numerical solutions of $u\left(x,t\right)$ and $\rho \left(x,t\right)$ in term of the point $\left(x_j,t_n\right)$ on the fine time mesh, then

$$\begin{array}{cc}\hfill & {\left(u_{F,j}^n\right)}_t+{\left({\rho }_{F,j}^{n+1}\right)}_{\widehat{x}}-{\left(u_{F,j}^n\right)}_{x\overline{x}t}+\frac{1}{6h}\left[f(u_{C,j-1}^{n+\frac{1}{2}},u_{C,j}^{n+\frac{1}{2}},u_{C,j+1}^{n+\frac{1}{2}})\right.\hfill \\ \hfill & +f_x(u_{C,j-1}^{n+\frac{1}{2}},u_{C,j}^{n+\frac{1}{2}},u_{C,j+1}^{n+\frac{1}{2}})(u_{F,j-1}^{n+\frac{1}{2}}-u_{C,j-1}^{n+\frac{1}{2}})\hfill \\ \hfill & +f_y(u_{C,j+1}^{n+\frac{1}{2}},u_{C,j}^{n+\frac{1}{2}},u_{C,j+1}^{n+\frac{1}{2}})(u_{F,j}^{n+\frac{1}{2}}-u_{C,j}^{n+\frac{1}{2}})\hfill \\ \hfill & \left.+f_z(u_{C,j-1}^{n+\frac{1}{2}},u_{C,j}^{n+\frac{1}{2}},u_{C,j+1}^{n+\frac{1}{2}})(u_{F,j+1}^{n+\frac{1}{2}}-u_{C,j+1}^{n+\frac{1}{2}})\right]=0,\hfill \end{array}$$

(8)

$${\left({\rho }_{F,j}^n\right)}_t+{\left(u_{F,j}^n\right)}_{\widehat{x}}=0,$$

(9)

$$\begin{array}{cc}& u_{F,0}^n=u_{F,J}^n=0,{\rho }_{F,0}^n={\rho }_{F,J}^n=0,\hfill \\ & u_{F,j}^0=u_0(x_L+jh),{\rho }_{F,j}^0={\rho }_0(x_L+jh),\hfill \\ & j=1,\cdots ,J-1,n=0,1,2,\cdots ,N,\hfill \end{array}$$

where $f(x,y,z)=(z-x)y+z^2-x^2$ and

$$f_x(x,y,z)=-y-2x,f_y(x,y,z)=z-x,f_z(x,y,z)=y+2z$$

are the three partial derivatives of $f(x,y,z)$ with respect to $x,y,z$.

Remark 2.

Similar to the Gauss-Seidel method applied to linear systems, we have modified our method to improve the accuracy of fine mesh solutions $u_F^{n+1}$ by using $u_F^n$ in calculation.

4. Convergence Analysis of the TT-M Finite Difference Scheme

The focus of this section is on performing convergence analysis of the nonlinear system specifically on the coarse time mesh.

Theorem 1.

Suppose that the exact solutions $v^n,{\phi }^n$ to the initial boundary value problem equation (1) is sufficiently smooth and let $u_C^n,{\rho }_C^n$ be the numerical solutions on the coarse time mesh. Then,

$$\parallel v^n-u_C^n{\parallel }_{\infty }\le O(h^2+{\tau }_C),\parallel {\phi }^n-{\rho }_C^n\parallel \le O(h^2+{\tau }_C).$$

Proof. 

Denote $e_{C,j}^{ks}=v_j^{ks}-u_{C,j}^{ks},{\eta }_{C,j}^{ks}={\phi }_j^{ks}-{\rho }_{C,j}^{ks},1\le j\le J-1,0\le k\le P$. Subtracting equation (4) from equation (2) and equation (5) from equation (3), we obtain

$$\begin{array}{cc}\hfill E{r}_{C,j}^{ks}=& {\left(e_{C,j}^{ks}\right)}_t+{\left({\eta }_{C,j}^{(k+1)s}\right)}_{\widehat{x}}-{\left(e_{C,j}^{ks}\right)}_{x\overline{x}t}\hfill \\ \hfill & +\frac{1}{3}\left\{v_j^{ks+\frac{1}{2}}{\left(v_j^{ks+\frac{1}{2}}\right)}_{\widehat{x}}+{\left[{\left(v_j^{ks+\frac{1}{2}}\right)}^2\right]}_{\widehat{x}}\right\}-\frac{1}{3}\left\{u_{C,j}^{ks+\frac{1}{2}}{\left(u_{C,j}^{ks+\frac{1}{2}}\right)}_{\widehat{x}}+{\left[{\left(u_{C,j}^{ks+\frac{1}{2}}\right)}^2\right]}_{\widehat{x}}\right\},\hfill \end{array}$$

(10)

$$E{s}_{C,j}^{ks}={\left({\eta }_{C,j}^{ks}\right)}_t+{\left(e_{C,j}^{ks}\right)}_{\widehat{x}}.$$

(11)

The proof contains two cases. Firstly, we consider the case of $n=ks(k=0,1,2,\cdots ,P)$, then $n+1=(k+1)s$. The initial and boundary condition satisfies

$$\begin{array}{c}{e}_{C,j}^0=0,{\eta }_{C,j}^0=0,\\ u_{C,0}^n=u_{C,J}^n=0,{\rho }_{C,0}^n={\rho }_{C,J}^n=0.\end{array}$$

Taking the inner product (·, ·) on both sides of equation (10) with $e_C^{n+1}+e_C^n$, we have

$$\begin{array}{cc}\hfill & (E{r}_C^n,e_C^{n+1}+e_C^n)=({\left(e_C^{n+1}\right)}_t,e_C^n+e_C^n)+({\left({\eta }_C^{n+1}\right)}_{\widehat{x}},e_C^{n+1}+e_C^n)\hfill \\ \hfill & -({\left(e_C^n\right)}_{x\overline{x}t},e_C^{ks+1}+e_C^n)+h\sum _{j=1}^{J-1}(\mathrm{I}+\mathrm{II})(e_{C,j}^{n+1}+e_{C,j}^n),\hfill \end{array}$$

(12)

where

$$\mathrm{I}=\frac{1}{3}\left[v_j^{n+\frac{1}{2}}{\left(v_j^{n+\frac{1}{2}}\right)}_{\widehat{x}}-u_{C,j}^{n+\frac{1}{2}}{\left(u_{C,j}^{n+\frac{1}{2}}\right)}_{\widehat{x}}\right],\mathrm{II}=\frac{1}{3}\left\{{\left[{\left(v_j^{n+\frac{1}{2}}\right)}^2\right]}_{\widehat{x}}-{\left[{\left(u_{C,j}^{n+\frac{1}{2}}\right)}^2\right]}_{\widehat{x}}\right\}.$$

Notice that

$$({\left(e_C^n\right)}_t,e_C^{n+1}+e_C^n)=\frac{1}{{\tau }_C}(\parallel e_C^{n+1}{\parallel }^2-\parallel e_C^n{\parallel }^2),$$

(13)

$$({\left({\eta }_C^{n+1}\right)}_{\widehat{x}},e_C^{n+1}+e_C^n)=-h\sum _{j=1}^{J-1}\left[{\eta }_{C,j}^{n+1}{(e_{C,j}^{n+1}+e_{C,j}^n)}_{\widehat{x}}\right],$$

(14)

$$-({\left(e_C^n\right)}_{x\overline{x}t},e_C^{n+1}+e_C^n)=\frac{1}{{\tau }_C}(\parallel e_{C,x}^{n+1}{\parallel }^2-\parallel e_{C,x}^n{\parallel }^2),$$

(15)

$$\begin{array}{cc}\hfill h\sum _{j=1}^{J-1}\mathrm{I}·(e_{C,j}^{n+1}+e_{C,j}^n)=& -\frac{2}{3}h\sum _{j=1}^{J-1}{(e_{C,j}^{n+\frac{1}{2}}·v_j^{n+\frac{1}{2}})}_{\widehat{x}}{e}_{C,j}^{n+\frac{1}{2}}\hfill \\ \hfill & -\frac{2}{3}h\sum _{j=1}^{J-1}{\left(e_{C,j}^{n+\frac{1}{2}}\right)}_{\widehat{x}}(e_{C,j+1}^{n+\frac{1}{2}}+e_{C,j-1}^{n+\frac{1}{2}})u_{C,j}^{n+\frac{1}{2}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill h\sum _{j=1}^{J-1}\mathrm{II}·(e_{C,j}^{n+1}+e_{C,j}^n)=& \frac{2}{3}h\sum _{j=1}^{J-1}{(v_j^{n+\frac{1}{2}}·e_{C,j}^{n+\frac{1}{2}})}_{\widehat{x}}{e}_{C,j}^{n+\frac{1}{2}}\hfill \\ \hfill & -\frac{2}{3}h\sum _{j=1}^{J-1}{u}_{C,j}^{n+\frac{1}{2}}{e}_{C,j}^{n+\frac{1}{2}}{\left(e_{C,j}^{n+\frac{1}{2}}\right)}_{\widehat{x}},\hfill \end{array}$$

(17)

then substituting equations (13)-(17) into (12), we have

$$\begin{array}{cc}\hfill & \parallel e_C^{n+1}{\parallel }^2+\parallel e_{C,x}^{n+1}{\parallel }^2=\parallel e_C^n{\parallel }^2+{\parallel e_{C,x}^n\parallel }^2+{\tau }_{C}h\sum _{j=1}^{J-1}\left[{\eta }_{C,j}^{n+1}{(e_{C,j}^{n+1}+e_{C,j}^n)}_{\widehat{x}}\right]\hfill \\ \hfill & +\frac{2}{3}{\tau }_{C}h\sum _{j=1}^{J-1}\left[{\left(e_{C,j}^{n+\frac{1}{2}}\right)}_{\widehat{x}}(e_{C,j+1}^{n+\frac{1}{2}}+e_{C,j-1}^{n+\frac{1}{2}})u_{C,j}^{n+\frac{1}{2}}+u_{C,j}^{n+\frac{1}{2}}{e}_{C,j}^{n+\frac{1}{2}}{\left(e_{C,j}^{n+\frac{1}{2}}\right)}_{\widehat{x}}\right]\hfill \\ \hfill & +{\tau }_C(E{r}_C^n,e_C^{n+1}+e_C^n).\hfill \end{array}$$

(18)

From Cauchy–Schwarz inequality, we obtain

$$\begin{array}{c}(E{r}_C^n,e_C^{n+1}+e_C^n)\le \parallel E{r}_C^n{\parallel }^2+\frac{1}{2}(\parallel e_C^{n+1}{\parallel }^2+\parallel e_C^n{\parallel }^2).\end{array}$$

Using Lemma 1, the equation (18) can be rewritten as

$$\begin{array}{cc}\hfill & \parallel e_C^{n+1}{\parallel }^2+\parallel e_{C,x}^{n+1}{\parallel }^2\le \parallel e_C^n{\parallel }^2+{\parallel e_{C,x}^n\parallel }^2\hfill \\ \hfill & +M{\tau }_C(\parallel {\eta }_C^{n+1}{\parallel }^2+\parallel e_C^{n+1}{\parallel }^2+\parallel e_C^n{\parallel }^2+\parallel e_{C,x}^{n+1}{\parallel }^2+\parallel e_{C,x}^n{\parallel }^2)+{\tau }_C{\parallel E{r}_C^n\parallel }^2.\hfill \end{array}$$

(19)

Similarly, taking the inner product (·, ·) on both sides of equation (11) with ${\eta }_C^{n+1}+{\eta }_C^n$, we obtain

$$\parallel {\eta }_C^{n+1}{\parallel }^2={\parallel {\eta }_C^n\parallel }^2-{\tau }_{C}h\sum _{j=1}^{J-1}{\left(e_{C,j}^n\right)}_{\widehat{x}}({\eta }_{C,j}^{n+1}+{\eta }_{C,j}^n)+{\tau }_C(E{s}_C^n,{\eta }_C^{n+1}+{\eta }_C^n).$$

(20)

From Cauchy–Schwarz inequality, we have

$$\begin{array}{c}(E{s}_C^n,{\eta }_C^{n+1}+{\eta }_C^n)\le \parallel E{s}_C^n{\parallel }^2+\frac{1}{2}(\parallel {\eta }_C^{n+1}{\parallel }^2+\parallel {\eta }_C^n{\parallel }^2).\end{array}$$

Using Lemma 1, the equation (20) can be rewritten as

$$\parallel {\eta }_C^{n+1}{\parallel }^2\le \parallel {\eta }_C^n{\parallel }^2+M{\tau }_C(\parallel e_{C,x}^n{\parallel }^2+\parallel {\eta }_C^{n+1}{\parallel }^2+\parallel {\eta }_C^n{\parallel }^2)+{\tau }_C{\parallel E{s}_C^n\parallel }^2.$$

(21)

Add equations (19) and (21), we get

$$\begin{array}{cc}\hfill & \parallel e_C^{n+1}{\parallel }^2+\parallel e_{C,x}^{n+1}{\parallel }^2+\parallel {\eta }_C^{n+1}{\parallel }^2\le \parallel e_C^n{\parallel }^2+\parallel e_{C,x}^n{\parallel }^2+{\parallel {\eta }_C^n\parallel }^2\hfill \\ \hfill & +M{\tau }_C(\parallel {\eta }_C^{n+1}{\parallel }^2+\parallel {\eta }_C^n{\parallel }^2+\parallel e_C^{n+1}{\parallel }^2+\parallel e_C^n{\parallel }^2+\parallel e_{C,x}^{n+1}{\parallel }^2+\parallel e_{C,x}^n{\parallel }^2)\hfill \\ \hfill & +{\tau }_C\parallel E{r}_C^n{\parallel }^2+{\tau }_C{\parallel E{s}_C^n\parallel }^2.\hfill \end{array}$$

(22)

Let $B_C^n=\parallel e_C^n{\parallel }^2+\parallel e_{C,x}^n{\parallel }^2+{\parallel {\eta }_C^n\parallel }^2$, then equation (22) becomes

$$B_C^{n+1}-B_C^n\le M{\tau }_C(B_C^{n+1}+B_C^n)+M{\tau }_C{(h^2+{\tau }_C)}^2,$$

and obtain

$$(1-M{\tau }_C)(B_C^{n+1}-B_C^n)\le 2M{\tau }_C{B}_C^n+M{\tau }_C{(h^2+{\tau }_C)}^2.$$

By taking ${\tau }_C$ small enough so that $(1-M{\tau }_C)>\lambda >0$, then

$$B_C^{n+1}-B_C^n\le M{\tau }_C{B}_C^n+M{\tau }_C{(h^2+{\tau }_C)}^2.$$

(23)

Summing from 0 to $P-1$ inequalities in equation (23), we have

$$B_C^P-B_C^0\le M{\tau }_C\sum _{n=1}^{P-1}{B}_C^n+M{\tau }_C{(h^2+{\tau }_C)}^2,$$

and using Lemma 2, get

$$B_C^P\le [B_C^0+M{(h^2+{\tau }_C)}^2]e^{MP{\tau }_C}.$$

(24)

From equation (24) and the initial and boundary condition, we have

$$\parallel e_C^n\parallel <O(h^2+{\tau }_C),\parallel e_{C,x}^n\parallel <O(h^2+{\tau }_C),\parallel {\eta }_C^n\parallel <O(h^2+{\tau }_C).$$

(25)

Then using Lemma 3, we obtain

$$\parallel e_C^n{\parallel }_{\infty }<O(h^2+{\tau }_C).$$

(26)

Secondly, we consider the case of $n=ks-l,(l=1,2,\cdots ,s-1$ and $k=1,2,\cdots ,P,ks-l=n)$. Based on the Lagrange’s interpolation formula, we get

$$\begin{array}{cc}\hfill v^{ks-l}& =\frac{t_{ks-l}-t_{ks}}{t_{(k-1)s}-t_{ks}}{v}^{(k-1)s}+\frac{t_{ks-l}-t_{(k-1)s}}{t_{ks}-t_{(k-1)s}}{v}^{ks}\hfill \\ \hfill & =\frac{l}{s}{v}^{(k-1)s}+(1-\frac{l}{s})v^{ks}+\frac{v^{\prime \prime }\left({\theta }_1\right)}{2}(t-t_{(k-1)s})(t-t_{ks}),{\theta }_1\in (t_{(k-1)s},t_{ks}),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\phi }^{ks-l}& =\frac{t_{ks-l}-t_{ks}}{t_{(k-1)s}-t_{ks}}{\phi }^{(k-1)s}+\frac{t_{ks-l}-t_{(k-1)s}}{t_{ks}-t_{(k-1)s}}{\phi }^{ks}\hfill \\ \hfill & =\frac{l}{s}{\phi }^{(k-1)s}+(1-\frac{l}{s}){\phi }^{ks}+\frac{{\phi }^{\prime \prime }\left({\theta }_2\right)}{2}(t-t_{(k-1)s})(t-t_{ks}),{\theta }_2\in (t_{(k-1)s},t_{ks}).\hfill \end{array}$$

(28)

Subtracting equation (27) from (6), we have

$$\begin{array}{cc}\hfill v^{ks-l}-u_C^{ks-l}=& \frac{l}{s}(v^{(k-1)s}-u_C^{(k-1)s})+(1-\frac{l}{s})(v^{ks}-u_C^{ks})\hfill \\ & +\frac{v^{\prime \prime }\left({\theta }_1\right)}{2}(t-t_{(k-1)s})(t-t_{ks}).\hfill \end{array}$$

Subtracting equation (28) from (7), we obtain

$$\begin{array}{cc}\hfill {\phi }^{ks-l}-{\rho }_C^{ks-l}=& \frac{l}{s}({\phi }^{(k-1)s}-{\rho }_C^{(k-1)s})+(1-\frac{l}{s})({\phi }^{ks}-{\rho }_C^{ks})\hfill \\ & +\frac{{\phi }^{\prime \prime }\left({\theta }_2\right)}{2}(t-t_{(k-1)s})(t-t_{ks}).\hfill \end{array}$$

Using (25), (26) and triangle inequality, we conclude

$$\parallel e_C^{ks-l}{\parallel }_{\infty }\le O(h^2+{\tau }_C),\parallel {\eta }_C^{ks-l}\parallel \le O(h^2+{\tau }_C).$$

We obtain the result of Theorem 1 by synthesizing the aforementioned two cases. Next, we give the convergence analysis of the scheme on the fine time mesh. □

Theorem 2.

Suppose that the exact solutions $v^n,{\phi }^n$ to the initial boundary value problem equation (1) is sufficiently smooth and let $u_F^n,{\rho }_F^n$ be the numerical solutions on the fine time mesh. Then,

$$\parallel v^n-u_F^n{\parallel }_{\infty }\le O(h^2+{\tau }_C^2+{\tau }_F),\parallel {\phi }^n-{\rho }_F^n\parallel \le O(h^2+{\tau }_C^2+{\tau }_F).$$

Proof. 

Assume $e_{F,j}^n=v_j^n-u_{F,j}^n,{\eta }_{F,j}^n={\phi }_j^n-{\rho }_{F,j}^n,1\le j\le J-1,0\le n\le N$, Subtracting equation (8) from equation (2) and equation (9) from equation (3), we obtain

$$E{r}_{F,j}^n={\left(e_{F,j}^n\right)}_t+{\left({\eta }_{F,j}^{n+1}\right)}_{\widehat{x}}-{\left(e_{F,j}^n\right)}_{x\overline{x}t}+\frac{1}{6h}[(f_x{e}_{F,j-1}^{n+\frac{1}{2}}+f_y{e}_{F,j}^{n+\frac{1}{2}}+f_z{e}_{F,j+1}^{n+\frac{1}{2}})+Q],$$

(29)

$$E{s}_{F,j}^n={\left({\eta }_{F,j}^n\right)}_t+{\left(e_{F,j}^n\right)}_{\widehat{x}},$$

(30)

where

$$\begin{array}{cc}\hfill Q=& \frac{1}{2}[(f_{xx}{\left(e_{C,j-1}^{n+\frac{1}{2}}\right)}^2+f_{yy}{\left(e_{C,j}^{n+\frac{1}{2}}\right)}^2+f_{zz}{\left(e_{C,j+1}^{n+\frac{1}{2}}\right)}^2]\hfill \\ & +[f_{xy}{e}_{C,j-1}^{n+\frac{1}{2}}{e}_{C,j}^{n+\frac{1}{2}}+f_{xz}{e}_{C,j-1}^{n+\frac{1}{2}}{e}_{C,j+1}^{n+\frac{1}{2}}+f_{yz}{e}_{C,j}^{n+\frac{1}{2}}{e}_{C,j+1}^{n+\frac{1}{2}}],\hfill \end{array}$$

and $f_{xx}=f_{xx}(\xi ,\epsilon ,\delta )$,$f_{yy}=f_{yy}(\xi ,\epsilon ,\delta ),f_{zz}=f_{zz}(\xi ,\epsilon ,\delta ),f_{xy}=f_{xy}(\xi ,\epsilon ,\delta ),f_{xz}=f_{xz}(\xi ,\epsilon ,\delta ),f_{yz}=f_{yz}(\xi ,\epsilon ,\delta )$ are the second order partial derivatives of $f(x,y,z)$, $\xi \in (v_{j-1}^n,u_{C,j-1}^n),\epsilon \in (v_j^n,u_{C,j}^n),\delta \in (v_{j+1}^n,u_{C,j+1}^n).$

Taking the inner product (·, ·) on both sides of equation (29) with $e_F^{n+1}+e_F^n$, we have

$$\begin{array}{cc}\hfill & \parallel e_F^{n+1}{\parallel }^2+\parallel e_{F,x}^{n+1}{\parallel }^2=\parallel e_F^n{\parallel }^2+{\parallel e_{F,x}^n\parallel }^2+{\tau }_{F}h\sum _{j=1}^{J-1}\left\{{\eta }_{F,j}^{n+1}\left[{\left(e_{F,j}^{n+1}\right)}_{\widehat{x}}+{\left(e_{F,j}^n\right)}_{\widehat{x}}\right]\right\}\hfill \\ \hfill & -\frac{{\tau }_F}{3}\sum _{j=1}^{J-1}(f_x{e}_{F,j-1}^{n+\frac{1}{2}}+f_y{e}_{F,j}^{n+\frac{1}{2}}+f_z{e}_{F,j+1}^{n+\frac{1}{2}})e_{F,j}^{n+\frac{1}{2}}-\frac{{\tau }_F}{3}\sum _{j=1}^{J-1}Q{e}_{F,j}^{n+\frac{1}{2}}+2{\tau }_F(E{r}_F^n,e_F^{n+\frac{1}{2}}).\hfill \end{array}$$

(31)

Using $f_y=\frac{1}{2}(f_x+f_z)$ and $f_{xx}=-2,f_{yy}=0,f_{zz}=2,f_{xy}=-1,f_{xz}=0,f_{yz}=1$, we obtain

$$\begin{array}{cc}\hfill & \sum _{j=1}^{J-1}(f_x{e}_{F,j-1}^{n+\frac{1}{2}}+f_y{e}_{F,j}^{n+\frac{1}{2}}+f_z{e}_{F,j+1}^{n+\frac{1}{2}})e_{F,j}^{n+\frac{1}{2}}\hfill \\ \hfill & =-(f_x{e}_{F,\overline{x}}^{n+\frac{1}{2}},e_F^{n+\frac{1}{2}})+\frac{3}{h}(f_y{e}_F^{n+\frac{1}{2}},e_F^{n+\frac{1}{2}})+(f_z{e}_{F,x}^{n+\frac{1}{2}},e_F^{n+\frac{1}{2}}),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & \sum _{j=1}^{J-1}Q{e}_{F,j}^{n+\frac{1}{2}}=\frac{1}{2}\sum _{j=1}^{J-1}[f_{xx}{\left(e_{C,j-1}^{n+\frac{1}{2}}\right)}^2+f_{yy}{\left(e_{C,j}^{n+\frac{1}{2}}\right)}^2+f_{zz}{\left(e_{C,j+1}^{n+\frac{1}{2}}\right)}^2]e_{F,j}^{n+\frac{1}{2}}\hfill \\ \hfill & +\sum _{j=1}^{J-1}[f_{xy}{e}_{C,j-1}^{n+\frac{1}{2}}{e}_{C,j}^{n+\frac{1}{2}}+f_{xz}{e}_{C,j-1}^{n+\frac{1}{2}}{e}_{C,j+1}^{n+\frac{1}{2}}+f_{yz}{e}_{C,j}^{n+\frac{1}{2}}{e}_{C,j+1}^{n+\frac{1}{2}}]e_{F,j}^{n+\frac{1}{2}}\hfill \\ \hfill & =({\left(e_C^{n+\frac{1}{2}}\right)}_x^2,e_F^{n+\frac{1}{2}})-({\left(e_C^{n+\frac{1}{2}}\right)}^2,e_{F,x}^{n+\frac{1}{2}})+(e_{C,x}^{n+\frac{1}{2}},e_C^{n+\frac{1}{2}}{e}_F^{n+\frac{1}{2}})-(e_C^{n+\frac{1}{2}},{\left(e_C^{n+\frac{1}{2}}{e}_F^{n+\frac{1}{2}}\right)}_x).\hfill \end{array}$$

(33)

Using Lemma 1 and Cauchy–Schwarz inequality, we have

$$\begin{array}{cc}\hfill & \frac{{\tau }_F}{3}(f_x{e}_{F,\overline{x}}^{n+\frac{1}{2}},e_F^{n+\frac{1}{2}})-\frac{{\tau }_F}{h}(f_y{e}_F^{n+\frac{1}{2}},e_F^{n+\frac{1}{2}})-\frac{{\tau }_F}{3}(f_z{e}_{F,x}^{n+\frac{1}{2}},e_F^{n+\frac{1}{2}})\hfill \\ \hfill & \le M{\tau }_F(\parallel e_{F,x}^{n+\frac{1}{2}}{\parallel }^2+\parallel e_F^{n+\frac{1}{2}}{\parallel }^2),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & -\frac{{\tau }_F}{3}({\left(e_C^{n+\frac{1}{2}}\right)}_x^2,e_F^{n+\frac{1}{2}})+\frac{{\tau }_F}{3}({\left(e_C^{n+\frac{1}{2}}\right)}^2,e_{F,x}^{n+\frac{1}{2}})\hfill \\ \hfill & -\frac{{\tau }_F}{3}(e_{C,x}^{n+\frac{1}{2}},e_C^{n+\frac{1}{2}}{e}_F^{n+\frac{1}{2}})+\frac{{\tau }_F}{3}(e_C^{n+\frac{1}{2}},{\left(e_C^{n+\frac{1}{2}}{e}_F^{n+\frac{1}{2}}\right)}_x)\hfill \\ \hfill & \le M{\tau }_F(\parallel e_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel e_C^{n+\frac{1}{2}}{\parallel }^2+\parallel e_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel e_{C,x}^{n+\frac{1}{2}}{\parallel }^2+\parallel e_F^{n+\frac{1}{2}}{\parallel }^2+\parallel e_{F,x}^{n+\frac{1}{2}}{\parallel }^2),\hfill \end{array}$$

(35)

$$2{\tau }_F(E{r}_F^n,e_F^{n+\frac{1}{2}})\le {\tau }_F\parallel E{r}_F^n{\parallel }^2+M{\tau }_F{\parallel e_F^{n+\frac{1}{2}}\parallel }^2.$$

(36)

Substituting equations (34)-(35) into (31), then

$$\begin{array}{cc}\hfill & \parallel e_F^{n+1}{\parallel }^2+{\parallel e_{F,x}^{n+1}\parallel }^2\hfill \\ \hfill & \le \parallel e_F^n{\parallel }^2+\parallel e_{F,x}^n{\parallel }^2+M{\tau }_F(\parallel {\eta }_F^{n+1}{\parallel }^2+\parallel e_F^{n+1}{\parallel }^2+\parallel e_F^n{\parallel }^2+\parallel e_{F,x}^{n+1}{\parallel }^2+\parallel e_{F,x}^n{\parallel }^2)\hfill \\ \hfill & +M{\tau }_F(\parallel e_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel e_C^{n+\frac{1}{2}}{\parallel }^2+\parallel e_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel e_{C,x}^{n+\frac{1}{2}}{\parallel }^2)+{\tau }_F{\parallel E{r}_F^n\parallel }^2.\hfill \end{array}$$

(37)

Taking the inner product (·, ·) on both sides of equation (30) with ${\eta }_F^{n+1}+{\eta }_F^n$, we obtain

$$\parallel {\eta }_F^{n+1}{\parallel }^2\le \parallel {\eta }_F^n{\parallel }^2+M{\tau }_F(\parallel e_{F,x}^n{\parallel }^2+\parallel {\eta }_F^{n+1}{\parallel }^2+\parallel {\eta }_F^n{\parallel }^2)+{\tau }_F{\parallel E{s}_F^n\parallel }^2.$$

(38)

Add equations (37) and (38), we have

$$\begin{array}{cc}\hfill & \parallel e_F^{n+1}{\parallel }^2+\parallel e_{F,x}^{n+1}{\parallel }^2+\parallel {\eta }_F^{n+1}{\parallel }^2\le \parallel e_F^n{\parallel }^2+\parallel e_{F,x}^n{\parallel }^2+{\parallel {\eta }_F^n\parallel }^2\hfill \\ \hfill & +M{\tau }_F(\parallel {\eta }_F^{n+1}{\parallel }^2+\parallel {\eta }_F^n{\parallel }^2+\parallel e_F^{n+1}{\parallel }^2+\parallel e_F^n{\parallel }^2+\parallel e_{F,x}^{n+1}{\parallel }^2+\parallel e_{F,x}^n{\parallel }^2)\hfill \\ \hfill & +M{\tau }_F(\parallel e_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel e_C^{n+\frac{1}{2}}{\parallel }^2+\parallel e_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel e_{C,x}^{n+\frac{1}{2}}{\parallel }^2)\hfill \\ \hfill & +{\tau }_F\parallel E{r}_F^n{\parallel }^2+{\tau }_F{\parallel E{s}_F^n\parallel }^2.\hfill \end{array}$$

(39)

Let $B_F^n=\parallel e_F^n{\parallel }^2+\parallel e_{F,x}^n{\parallel }^2+{\parallel {\eta }_F^n\parallel }^2$, then

$$\begin{array}{c}{B}_F^{n+1}-B_F^n\le M{\tau }_F(B_F^{n+1}+B_F^n)+M{\tau }_F(\parallel e_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel e_C^{n+\frac{1}{2}}{\parallel }^2+\parallel e_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel e_{C,x}^{n+\frac{1}{2}}{\parallel }^2)\\ +{\tau }_F\parallel E{r}_F^n{\parallel }^2+{\tau }_F{\parallel E{s}_F^n\parallel }^2,\end{array}$$

and obtain

$$(1-M{\tau }_F)(B_F^{n+1}-B_F^n)\le 2M{\tau }_F{B}_F^n+M{\tau }_F(h^4+{\tau }_C^4+{\tau }_F^2).$$

By taking ${\tau }_F$ small enough so that $(1-M{\tau }_F)>\lambda >0$, then

$$B_F^{n+1}-B_F^n\le M{\tau }_F(h^4+{\tau }_C^4+{\tau }_F^2)+M{\tau }_F{B}_F^n.$$

(40)

Summing from 0 to $N-1$ inequalities in equation (40), we obtain

$$B_F^N\le B_F^0+M{\tau }_F(h^4+{\tau }_C^4+{\tau }_F^2)+M{\tau }_F\sum _{n=0}^{N-1}{B}_F^n.$$

(41)

Using Lemma 2, we get

$$B_F^N\le [B_F^0+M{\tau }_F(h^4+{\tau }_C^4+{\tau }_F^2)]e^{MN{\tau }_F}.$$

(42)

From equation (42) and the initial and boundary condition, we have

$$\parallel e_F^n\parallel \le O(h^2+{\tau }_C^2+{\tau }_F),\parallel e_{F,x}^n\parallel \le O(h^2+{\tau }_C^2+{\tau }_F),\parallel {\eta }_F^n\parallel <O(h^2+{\tau }_C^2+{\tau }_F).$$

(43)

Using Lemma 3, it lead to

$$\parallel e_F^n{\parallel }_{\infty }\le O(h^2+{\tau }_C^2+{\tau }_F).$$

(44)

This completes the proof of Theorem 2. □

5. Numerical Results

This section provides some numerical examples aimed at demonstrating the accuracy and computational time of the TT-M finite difference scheme that was discussed in Section 3. We consider the SRLW equation as the following form:

$$\begin{array}{cc}{u}_t+{\rho }_x+u{u}_x-u_{xxt}=0,\hfill & -40\le x\le 40,0<t\le 4,\\ {\rho }_t+u_x=0,\hfill & -40\le x\le 40,0<t\le 4,\\ u\left(x_L,t\right)=u\left(x_R,t\right)=0,\rho \left(x_L,t\right)=\rho \left(x_R,t\right)=0,\hfill & 0<t\le 4,\\ u(x,0)=u_0\left(x\right),\rho (x,0)={\rho }_0\left(x\right),\hfill & -40\le x\le 40.\end{array}$$

The exact solitary wave solution [4] of the SRLW equation (1) has the following form

$$\begin{array}{cc}& u(x,t)=\frac{3\left(v^2-1\right)}{v}{sech}^2\left(\sqrt{\frac{v^2-1}{4v^2}}(x-vt)\right)\hfill \\ & \rho (x,t)=\frac{3\left(v^2-1\right)}{v^2}{sech}^2\left(\sqrt{\frac{v^2-1}{4v^2}}(x-vt)\right).\hfill \end{array}$$

In this section, we chose $v=1.5$ associated with this equation, which takes the form

$$u(x,t)=\frac{5}{2}{sech}^2\frac{\sqrt{5}}{6}\left(x-\frac{3}{2}t\right),\rho (x,t)=\frac{5}{3}{sech}^2\frac{\sqrt{5}}{6}\left(x-\frac{3}{2}t\right),$$

and consider the following initial conditions

$$u_0\left(x\right)=\frac{5}{2}{sech}^2\frac{\sqrt{5}}{6}x,{\rho }_0\left(x\right)=\frac{5}{3}{sech}^2\frac{\sqrt{5}}{6}x.$$

5.1. Error and convergence rate

We define the error and convergence rate by the following formula:

$$\begin{array}{c}{e}_m(h,\tau )=\parallel v^n-u_m^n{\parallel }_{\infty },{\eta }_m(h,\tau )=\parallel {\phi }^n-{\rho }_m^n\parallel ,\\ uRat{e}_m^x={log}_2\left(\frac{e_m(2h,4\tau )}{e_m(h,\tau )}\right),\rho Rat{e}_m^x={log}_2\left(\frac{{\eta }_m(2h,4\tau )}{{\eta }_m(h,\tau )}\right),\\ uRat{e}_m^t={log}_2\left(\frac{e_m(2h,2\tau )}{e_m(h,\tau )}\right),\rho Rat{e}_m^t={log}_2\left(\frac{{\eta }_m(2h,2\tau )}{{\eta }_m(h,\tau )}\right),\end{array}$$

where m represents the TT-M finite difference scheme or the standard nonlinear finite difference (SNFD) scheme. We set ${\tau }_C=4{\tau }_F$ in the entire numerical illustration process.

Table 1 and Table 2 present discrete norm errors, convergence rates, and the time cost for both the TT-M finite difference scheme and the SNFD scheme. To demonstrate the accuracy of the proposed method, we computed the error of TT-M finite difference scheme at the final time $t=4$ for various mesh steps and compare it to the errors obtained by the SNFD scheme. The results from the new scheme show nearly identical significant digits as those obtained by the SNFD scheme. In term of the convergence rate, the results indicate that both the SNFD scheme and the TT-M finite difference scheme achieve approximately second-order convergence in space when ${\tau }_F=h^2$ and first-order in time when $h={\tau }_F$, which confirming the theoretical results.

Figure 1 illustrates the exact and numerical solutions of $u(x,t)$ and $\rho (x,t)$ at $t=4$ obtained by the TT-M finite difference scheme. The results indicate the excellent correspondence between our numerical solution and the exact solution. Furthermore, the CPU times of the two schemes are plotted in Figure 2 under ${\tau }_F=h^2$ and $h={\tau }_F$, respectively. It is worth noting that the TT-M finite difference scheme can significantly decrease computation time. To sum up, the computational performance of the new scheme is obviously better than that of the SNFD scheme.

5.2. Conservative approximations

To further verify the accuracy of the new scheme, we calculate four conservation laws of the SRLW equation (1), such as:

$$\begin{array}{cc}& Q_1=\frac{1}{2}{\int }_{-\infty }^{\infty }\rho dx,Q_2=\frac{1}{2}{\int }_{-\infty }^{\infty }udx,\hfill \\ & Q_3=\frac{1}{2}{\int }_{-\infty }^{\infty }(u^2+u_x^2+{\rho }^2)dx,Q_4=\frac{1}{2}{\int }_{-\infty }^{\infty }(\rho u+\frac{1}{6}{u}^3)dx.\hfill \end{array}$$

Afterwards, employing discrete forms, we are able to compute four approximate conservative quantities which can be represented as

$$\begin{array}{cc}& Q_1=\frac{h}{2}\sum _{j=0}^{J-1}{\rho }_j^n,\hfill \\ & Q_2=\frac{h}{2}\sum _{j=0}^{J-1}{u}_j^n,\hfill \\ & Q_3=\frac{h}{2}\sum _{j=0}^{J-1}{\left(u_j^n\right)}^2+\frac{1}{2h}\sum _{j=0}^{J-1}{(u_{j+1}^n-u_j^n)}^2+\frac{h}{2}\sum _{j=0}^{J-1}{\left({\rho }_j^n\right)}^2,\hfill \\ & Q_4=\frac{h}{2}\sum _{j=0}^{J-1}{\rho }_j^n{u}_j^n+\frac{h}{12}\sum _{j=0}^{J-1}{\left(u_j^n\right)}^3.\hfill \end{array}$$

The quantities values are recorded in Table 3, Table 4, Table 5 and Table 6. In Table 3 and Table 4, regardless of the time step and grid spacing, the quantities $Q_1$ and $Q_2$ remain well-preserved at various times. In Table 5, for the case $h=1/2$ and ${\tau }_F=1/4$, one can see that the quantity $Q_3$ experiences a slight increase as time increases, however, as the spatial and temporal step sizes decrease, the variation of $Q_3$ becomes extremely small. In Table 6, it has been found that for quantity $Q_4$, there was a minor decline under different mesh steps, but it gradually rebounded over time. Meanwhile, as the spatial and temporal step sizes decrease, the $Q_4$ increases slightly. Figure 3 plots the variation curves of four quantities for the case $h=1/8$ and ${\tau }_F=1/64$, which visually demonstrate that our scheme preserves the four conservation laws.

6. Conclusions

The paper presents a new time two-mesh finite difference scheme for the nonlinear symmetric regularized long wave equation with a nonlinear term including derivatives. The time interval is divied into coarse and fine meshes, then the Lagrange’s linear interpolation formula and Taylor’s formula are utilized to construct the three steps time two-mesh finite difference scheme. The convergence of the new scheme are also analyzed and theoretical results are verified by some numerical examples. Compared to the standard nonlinear finite difference scheme, our scheme not only maintains accuracy but also reduces CPU time. Therefore, the TT-M finite difference scheme is a promising method for solving the SRLW equation.