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

Abstract

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity $u(x,t)$ and density $\rho (x,t)$ in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with $O({\tau }_C^2+{\tau }_F+h^2)$ in the discrete $L_{\infty }$-norm for $u(x,t)$ and in the discrete $L_2$-norm for $\rho (x,t)$ are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time.

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 acoustics, 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:

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

(1)

where $u(x,t)$ and $\rho (x,t)$ are the fluid velocity and the density, respectively.

The SRLW equation has attracted significant attention and has been extensively studied in the literature. The existence of global attractors of the SRLW equation was studied in [5]. The travelling and solitary wave solutions of the SRLW equation were investigated by several methods including the exp-function method [6], the ($G^{\prime }/G$)-expansion method [7,8], the tanh-function method [9,10], the Lie symmetry approach [11], the extended simple equation method [12], the analytical method [13], the tan ($(\varphi (\xi )/2)$)-expansion method [14] and the sine–cosine method [15]. In [16,17], the spectral method and the Fourier pseudo-spectral method with a constraint operator were developed as approximations for the nonlinear term of the SRLW equation. Numerical solutions to the SRLW equation have also been studied by numerous methods, ranging from conservative finite difference schemes to mixed finite element methods. In their study, Wang et al. [18] introduced three conservative finite difference schemes that achieve second-order accuracy in both spatial and temporal domains. Their results indicate that each scheme is able to preserve energy conservation, but only the first scheme is capable of preserving mass conservation. Yimnet et al. [19] 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. A coupled conservative three-level implicit scheme achieving fourth-order convergence rate was developed by Hu et al. [20]. Li [21] 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. [22] 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. [23] applied a mixed finite element method to solve the dissipative SRLW equations with damping term. He et al. [24] 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.

In terms of numerical computation, the time two-mesh (TT-M) method, when combined with either the finite element method or the finite difference method, offers better computational efficiency in solving a broad range of nonlinear partial differential equations. For instance, Liu et al. [25] proposed the fast TT-M finite element method for solving the fractional water wave model and applied it successfully to other fractional models. Yin et al. [26] developed a TT-M finite element algorithm for solving a space fractional Allen-Cahn model and analyzed the problem of parameter selection in detail. The TT-M finite element method was also leveraged by Liu et al. [27] to numerically solve the 2D Gray-Scott model with space fractional derivatives. A nonlinear distributed order diffusion model was efficiently solved using the TT-M algorithm in conjunction with the $H_1$-Galerkin mixed finite element method by Wen et al. [28], both smooth and non-smooth solutions were considered. Additionally, Tian et al. [29] developed a finite element method equipped with the TT-M technique to solve the coupled Schrödinger-Boussinesq equations. Moreover, some studies have investigated combining the TT-M and finite difference methods to solve nonlinear fractional partial differential equations, such as the works of Qiu and Xu et al. [30,31], who proposed and analyzed a TT-M algorithm based on finite difference methods. The TT-M technique was also employed by Niu et al. [32] to develop a fast high-order compact difference scheme for the nonlinear distributed order fractional Sobolev model in porous media. Furthermore, He et al. [33] extended the application of the TT-M method by studying a primary scheme of second-order convergence in time and fourth-order in space for solving the nonlinear Schrödinger equation with a time two-mesh high-order compact difference scheme. 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 SRLW Equation (1) is a coupled system and most existing schemes found in the literature are also coupled and require nonlinear implementation, which results in prolonged CPU processing time. The objective of this paper is to develop a novel difference scheme that offers the following three main advantages for solving the SRLW equation: (i) Based on the time two-mesh technique, we proposed a scheme that achieves decoupling and the nonlinear term of the system is linearized by using Taylor’s formula for a function with three variables, which is different from the literature [32,33]. In [32,33], the time two-mesh scheme is formulated by using Taylor’s formula for a function with one or two variables. As a result, our scheme becomes a linearized system in approximate numerical solution and can reduce the computational time. (ii) The scheme’s convergence and stability have been verified through detailed proof, and the theoretical analysis process is more complex than that of existing methods, since the linearized scheme contains a function of three variables. (iii) The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation for solutions. The conservation law of four quantities is preserved in the new scheme at the discrete level.

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 and stability of the scheme is analyzed. In Section 5, some numerical results are provided to test the theoretical results and computational efficiency of the scheme. Finally, in Section 6, we provide the conclusions of the paper.

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 a different value in a different place.

Since $u\to 0$ for $x\to +\infty$ or $x\to -\infty$, we may assume $u_{-1}^n=u_{J+1}^n=0,1\le n\le N$ 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 present some useful lemmas.

Lemma 1

(See [24]). For any mesh functions $u^n,w^n\in H_{h,0}$, we have

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

Lemma 2

(See [33,34]). Assume that a sequence of non-negative 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 [22,34]). 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 of them is divided into $s\left(2\le s\in Z^{+}\right)$ fine time intervals. The coarse time mesh with the nodes $t_{ks}=k{\tau }_C(k=1,\dots ,P)$ satisfying $0=t_0<t_s<t_{2s}<\cdots <t_{Ps}=T$ and the fine time mesh with the nodes $t_n=n{\tau }_F(n=1,2,\dots ,Ps=N)$ 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) are 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

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

(2)

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

(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 the coarse time nonlinear finite difference scheme is given as

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

(4)

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

(5)

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

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 $t_{ks-l}(l=1,2,\dots ,s-1$ and $k=1,2,\dots ,P,ks-l=n)$, 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 not needed for computation since they are not used in step 3.

Step 3: based on all the coarse numerical solutions $u_{C,j}^n(n=0,1,2,\dots ,Ps=N,$ $j=1,2,\dots ,J-1)$ 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

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

(8)

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

(9)

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

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.

Our method, similarly to the Gauss-Seidel method applied to linear systems, has been modified in order to enhance the accuracy of fine mesh solutions $u_F^{n+1}$ by using $u_F^n$ in calculation.

4. Convergence Analysis and Stability 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

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

(10)

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

(11)

The proof contains two cases. Firstly, we consider the case of $n=ks(k=0,1,2,\dots ,P)$, 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

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

(12)

where

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

Notice that

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

(13)

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

(14)

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

(15)

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

(16)

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

(17)

then substituting Equations (13)–(17) into (12), we have

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

(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

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

(20)

From the 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 obtain

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

(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

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

and using Lemma 2, we obtain

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

(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 $n=ks-l,(l=1,2,\dots ,s-1$ and $k=1,2,\dots ,P,ks-l=n)$. Based on the Lagrange’s interpolation formula, we obtain

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

(27)

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

(28)

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

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

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

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

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

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

(29)

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

(30)

where

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

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

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

(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)

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

(33)

Using Lemma 1 and the 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)

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

(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) and (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

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

(41)

Using Lemma 2, we obtain

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

(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 led to

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

(44)

This completes the proof of Theorem 2. □

Next, we prove stability of the scheme on the coarse time mesh.

Theorem 3.

Suppose that $u_0\in H_{h,0}\left[x_L,x_R\right]$, if ${\tau }_C$ and h are sufficiently small, then the proposed scheme (4)–(5) is stable with respect to the initial conditions.

Proof. 

The proof contains two cases. Firstly, we consider the case of $n=ks(k=0,1,2,\dots ,P)$. Taking the inner product (·, ·) on both sides of Equation (4) with $u_C^{n+1}+u_C^n$, we have

$$\begin{array}{cc}\hfill & {∥u_C^{n+1}∥}^2-{∥u_C^n∥}^2+{∥u_{C,x}^{n+1}∥}^2-{∥u_{C,x}^n∥}^2\hfill \\ \hfill & +{\tau }_{C}h\sum _{j=1}^{J-1}[-{\rho }_{C,j}^{n+1}{(u_{C,j}^{n+1})}_{\widehat{x}}-{\rho }_{C,j}^{n+1}{(u_{C,j}^n)}_{\widehat{x}}]=0.\hfill \end{array}$$

(45)

Taking the inner product (·, ·) on both sides of Equation (5) with ${\rho }_C^{n+1}+{\rho }_C^n$, we obtain

$$\parallel {\rho }_C^{n+1}{\parallel }^2-{|{\rho }_C^n\parallel }^2+{\tau }_{C}h\sum _{j=1}^{J-1}[{(u_{C,j}^n)}_{\widehat{x}}{\rho }_{C,j}^{n+1}+{(u_{C,j}^n)}_{\widehat{x}}{\rho }_{C,j}^n]=0.$$

(46)

Add Equations (45) and (46), we obtain

$$\begin{array}{cc}\hfill & \parallel u_C^{n+1}{\parallel }^2+\parallel u_{C,x}^{n+1}{\parallel }^2+\parallel {\rho }_C^{n+1}{\parallel }^2=\parallel u_C^n{\parallel }^2+\parallel u_{C,x}^n{\parallel }^2+{\parallel {\rho }_C^n\parallel }^2\hfill \\ \hfill & +{\tau }_{C}h\sum _{j=1}^{J-1}{\rho }_{C,j}^{n+1}{(u_{C,j}^{n+1})}_{\widehat{x}}-{\tau }_{C}h\sum _{j=1}^{J-1}{(u_{C,j}^n)}_{\widehat{x}}{\rho }_{C,j}^n\hfill \\ \hfill & \le \parallel u_C^n{\parallel }^2+\parallel u_{C,x}^n{\parallel }^2+{|{\rho }_C^n\parallel }^2\hfill \\ \hfill & +M{\tau }_C(\parallel {\rho }_C^{n+1}{\parallel }^2+\parallel {\rho }_C^n{\parallel }^2+\parallel u_C^{n+1}{\parallel }^2+\parallel u_C^n{\parallel }^2+\parallel u_{C,x}^{n+1}{\parallel }^2+\parallel u_{C,x}^n{\parallel }^2).\hfill \end{array}$$

(47)

Let $D_C^n=\parallel u_C^n{\parallel }^2+\parallel u_{C,x}^n{\parallel }^2+{\parallel {\rho }_C^n\parallel }^2$, then Equation (47) becomes

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

and obtain

$$(1-M{\tau }_C)(D_C^{n+1}-D_C^n)\le 2M{\tau }_C{D}_C^n.$$

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

$$D_C^{n+1}-D_C^n\le M{\tau }_C{D}_C^n.$$

(48)

Summing from 0 to $P-1$ inequalities in Equation (48), we have

$$D_C^N\le D_C^0+M{\tau }_C\sum _{n=0}^{N-1}{D}_C^n,$$

and using Lemma 2, we obtain

$$D_C^N\le D_C^0{e}^{MT},$$

(49)

that is

$$\parallel u_C^n{\parallel }^2+\parallel u_{C,x}^n{\parallel }^2+\parallel {\rho }_C^n{\parallel }^2\le M(\parallel u_C^0{\parallel }^2+\parallel u_{C,x}^0{\parallel }^2+\parallel {\rho }_C^0{\parallel }^2).$$

Secondly, we consider the case of $n=ks-l,(l=1,2,\dots ,s-1$ and $k=1,2,\dots ,P,ks-l=n)$. From Equations (6) and (7), we obtain

$$\begin{array}{cc}& \parallel u_C^{ks-l}\parallel \le \frac{l}{s}\parallel u_C^{(k-1)s}\parallel +(1-\frac{l}{s})\parallel u_C^{ks}\parallel \le M(\parallel u_C^0\parallel +\parallel u_{C,x}^0\parallel +\parallel {\rho }_C^0\parallel ),\hfill \\ & \parallel {\rho }_C^{ks-l}\parallel \le \frac{l}{s}\parallel {\rho }_C^{(k-1)s}\parallel +(1-\frac{l}{s})\parallel {\rho }_C^{ks}\parallel \le M(\parallel u_C^0\parallel +\parallel u_{C,x}^0\parallel +\parallel {\rho }_C^0\parallel ).\hfill \end{array}$$

The results obtained in the aforementioned two cases mean that the solution of scheme (4)–(5) is stable on the coarse time mesh. This completes the proof of Theorem 3. □

Next, we present stability analysis of the scheme on the fine time mesh.

Theorem 4.

Suppose that $u_0\in H_{h,0}\left[x_L,x_R\right]$, if ${\tau }_F$ and h are sufficiently small, then the proposed scheme (8)–(9) is stable with respect to the initial conditions.

Proof. 

Taking the inner product (·, ·) on both sides of Equation (8) with $u_F^{n+1}+u_F^n$, we have

$$\begin{array}{cc}\hfill \frac{1}{{\tau }_F}& (\parallel u_F^{n+1}{\parallel }^2+\parallel u_{F,x}^{n+1}{\parallel }^2)=\frac{1}{{\tau }_F}(\parallel u_F^n{\parallel }^2+\parallel u_{F,x}^n{\parallel }^2)\hfill \\ \hfill & -h\sum _{j=1}^{J-1}[-{\rho }_{F,j}^{n+1}{(u_{F,j}^{n+1})}_{\widehat{x}}-{\rho }_{F,j}^{n+1}{(u_{F,j}^n)}_{\widehat{x}}]\hfill \\ \hfill & -\frac{1}{3}\sum _{j=1}^{J-1}[(u_{C,j+1}^{n+\frac{1}{2}}-u_{C,j-1}^{n+\frac{1}{2}})u_{C,j}^{n+\frac{1}{2}}+{(u_{C,j+1}^{n+\frac{1}{2}})}^2-{(u_{C,j-1}^{n+\frac{1}{2}})}^2]u_{F,j}^{n+\frac{1}{2}}\hfill \\ \hfill & -\frac{1}{3}\sum _{j=1}^{J-1}(f_x{u}_{F,j-1}^{n+\frac{1}{2}}+f_y{u}_{F,j}^{n+\frac{1}{2}}+f_z{u}_{F,j+1}^{n+\frac{1}{2}})u_{F,j}^{n+\frac{1}{2}}\hfill \\ \hfill & +\frac{1}{3}\sum _{j=1}^{J-1}(f_x{u}_{C,j-1}^{n+\frac{1}{2}}+f_y{u}_{C,j}^{n+\frac{1}{2}}+f_z{u}_{C,j+1}^{n+\frac{1}{2}})u_{F,j}^{n+\frac{1}{2}}.\hfill \end{array}$$

(50)

From the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{cc}\hfill & -\frac{1}{3}\sum _{j=1}^{J-1}[(u_{C,j+1}^{n+\frac{1}{2}}-u_{C,j-1}^{n+\frac{1}{2}})u_{C,j}^{n+\frac{1}{2}}+{(u_{C,j+1}^{n+\frac{1}{2}})}^2-{(u_{C,j-1}^{n+\frac{1}{2}})}^2]u_{F,j}^{n+\frac{1}{2}}\hfill \\ \hfill & \le M(\parallel u_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel u_C^{n+\frac{1}{2}}{\parallel }^2+\parallel u_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel u_{C,x}^{n+\frac{1}{2}}{\parallel }^2+\parallel u_F^{n+\frac{1}{2}}{\parallel }^2+\parallel u_{F,x}^{n+\frac{1}{2}}{\parallel }^2),\hfill \end{array}$$

(51)

$$-\frac{1}{3}\sum _{j=1}^{J-1}(f_x{u}_{F,j-1}^{n+\frac{1}{2}}+f_y{u}_{F,j}^{n+\frac{1}{2}}+f_z{u}_{F,j+1}^{n+\frac{1}{2}})u_{F,j}^{n+\frac{1}{2}}\le M(\parallel u_F^{n+\frac{1}{2}}{\parallel }^2+\parallel u_{F,x}^{n+\frac{1}{2}}{\parallel }^2),$$

(52)

$$\begin{array}{cc}\hfill & \frac{1}{3}\sum _{j=1}^{J-1}(f_x{u}_{C,j-1}^{n+\frac{1}{2}}+f_y{u}_{C,j}^{n+\frac{1}{2}}+f_z{u}_{C,j+1}^{n+\frac{1}{2}})u_{F,j}^{n+\frac{1}{2}}\hfill \\ \hfill & \le M(\parallel u_C^{n+\frac{1}{2}}{\parallel }^2+\parallel u_{C,x}^{n+\frac{1}{2}}{\parallel }^2+\parallel u_F^{n+\frac{1}{2}}{\parallel }^2+\parallel u_{F,x}^{n+\frac{1}{2}}{\parallel }^2),\hfill \end{array}$$

(53)

then substituting Equations (51)–(53) into (50), we have

$$\begin{array}{cc}\hfill & \parallel u_F^{n+1}{\parallel }^2+\parallel u_{F,x}^{n+1}{\parallel }^2\le \parallel u_F^n{\parallel }^2+{\parallel u_{F,x}^n\parallel }^2+{\tau }_{F}h\sum _{j=1}^{J-1}[{\rho }_{F,j}^{n+1}{(u_{F,j}^{n+1})}_{\widehat{x}}+{\rho }_{F,j}^{n+1}{(u_{F,j}^n)}_{\widehat{x}}]\hfill \\ \hfill & +M{\tau }_F(\parallel u_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel u_C^{n+\frac{1}{2}}{\parallel }^2+\parallel u_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel u_{C,x}^{n+\frac{1}{2}}{\parallel }^2+\parallel u_C^{n+\frac{1}{2}}{\parallel }^2+\parallel u_{C,x}^{n+\frac{1}{2}}{\parallel }^2)\hfill \\ \hfill & +M{\tau }_F(\parallel u_F^{n+\frac{1}{2}}{\parallel }^2+\parallel u_{F,x}^{n+\frac{1}{2}}{\parallel }^2).\hfill \end{array}$$

(54)

Taking the inner product (·, ·) on both sides of Equation (9) with ${\rho }_F^{n+1}+{\rho }_F^n$, we obtain

$$\parallel {\rho }_F^{n+1}{\parallel }^2={\parallel {\rho }_F^n\parallel }^2-{\tau }_{F}h\sum _{j=1}^{J-1}[{(u_{F,j}^n)}_{\widehat{x}}{\rho }_{F,j}^{n+1}+{(u_{F,j}^n)}_{\widehat{x}}{\rho }_{F,j}^n].$$

(55)

Add Equations (54) and (55), we obtain

$$\begin{array}{cc}\hfill & \parallel u_F^{n+1}{\parallel }^2+\parallel u_{F,x}^{n+1}{\parallel }^2+\parallel {\rho }_F^{n+1}{\parallel }^2\le \parallel u_F^n{\parallel }^2+\parallel u_{F,x}^n{\parallel }^2+{\parallel {\rho }_F^n\parallel }^2\hfill \\ \hfill & +M{\tau }_F(\parallel u_F^{n+1}{\parallel }^2+\parallel u_{F,x}^n{\parallel }^2+\parallel u_{F,x}^{n+1}{\parallel }^2+\parallel u_{F,x}^n{\parallel }^2+\parallel {\rho }_F^{n+1}{\parallel }^2+\parallel {\rho }_F^n{\parallel }^2)\hfill \\ \hfill & +M{\tau }_F(\parallel u_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel u_C^{n+\frac{1}{2}}{\parallel }^2+\parallel u_C^{n+\frac{1}{2}}{\parallel }_{\infty }^2\parallel u_{C,x}^{n+\frac{1}{2}}{\parallel }^2+\parallel u_C^{n+\frac{1}{2}}{\parallel }^2+\parallel u_{C,x}^{n+\frac{1}{2}}{\parallel }^2).\hfill \end{array}$$

(56)

Let $D_F^n=\parallel u_F^n{\parallel }^2+\parallel u_{F,x}^n{\parallel }^2+{\parallel {\rho }_F^n\parallel }^2$, then combine with the result of Theorem 3, the Equation (56) becomes

$$\begin{array}{c}\hfill D_F^{n+1}-D_F^n\le M{\tau }_F(D_F^{n+1}+D_F^n)+M{\tau }_F(\parallel u_C^0{\parallel }^2+\parallel u_{C,x}^0{\parallel }^2),\end{array}$$

and obtain

$$\begin{array}{c}\hfill \left(1-M{\tau }_F\right)(D_F^{n+1}-D_F^n)\le 2M{\tau }_F{D}_F^n+M{\tau }_F(\parallel u_C^0{\parallel }^2+\parallel u_{C,x}^0{\parallel }^2).\end{array}$$

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

$$\begin{array}{c}\hfill D_F^{n+1}-D_F^n\le M{\tau }_F{D}_F^n+M{\tau }_F(\parallel u_C^0{\parallel }^2+\parallel u_{C,x}^0{\parallel }^2).\end{array}$$

(57)

Summing from 0 to $N-1$ inequalities in Equation (57), we obtain

$$\begin{array}{c}\hfill \sum _{n=0}^{N-1}(D_F^{n+1}-D_F^n)\le M{\tau }_F\sum _{n=0}^{N-1}{D}_F^n+M(\parallel u_C^0{\parallel }^2+\parallel u_{C,x}^0{\parallel }^2),\end{array}$$

(58)

and then

$$\begin{array}{c}\hfill D_F^N\le D_F^0+M{\tau }_F\sum _{n=0}^{N-1}{D}_F^n+M(\parallel u_C^0{\parallel }^2+\parallel u_{C,x}^0{\parallel }^2).\end{array}$$

(59)

Using Lemma 2, we obtain

$$\begin{array}{c}\hfill D_F^N\le [D_F^0+M\parallel u_C^0{\parallel }^2+\parallel u_{C,x}^0{\parallel }^2)]e^{MT},\end{array}$$

(60)

that is

$$\begin{array}{c}\hfill \parallel u_F^n{\parallel }^2+\parallel u_{F,x}^n{\parallel }^2+\parallel {\rho }_F^n{\parallel }^2\le M(\parallel u_F^0{\parallel }^2+\parallel u_{F,x}^0{\parallel }^2+\parallel {\rho }_F^0{\parallel }^2+\parallel u_C^0{\parallel }^2+\parallel u_{C,x}^0{\parallel }^2).\end{array}$$

(61)

Notice that $\parallel u_C^0{\parallel }^2={\parallel u_F^0\parallel }^2$, $\parallel u_{C,x}^0{\parallel }^2={\parallel u_{F,x}^0\parallel }^2$, then

$$\begin{array}{c}\hfill \parallel u_F^n{\parallel }^2+\parallel u_{F,x}^n{\parallel }^2+\parallel {\rho }_F^n{\parallel }^2\le M(\parallel u_F^0{\parallel }^2+\parallel u_{F,x}^0{\parallel }^2+\parallel {\rho }_F^0{\parallel }^2),\end{array}$$

(62)

which indicates that the solution of scheme (8)–(9) is stable on the fine time mesh. This completes the proof of Theorem 4. □

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. All simulations were implemented using a Intel(R) i7-10710U 1.61 GHz CPU and 16 GB memory personal computer running Windows 10 and Matlab R2019b. The SRLW equation is represented in the following formation:

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

and consider the following initial conditions

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

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

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

which takes the form of

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

when we set $v=1.5$.

Next, we presented some values of error, convergence rate, conservation laws and long time simulation of the proposed scheme.

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 in [22]. We set ${\tau }_C=4{\tau }_F$ in the whole numerical illustration process.

In order to illustrate the advantage of this approach, we calculated the error, convergence rate and computational time of the presented scheme and compared these numerical results with those obtained from scheme in [22].

Table 1. The errors and convergence rates with ${\tau }_F=h^2$.

SNFD Scheme in [22]
$\left(\mathit{h},{\mathit{\tau }}_{\mathit{F}}\right)$	${\mathit{e}}_{\mathit{SNFD}}(\mathit{h},{\mathit{\tau }}_{\mathit{F}})$	${\mathit{uRate}}_{\mathit{S}\mathit{N}\mathit{F}\mathit{D}}^{\mathit{x}}$	${\mathit{\eta }}_{\mathit{SNFD}}(\mathit{h},{\mathit{\tau }}_{\mathit{F}})$	$\mathit{\rho }$${\mathit{Rate}}_{\mathit{S}\mathit{N}\mathit{F}\mathit{D}}^{\mathit{x}}$	CPU(s)
$\left(\frac{1}{2},\frac{1}{4}\right)$	$8.2798\times {10}^{-2}$	—	$2.9797\times {10}^{-1}$	—	0.14
$\left(\frac{1}{4},\frac{1}{16}\right)$	$2.0361\times {10}^{-2}$	2.02	$7.3503\times {10}^{-2}$	2.02	0.51
$\left(\frac{1}{8},\frac{1}{64}\right)$	$5.0629\times {10}^{-3}$	2.01	$1.8306\times {10}^{-2}$	2.01	12.30
$\left(\frac{1}{16},\frac{1}{256}\right)$	$1.2641\times {10}^{-3}$	2.00	$4.5719\times {10}^{-3}$	2.00	136.31
$\left(\frac{1}{32},\frac{1}{1024}\right)$	$3.1597\times {10}^{-4}$	2.00	$1.1427\times {10}^{-3}$	2.00	1943.47
Present scheme
$\left(\mathit{h},{\mathit{\tau }}_{\mathit{F}}\right)$	${\mathit{e}}_{\mathit{TT}-\mathit{M}}(\mathit{h},{\mathit{\tau }}_{\mathit{F}})$	${\mathit{uRate}}_{\mathit{TT}-\mathit{M}}^{\mathit{x}}$	${\mathit{\eta }}_{\mathit{TT}-\mathit{M}}(\mathit{h},{\mathit{\tau }}_{\mathit{F}})$	$\mathit{\rho }$${\mathit{Rate}}_{\mathit{TT}-\mathit{M}}^{\mathit{x}}$	CPU(s)
$\left(\frac{1}{2},\frac{1}{4}\right)$	$8.3934\times {10}^{-2}$	—	$3.0015\times {10}^{-1}$	—	0.10
$\left(\frac{1}{4},\frac{1}{16}\right)$	$2.0360\times {10}^{-2}$	2.04	$7.3555\times {10}^{-2}$	2.03	0.30
$\left(\frac{1}{8},\frac{1}{64}\right)$	$5.0616\times {10}^{-3}$	2.01	$1.8308\times {10}^{-2}$	2.01	7.53
$\left(\frac{1}{16},\frac{1}{256}\right)$	$1.2639\times {10}^{-3}$	2.00	$4.5721\times {10}^{-3}$	2.00	76.99
$\left(\frac{1}{32},\frac{1}{1024}\right)$	$3.1596\times {10}^{-4}$	2.00	$1.1427\times {10}^{-3}$	2.00	1297.88

and

Table 2. The errors and convergence rates with $h={\tau }_F$.

SNFD Scheme in [22]
$\left(\mathit{h},{\mathit{\tau }}_{\mathit{F}}\right)$	${\mathit{e}}_{\mathit{SNFD}}(\mathit{h},{\mathit{\tau }}_{\mathit{F}})$	${\mathit{uRate}}_{\mathit{S}\mathit{N}\mathit{F}\mathit{D}}^{\mathit{t}}$	${\mathit{\eta }}_{\mathit{SNFD}}(\mathit{h},{\mathit{\tau }}_{\mathit{F}})$	$\mathit{\rho }$${\mathit{Rate}}_{\mathit{S}\mathit{N}\mathit{F}\mathit{D}}^{\mathit{t}}$	CPU(s)
$\left(\frac{1}{8},\frac{1}{8}\right)$	$3.1920\times {10}^{-2}$	—	$1.1381\times {10}^{-1}$	—	2.47
$\left(\frac{1}{16},\frac{1}{16}\right)$	$1.5552\times {10}^{-2}$	1.04	$5.5256\times {10}^{-2}$	1.04	14.09
$\left(\frac{1}{32},\frac{1}{32}\right)$	$7.6748\times {10}^{-3}$	1.02	$2.7221\times {10}^{-2}$	1.02	109.20
$\left(\frac{1}{64},\frac{1}{64}\right)$	$3.8121\times {10}^{-3}$	1.01	$1.3509\times {10}^{-2}$	1.01	770.01
$\left(\frac{1}{128},\frac{1}{128}\right)$	$1.8997\times {10}^{-3}$	1.00	$6.7298\times {10}^{-3}$	1.00	5792.53
Present scheme
$\left(\mathit{h},{\mathit{\tau }}_{\mathit{F}}\right)$	${\mathit{e}}_{\mathit{TT}-\mathit{M}}(\mathit{h},{\mathit{\tau }}_{\mathit{F}})$	${\mathit{uRate}}_{\mathit{TT}-\mathit{M}}^{\mathit{t}}$	${\mathit{\eta }}_{\mathit{TT}-\mathit{M}}(\mathit{h},{\mathit{\tau }}_{\mathit{F}})$	$\mathit{\rho }$${\mathit{Rate}}_{\mathit{TT}-\mathit{M}}^{\mathit{t}}$	CPU(s)
$\left(\frac{1}{8},\frac{1}{8}\right)$	$3.1962\times {10}^{-2}$	—	$1.1414\times {10}^{-1}$	—	1.49
$\left(\frac{1}{16},\frac{1}{16}\right)$	$1.5540\times {10}^{-2}$	1.04	$5.5323\times {10}^{-2}$	1.04	7.10
$\left(\frac{1}{32},\frac{1}{32}\right)$	$7.6694\times {10}^{-3}$	1.02	$2.7237\times {10}^{-2}$	1.02	51.40
$\left(\frac{1}{64},\frac{1}{64}\right)$	$3.8102\times {10}^{-3}$	1.01	$1.3513\times {10}^{-2}$	1.01	414.33
$\left(\frac{1}{128},\frac{1}{128}\right)$	$1.8991\times {10}^{-3}$	1.00	$6.7308\times {10}^{-3}$	1.01	3289.71

present discrete norm errors, the corresponding convergence rates and the time cost for both the TT-M finite difference scheme and the SNFD scheme in [22] under ${\tau }_F=h^2$ and $h={\tau }_F$, respectively. We recorded the error of both schemes at the final time $t=4$ for various mesh steps. The results from the new scheme show nearly identical important values as those obtained by the SNFD scheme in [22]. In term of the convergence rate, the results indicate that both the TT-M finite difference scheme and the SNFD 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 in Theorem 2. Especially, the proposed scheme achieves a remarkable reduction in computation time of over 30 percent as compared to the SNFD scheme under ${\tau }_F=h^2$ and nearly half the computational time of the SNFD scheme under $h={\tau }_F$. The obtained results demonstrate a clear improvement achieved by our scheme over the previous method reported by [22].

The 3D images of numerical solutions of $u(x,t)$ and $\rho (x,t)$ computed by the present scheme were shown in Figure 1. From the graph, one can see the propagation state of waves over a period of time. Figure 2 illustrates the exact and numerical solutions of $u(x,t)$ and $\rho (x,t)$ at $t=4$ obtained by the present scheme. It is evident that our numerical solutions exhibit an excellent correspondence with the exact solution. Furthermore, the CPU times of the two schemes are plotted in Figure 3 under ${\tau }_F=h^2$ and $h={\tau }_F$, respectively. From the figure, it is also can be seen that the present scheme can significantly decrease computation time. To sum up, compared with the method in [22], the scheme proposed in this paper not only ensures error and convergence order, but also greatly reduces computational time.

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

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

The values of four quantities are recorded in

Table 3. Quantity $Q_1$ under different mesh steps h and ${\tau }_F$ at various times.

TT-M Finite Difference Scheme
	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{16}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{64}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{16}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{256}}\mathbf{)}$
$t=0.0$	4.4721359549	4.4721359549	4.4721359549	4.4721359549
$t=0.5$	4.4721359549	4.4721359549	4.4721359549	4.4721359549
$t=1.0$	4.4721359549	4.4721359549	4.4721359549	4.4721359549
$t=1.5$	4.4721359549	4.4721359549	4.4721359549	4.4721359549
$t=2.0$	4.4721359549	4.4721359549	4.4721359549	4.4721359549
$t=2.5$	4.4721359549	4.4721359549	4.4721359549	4.4721359549
$t=3.0$	4.4721359549	4.4721359549	4.4721359549	4.4721359549
$t=3.5$	4.4721359549	4.4721359549	4.4721359549	4.4721359549
$t=4.0$	4.4721359549	4.4721359549	4.4721359549	4.4721359549

,

Table 4. Quantity $Q_2$ under different mesh steps h and ${\tau }_F$ at various times.

TT-M Finite Difference Scheme
	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{16}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{64}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{16}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{256}}\mathbf{)}$
$t=0.0$	6.7082039324	6.7082039324	6.7082039324	6.7082039324
$t=0.5$	6.7082039324	6.7082039324	6.7082039324	6.7082039324
$t=1.0$	6.7082039324	6.7082039324	6.7082039324	6.7082039324
$t=1.5$	6.7082039324	6.7082039324	6.7082039324	6.7082039324
$t=2.0$	6.7082039324	6.7082039324	6.7082039324	6.7082039324
$t=2.5$	6.7082039324	6.7082039324	6.7082039324	6.7082039324
$t=3.0$	6.7082039324	6.7082039324	6.7082039324	6.7082039324
$t=3.5$	6.7082039324	6.7082039324	6.7082039324	6.7082039324
$t=4.0$	6.7082039323	6.7082039323	6.7082039323	6.7082039323

,

Table 5. Quantity $Q_3$ under different mesh steps h and ${\tau }_F$ at various times.

TT-M Finite Difference Scheme
	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{16}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{64}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{16}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{256}}\mathbf{)}$
$t=0.0$	17.3814360100	17.3890764778	17.3909982095	17.3914793723
$t=0.5$	17.3879364459	17.3894925427	17.3910247524	17.3914810404
$t=1.0$	17.4021575260	17.3904434452	17.3910850544	17.3914848239
$t=1.5$	17.4178948195	17.3914089843	17.3911453660	17.3914885933
$t=2.0$	17.4279575149	17.3919924045	17.3911803618	17.3914907570
$t=2.5$	17.4320624736	17.3921089940	17.3911854445	17.3914910387
$t=3.0$	17.4298852213	17.3919072419	17.3911710024	17.3914901056
$t=3.5$	17.4257242289	17.3915916231	17.3911504036	17.3914888028
$t=4.0$	17.4208004460	17.3913048704	17.3911324701	17.3914876800

and

Table 6. Quantity $Q_4$ under different mesh steps h and ${\tau }_F$ at various times.

TT-M Finite Difference Scheme
	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{16}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{64}}\mathbf{)}$	$\mathbf{(}\frac{\mathbf{1}}{\mathbf{16}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{256}}\mathbf{)}$
$t=0.0$	29.8645685095	29.8270800111	29.8174683777	29.8150480627
$t=0.5$	29.5194998279	29.7377964805	29.7949677954	29.8094115019
$t=1.0$	29.2953094352	29.6804305056	29.7805099425	29.8057894879
$t=1.5$	29.2373157339	29.6665651150	29.7770997278	29.8049405034
$t=2.0$	29.3052034101	29.6864373371	29.7822468973	29.8062391542
$t=2.5$	29.4333002603	29.7204018949	29.7909264484	29.8084218678
$t=3.0$	29.5520897511	29.7519209103	29.7989376025	29.8104338587
$t=3.5$	29.6355327196	29.7733189521	29.8043614200	29.8117950932
$t=4.0$	29.6765644108	29.7843463859	29.8071613054	29.8124980105

. 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 4 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.

5.3. Long-Time Simulation

Using the parameters $h=0.1$, ${\tau }_F=0.01$, $x_L=-20$, $x_R=100$ and $T=40$, the long time waveforms of $u(x,t)$ and $\rho (x,t)$ generated by the present scheme are illustrated in Figure 5. The waveforms at t = 20 and 40 demonstrate a significant level of agreement with the waveforms at t = 0, which also indicates the accuracy of the scheme. Figure 6 displays the computational times of both the SNFD scheme and proposed scheme at long time. From the graph, it can be seen that the computational time difference between the two schemes grows increasingly as the simulation time increases, which indicates that the longer the simulation time, the more computational time can be saved by the scheme proposed in this paper. This once again proves the advantage of our scheme compared to the SNFD scheme in [22].

6. Conclusions

This paper presents a new time two-mesh finite difference scheme for solving the symmetric regularized long wave equation, which contains a nonlinear derivative term in its formulation. Based on the previous work, our aim is to obtain a scheme that is not only error-preserving but also can save more computational time compared to the scheme in [22]. To achieve this goal, we have constructed a TT-M finite difference scheme that mainly consists of three steps. Firstly, the time interval is divided into coarse and fine time meshes, then the nonlinear system is solved on the coarse time mesh; secondly, coarse numerical solutions on the fine time mesh are computed using an interpolation formula based on the solutions derived in the step one; lastly, the nonlinear term of the SRLW equation is linearized using Taylor’s formula for a function with three variables and constructed the TT-M finite difference linear numerical scheme on the fine time mesh. In terms of theoretical results, we investigated the convergence and stability of numerical solutions. We observed that the solutions obtained from our proposed scheme converged to the exact solutions of the SRLW equation, and the rate of convergence is $O({\tau }_C^2+{\tau }_F+h^2)$. Additionally, ${\tau }_F$ and h are sufficiently small, then the proposed scheme (8)–(9) is stable with respect to the initial conditions on the fine time mesh. The numerical results demonstrated that the proposed method effectively supported the analysis of the convergence rate. Meanwhile, one can observe that the errors of $u(x,t)$ and $\rho (x,t)$ in both the SNFD scheme in [22] and the present scheme are nearly identical, but the proposed scheme can reduce the computational time compared with the SNFD scheme. In addition, we calculated four conservation laws of the SRLW equation, regardless of the time step and grid spacing; the four quantities remain well-preserved at various times. The long time simulation example of the proposed scheme is also conducted in this part, the result shows that the waveform always maintains its original state in a long time. The error increases gently as simulation time is extended, which is a limitation of the proposed scheme and it will be improved by other methods in our future work. These findings demonstrate that the proposed method is more effective and yields a better improvement in solving the SRLW equation.