Skip to main content

A high-accuracy compact conservative scheme for generalized regularized long-wave equation

Abstract

In this article, we develop a high-order compact conservative numerical scheme to solve the initial-boundary problem of GRLW equation. The proposed scheme is three-level and linear-implicit based on a finite difference method. A detailed numerical analysis of the scheme is presented including a convergence analysis result. Some numerical examples are provided to show the present scheme is efficient, reliable, and of high accuracy.

1 Introduction

The Cauchy problem of the generalized regularized long-wave (GRLW) equation reads

$$\begin{aligned} &u_{t}-\beta u_{xxt}+u_{x}+\alpha \bigl(u^{p}\bigr)_{x}=0, \end{aligned}$$
(1.1)
$$\begin{aligned} &u(x,0)=u_{0}(x), \end{aligned}$$
(1.2)

where α, β are positive constants and p is a positive integer [1]. The GRLW equation was first put forward by Peregrine [2] and Benjamin et al. [3] as a model for small-amplitude long waves on the surface of water in a channel. Many authors [4–8] have recently studied models for long waves in nonlinear dispersive systems. When \(p=2\), (1.1) is usually called the RLW equation. When \(p=3\), (1.1) is called a modified regularized long-wave (MRLW) equation. Various numerical techniques have been developed to solve the equation. These partly include the finite difference method, finite element methods, the least squares method, and a collocation method with quadratic B-splines, cubic B-splines and septic splines; we refer to [9–20], and references therein.

In general, the solutions of the system (1.1)-(1.2) decays rapidly to zero for \(\vert x\vert \gg0\). Therefore, numerically we can solve the system (1.1)-(1.2) in a compact domain \(\Omega =(x_{l},x_{r})\) with \(-x_{l}\gg0\) and \(x_{r}\gg0\). We can add the boundary conditions to the Cauchy problem (1.1)-(1.2),

$$ u(x_{l},t)=u(x_{r},t)=0. $$
(1.3)

It is well known that the system (1.1)-(1.2) possesses the following conservative law:

$$ E(t)=\Vert u\Vert ^{2}_{L_{2}}+\Vert u_{x} \Vert ^{2}_{L_{2}}=E(0). $$
(1.4)

In [1], Zhang considered a linear conservative scheme for GRLW equation, however, the accuracy of the scheme is only second-order. Recently, there has been growing interest in high-order compact methods to solve the partial differential equations [21–31], where fourth-order compact finite difference approximation solutions for the transient wave equations, a N-carrier system, the Klein-Gordon equation, the Sine-Gordon equation, the one-dimensional heat and advection-diffusion equations, the Schrödinger equation, the Klein-Gordon-Schrödinger equation and the RLW equation were shown, respectively. These numerical methods may give us many enlightenments to design a new numerical scheme for the GRLW equation. For a wide and most complete vision concerning the importance, the breadth, and the interest of the topics covered, we should also recall the study done on the long waves in [32–36].

The main purpose of this paper is to construct a new numerical scheme which has the following advantages:

  1. 1.

    Coupling with the Richardson extrapolation, the new scheme is high-accuracy and without refined mesh; it has an accuracy of \(O(\tau ^{2}+h^{4})\).

  2. 2.

    The new scheme is linearized and preserves the original conservative property.

  3. 3.

    The coefficient matrices of the scheme is symmetric and pentadiagonal, and the Thomas algorithm can be employed to solve it effectively.

  4. 4.

    Useful numerical examples are given to show the efficiency of the scheme.

The rest of this paper is organized as follows. In Section 2, a high-accuracy linear-compact difference scheme for the GRLW equation is described. In Section 3, we discuss the solvability of the scheme and the estimate of the difference solution. In Section 4, the fourth-order convergence and stability of the scheme are proved by the discrete energy method. Numerical results are reported in Section 5.

2 High-accuracy compact scheme and its discrete conservative law

In this section, we describe a high-order linear-compact difference scheme for (1.1)-(1.2).

Let \(h=\frac{x_{r}-x_{l}}{J}\) and \(\tau=\frac{T}{N}\) be the uniform step size in the spatial and the temporal direction, respectively. Denote \(x_{j}=jh\) (\(0\le j\le J\)), \(t_{n}=n\tau\) (\(0\le n\le N\)), \(u^{n}_{j}\approx u(x_{j},t_{n})\) and \(Z^{0}_{h}=\{u=(u_{j})|u_{-1}=u_{0}=u_{J}=u_{J+1}=0, j=0,1,\ldots, J\}\). For simplicity, we introduce the following notations of the difference operators:

$$\begin{aligned}& \delta_{x}u^{n}_{j}=\frac{u^{n}_{j+1}-u^{n}_{j}}{h},\qquad \delta_{\bar{x}}u^{n}_{j}=\frac{u^{n}_{j}-u^{n}_{j-1}}{h},\qquad \delta_{\hat{x}}u^{n}_{j}=\frac{u^{n}_{j+1}-u^{n}_{j-1}}{2h},\qquad \delta_{\ddot{x}}u^{n}_{j}=\frac{u^{n}_{j+2}-u^{n}_{j-2}}{4h}, \\& \delta_{\hat{t}}u^{n}_{j}=\frac{u^{n+1}_{j}-u^{n-1}_{j}}{2\tau}, \qquad \delta _{t}u^{n}_{j}=\frac{u^{n+1}_{j}-u^{n}_{j}}{\tau},\qquad \bar{u}^{n}_{j}=\frac {u^{n+1}_{j}+u^{n-1}_{j}}{2}. \end{aligned}$$

In the paper, C denotes a general positive constant which may have different values in different occurrences.

Based on the notations above, we consider the following high-accuracy linear-compact scheme for the initial-boundary problem (1.1)-(1.3),

$$\begin{aligned}& \begin{aligned}[b] &\delta_{\hat{t}}u^{n}_{j}-\frac{4}{3}\beta \delta_{x}\delta_{\bar{x}}\delta _{\hat{t}}u^{n}_{j}+ \frac{1}{3}\beta\delta_{\hat{x}}\delta_{\hat{x}}\delta _{\hat{t}}u^{n}_{j}+\frac{4}{3} \delta_{\hat{x}}\bigl(u^{n}_{j}\bigr)-\frac{1}{3} \delta _{\ddot{x}}\bigl(u^{n}_{j}\bigr) \\ & \quad {}+\frac{p}{p+1}\alpha\biggl[\frac{4}{3}\bigl\{ \bigl( \delta_{\hat{x}}\bar {u}^{n}_{j}\bigr) \bigl(u^{n}_{j}\bigr)^{p-1}+\delta_{\hat{x}} \bigl[\bigl(u^{n}_{j}\bigr)^{p-1}\bigl( \bar{u}^{n}_{j}\bigr)\bigr]\bigr\} \\ & \quad {}-\frac{1}{3}\bigl\{ \bigl(\delta_{\ddot{x}}\bar{u}^{n}_{j}\bigr) \bigl(u^{n}_{j}\bigr)^{p-1}+\delta_{\ddot {x}} \bigl[\bigl(u^{n}_{j}\bigr)^{p-1}\bigl( \bar{u}^{n}_{j}\bigr)\bigr]\bigr\} \biggr]=0,\\ &\quad 1\le j\le J-1, 1\le n\le N-1, \end{aligned} \end{aligned}$$
(2.1)
$$\begin{aligned}& \begin{aligned}[b] &u^{1}_{j}-\frac{4}{3}\beta\delta_{x} \delta_{\bar{x}}u^{1}_{j}+\frac{1}{3}\beta \delta_{\hat{x}}\delta_{\hat{x}}u^{1}_{j}\\ &\quad =u_{0}(x_{j})- \frac{d^{2} u_{0}}{dx^{2}}(x_{j})-\tau\frac{du_{0}}{dx}(x_{j})-\tau \alpha p u_{0}^{p-1}(x_{j})\frac{du_{0}}{dx}(x_{j}), \end{aligned} \end{aligned}$$
(2.2)
$$\begin{aligned}& u^{0}_{j}=u_{0}(x_{j}),\quad 1\le j\le J, \end{aligned}$$
(2.3)
$$\begin{aligned}& u^{n}_{0}=u^{n}_{J}=0,\quad 0\le n\le N. \end{aligned}$$
(2.4)

For convenience, the last term of (2.1) is defined by

$$ \kappa\bigl(u^{n},\bar{u}^{n}\bigr)=\kappa_{1} \bigl(u^{n},\bar{u}^{n}\bigr)+\kappa_{2} \bigl(u^{n},\bar {u}^{n}\bigr), $$
(2.5)

where

$$\begin{aligned} &\kappa_{1}\bigl(u^{n},\bar{u}^{n}\bigr)= \frac{4p}{3(p+1)}\alpha\bigl\{ \bigl(\delta_{\hat{x}}\bar {u}^{n} \bigr) \bigl(u^{n}\bigr)^{p-1}+\delta_{\hat{x}}\bigl[ \bigl(u^{n}\bigr)^{p-1}\bigl(\bar{u}^{n}\bigr)\bigr] \bigr\} , \\ &\kappa_{2}\bigl(u^{n},\bar{u}^{n}\bigr)=- \frac{p}{3(p+1)}\alpha\bigl\{ \bigl(\delta_{\ddot{x}}\bar {u}^{n} \bigr) \bigl(u^{n}\bigr)^{p-1}+\delta_{\ddot{x}}\bigl[ \bigl(u^{n}\bigr)^{p-1}\bigl(\bar{u}^{n}\bigr)\bigr] \bigr\} . \end{aligned}$$

Theorem 2.1

Suppose \(u_{0}\in H^{1}_{0}[x_{l},x_{r}]\), then the scheme (2.1) admits the following invariant:

$$\begin{aligned} E^{n}={}&\frac{1}{2}\bigl(\bigl\Vert u^{n+1}\bigr\Vert ^{2}+\bigl\Vert u^{n}\bigr\Vert ^{2} \bigr)+\frac{2}{3}\beta\bigl(\bigl\Vert \delta _{x}u^{n+1} \bigr\Vert ^{2}+\bigl\Vert \delta_{x}u^{n}\bigr\Vert ^{2}\bigr)-\frac{1}{6}\beta\bigl(\bigl\Vert \delta_{\hat {x}}u^{n+1}\bigr\Vert ^{2}+\bigl\Vert \delta_{\hat{x}}u^{n}\bigr\Vert ^{2}\bigr) \\ &{}+\frac{4}{3}h\tau\sum_{j=1}^{J-1} \delta_{x}u^{n}_{j}u^{n+1}_{j}- \frac {1}{3}h\tau\sum_{j=1}^{J-1} \delta_{\ddot {x}}u^{n}_{j}u^{n+1}_{j}=E^{n-1}= \cdots=E^{0}. \end{aligned}$$
(2.6)

Proof

Taking in (2.1) the inner product with \(2\bar {u}^{n}\) and using the boundary condition (2.4) yield

$$\begin{aligned} &\frac{1}{2\tau}\bigl(\bigl\Vert u^{n+1}\bigr\Vert ^{2}-\bigl\Vert u^{n-1}\bigr\Vert ^{2}\bigr)+ \frac{2}{3\tau}\beta \bigl(\bigl\Vert \delta_{x}u^{n+1} \bigr\Vert ^{2}-\bigl\Vert \delta_{x}u^{n-1}_{x} \bigr\Vert ^{2}\bigr) \\ &\quad{}-\frac{1}{6\tau}\beta \bigl(\bigl\Vert \delta_{\hat{x}}u^{n+1}\bigr\Vert ^{2}-\bigl\Vert \delta_{\hat{x}}u^{n-1}_{x}\bigr\Vert ^{2} \bigr) \\ &\quad{} +\frac{4}{3}\bigl(\delta_{x}u^{n},2 \bar{u}^{n}\bigr)-\frac{1}{3}\bigl(\delta_{\ddot {x}}u^{n},2 \bar{u}^{n}\bigr)+\bigl(\kappa_{1}\bigl(u^{n}, \bar{u}^{n}\bigr)+\kappa_{2}\bigl(u^{n},\bar {u}^{n}\bigr),2\bar{u}^{n}\bigr)=0. \end{aligned}$$
(2.7)

Notice that

$$ \frac{4}{3}\bigl(\delta_{x}u^{n},2 \bar{u}^{n}\bigr)=\frac{4}{3}h\sum_{j=1}^{J-1} \bigl(\delta _{x}u^{n}_{j}u^{n+1}_{j}-u^{n}_{j} \delta_{x}u^{n-1}_{j}\bigr) $$
(2.8)

and

$$ \frac{1}{3}\bigl(\delta_{\ddot{x}}u^{n},2 \bar{u}^{n}\bigr)=\frac{1}{3}h\sum_{j=1}^{J-1} \bigl(\delta_{\ddot{x}}u^{n}_{j}u^{n+1}_{j}-u^{n}_{j} \delta_{\ddot {x}}u^{n-1}_{j}\bigr). $$
(2.9)

Now, computing the last term of the left-hand side in (2.7), we have

$$\begin{aligned} \bigl(\kappa_{1}\bigl(u^{n},\bar{u}^{n}\bigr),2 \bar{u}^{n}\bigr)={}&\frac{8p}{3(p+1)}\alpha h\sum _{j=1}^{J-1}\bigl\{ \bigl(u^{n}_{j} \bigr)^{p-1}\delta_{\hat{x}}\bigl(\bar{u}^{n}_{j} \bigr)+\delta _{\hat{x}}\bigl[\bigl(u^{n}_{j} \bigr)^{p-1}\bar{u}^{n}_{j}\bigr]\bigr\} \bar{u}^{n}_{j} \\ ={}& \frac{4p}{3(p+1)}\alpha\sum_{j=1}^{J-1} \bigl[\bigl(u^{n}_{j}\bigr)^{p-1}\bar {u}^{n}_{j+1}\bar{u}^{n}_{j}- \bigl(u^{n}_{j+1}\bigr)^{p-1}\bar{u}^{n}_{j+1} \bar {u}^{n}_{j}\bigr] \\ &{}-\frac{4p}{3(p+1)}\alpha\sum_{j=1}^{J-1} \bigl[\bigl(u^{n}_{j-1}\bigr)^{p-1}\bar {u}^{n}_{j}\bar{u}^{n}_{j-1}- \bigl(u^{n}_{j}\bigr)^{p-1}\bar{u}^{n}_{j} \bar {u}^{n}_{j-1}\bigr] \\ ={}&0. \end{aligned}$$
(2.10)

Similarly to the proof of (2.10), we get

$$ \bigl(\kappa_{2}\bigl(u^{n},\bar{u}^{n}\bigr),2 \bar{u}^{n}\bigr)=0. $$
(2.11)

Substituting (2.8)-(2.11) into (2.7). Let

$$\begin{aligned} E^{n}={}&\frac{1}{2}\bigl(\bigl\Vert u^{n+1}\bigr\Vert ^{2}+\bigl\Vert u^{n}\bigr\Vert ^{2} \bigr)+\frac{2}{3}\beta\bigl(\bigl\Vert \delta _{x}u^{n+1} \bigr\Vert ^{2}+\bigl\Vert \delta_{x}u^{n}\bigr\Vert ^{2}\bigr)-\frac{1}{6}\beta\bigl(\bigl\Vert \delta_{\hat {x}}u^{n+1}\bigr\Vert ^{2}+\bigl\Vert \delta_{\hat{x}}u^{n}\bigr\Vert ^{2}\bigr) \\ &{}+\frac{4}{3}h\tau\sum_{j=1}^{J-1} \delta_{x}u^{n}_{j}u^{n+1}_{j}- \frac {1}{3}h\tau\sum_{j=1}^{J-1} \delta_{\ddot{x}}u^{n}_{j}u^{n+1}_{j}. \end{aligned}$$

By the definition of \(E^{n}\), (2.6) follows. □

3 Solvability and estimate for the difference solution

In this section, we shall discuss the estimate for the difference solution and the solvability of the difference scheme (2.1). For \(\forall v^{n}\), \(w^{n}\in Z^{0}_{h}\), we define the discrete inner products and norms on \(Z^{0}_{h}\) via

$$\begin{aligned}& \bigl(v^{n},w^{n}\bigr)=h\sum _{j=1}^{J-1}v^{n}_{j} {w^{n}_{j}}, \qquad \bigl(\delta_{x}v^{n}, \delta_{x}w^{n}\bigr)_{l}=h\sum _{j=0}^{J-1}\delta _{x}v^{n}_{j} \delta_{x}{w^{n}_{j}}, \qquad \bigl\Vert v^{n}\bigr\Vert ^{2}=\bigl(v^{n},v^{n} \bigr), \\& \bigl\Vert \delta_{x}v^{n}\bigr\Vert =\sqrt{\bigl( \delta_{x}v^{n},\delta_{x}v^{n} \bigr)_{l}},\qquad \bigl\Vert \delta_{\ddot {x}}v^{n}\bigr\Vert =\sqrt{\bigl(\delta_{\ddot{x}}v^{n},\delta_{\ddot{x}}v^{n} \bigr)_{l}},\qquad \bigl\Vert v^{n}\bigr\Vert _{\infty}= \max_{1\le j\le J-1}\bigl\vert v^{n}_{j}\bigr\vert . \end{aligned}$$

To analyze the estimates of difference solution for the scheme (2.1)-(2.4), the following lemmas should be introduced.

Lemma 3.1

([31])

For a mesh function \(u\in Z^{0}_{h}\), by the Cauchy-Schwarz inequality, we have

$$\begin{aligned} \Vert \delta_{\ddot{x}}u\Vert ^{2}\leq \Vert \delta_{\hat{x}}u\Vert ^{2}\leq \Vert \delta_{x}u \Vert ^{2}. \end{aligned}$$

Lemma 3.2

(Discrete Sobolev’s inequality [37])

There exist two constants \(C_{1}\) and \(C_{2}\) such that

$$\begin{aligned} \bigl\Vert u^{n}\bigr\Vert _{\infty}\le C_{1}\bigl\Vert u^{n}\bigr\Vert +C_{2}\bigl\Vert \delta_{x}u^{n}\bigr\Vert ^{2}. \end{aligned}$$

Theorem 3.1

Suppose that \(u_{0}\in H^{1}\), then there is the estimation for the solution \(u^{n}\) of the scheme (2.1): \(\Vert u^{n}\Vert \le C\), \(\Vert \delta_{x} u^{n}\Vert \le C\), which yields \(\Vert u^{n}\Vert _{\infty}\le C\).

Proof

It follows from (2.6) and the Cauchy-Schwartz inequality that

$$\begin{aligned} &\frac{1}{2}\bigl(\bigl\Vert u^{n+1}\bigr\Vert ^{2}+\bigl\Vert u^{n}\bigr\Vert ^{2}\bigr)+ \frac{2}{3}\beta\bigl(\bigl\Vert \delta _{x}u^{n+1} \bigr\Vert ^{2}+\bigl\Vert \delta_{x}u^{n}\bigr\Vert ^{2}\bigr)-\frac{1}{6}\beta\bigl(\bigl\Vert \delta_{\hat {x}}u^{n+1}\bigr\Vert ^{2}+\bigl\Vert \delta_{\hat{x}}u^{n}\bigr\Vert ^{2}\bigr) \\ &\quad \leq C+\frac{4}{3}h\tau\sum_{j=1}^{J-1} \bigl\vert \delta_{x}u^{n}_{j}u^{n+1}_{j} \bigr\vert +\frac {1}{3}h\tau\sum_{j=1}^{J-1} \bigl\vert \delta_{\ddot{x}}u^{n}_{j}u^{n+1}_{j} \bigr\vert \\ &\quad \leq C+\frac{2}{3}\tau\bigl(\bigl\Vert \delta_{x}u^{n} \bigr\Vert ^{2}+\bigl\Vert u^{n+1}\bigr\Vert ^{2}\bigr)+\frac {1}{6}\tau\bigl(\bigl\Vert \delta_{\ddot{x}}u^{n}\bigr\Vert ^{2}+\bigl\Vert u^{n+1}\bigr\Vert ^{2}\bigr). \end{aligned}$$
(3.1)

According to Lemma 3.1, we obtain from (3.1)

$$\begin{aligned} \frac{1}{2}\biggl[\biggl(1-\frac{5}{3}\tau\biggr)\bigl\Vert u^{n+1}\bigr\Vert ^{2}+\bigl\Vert u^{n}\bigr\Vert ^{2}\biggr]+\frac {1}{2}\biggl[\beta\bigl\Vert \delta_{x}u^{n+1}\bigr\Vert ^{2}+\biggl(\beta- \frac{5}{3}\tau\biggr)\bigl\Vert \delta _{x}u^{n} \bigr\Vert ^{2}\biggr]\leq C. \end{aligned}$$
(3.2)

This implies for small Ï„ which satisfies \(\beta-\frac{5}{3}\tau> 0\) that we have

$$\begin{aligned} \bigl\Vert u^{n}\bigr\Vert \le C,\qquad \bigl\Vert \delta_{x}u^{n}\bigr\Vert \le C. \end{aligned}$$
(3.3)

Using Lemma 3.2, we obtain

$$\begin{aligned} \bigl\Vert u^{n}\bigr\Vert _{\infty}\le C. \end{aligned}$$
(3.4)

 □

Remark 3.1

Theorem 3.1 implies that the scheme (2.1) is unconditionally stable.

Theorem 3.2

The difference scheme (2.1) is uniquely solvable.

Proof

Let us prove the unique solvability by induction. It is obvious that \(u^{0}\) and \(u^{1}\) are uniquely determined by (2.3) and (2.2), respectively. Suppose that \(u^{0},u^{1},\ldots,u^{n}\) be solved uniquely. Consider \(u^{n+1}\) in (2.1) which satisfies

$$\begin{aligned} &\frac{1}{2\tau}u^{n+1}_{j}-\frac{2}{3\tau}\beta \delta_{x}\delta_{\bar {x}}u^{n+1}_{j}+ \frac{1}{6\tau}\delta_{\hat{x}}\delta_{\hat {x}}\bigl(u^{n+1}_{j} \bigr)+\frac{2p}{3(p+1)}\alpha\bigl\{ \bigl(u^{n}_{j} \bigr)^{p-1}\delta_{\hat {x}}u^{n+1}_{j}+ \delta_{\hat{x}}\bigl[\bigl(u^{n}_{j} \bigr)^{p-1}u_{j}^{n+1}\bigr]\bigr\} \\ &\quad {}-\frac{p}{6(p+1)}\bigl\{ \bigl(u^{n}_{j} \bigr)^{p-1}\delta_{\ddot{x}}u^{n+1}_{j}+\delta _{\ddot{x}}\bigl[\bigl(u^{n}_{j}\bigr)^{p-1}u_{j}^{n+1} \bigr]\bigr\} =0. \end{aligned}$$
(3.5)

Taking the inner product of (3.5) with \(u^{n+1}\), we obtain

$$\begin{aligned} \frac{1}{2\tau}\bigl\Vert u^{n+1}\bigr\Vert ^{2}+ \frac{2}{3\tau}\beta\bigl\Vert \delta _{x}u^{n+1}\bigr\Vert ^{2}-\frac{1}{6\tau}\beta\bigl\Vert \delta_{\hat {x}}u^{n+1} \bigr\Vert ^{2}+\bigl(I-\mathit{II},u^{n+1}\bigr)=0, \end{aligned}$$
(3.6)

where

$$\begin{aligned}& I=\frac{2p}{3(p+1)}\alpha\bigl\{ \bigl(u^{n}_{j} \bigr)^{p-1}\delta_{\hat {x}}u^{n+1}_{j}+ \delta_{\hat{x}}\bigl[\bigl(u^{n}_{j} \bigr)^{p-1}u_{j}^{n+1}\bigr]\bigr\} , \\& \mathit{II}=\frac {p}{6(p+1)}\bigl\{ \bigl(u^{n}_{j} \bigr)^{p-1}\delta_{\ddot{x}}u^{n+1}_{j}+ \delta_{\ddot {x}}\bigl[\bigl(u^{n}_{j} \bigr)^{p-1}u_{j}^{n+1}\bigr]\bigr\} . \end{aligned}$$

Similarly to the proof of (2.10), we get

$$\begin{aligned} \bigl(I,u^{n+1}\bigr)=0, \qquad \bigl(\mathit{II},u^{n+1}\bigr)=0. \end{aligned}$$
(3.7)

It follows from (3.6)-(3.7) and Lemma 3.1 that

$$\begin{aligned} \frac{1}{2\tau}\bigl\Vert u^{n+1}\bigr\Vert ^{2}+ \frac{1}{2\tau}\beta\bigl\Vert \delta _{x}u^{n+1}\bigr\Vert ^{2}\leq0. \end{aligned}$$
(3.8)

That is, (3.5) has only a trivial solution. Hence, (2.1) determines \(u^{n+1}_{j}\) uniquely. This completes the proof of Theorem 3.2. □

4 Convergence and stability of the difference scheme

First, we shall consider the truncation error of the difference scheme (2.8)-(2.10). Let \(v^{n}_{j}=u(x_{j},t_{n})\). We define the truncation error as follows:

$$\begin{aligned}& \begin{aligned}[b] &Er^{n}_{j}={}\delta_{\hat{t}}v^{n}_{j}- \frac{4}{3}\beta\delta_{x}\delta_{\bar {x}} \delta_{\hat{t}}v^{n}_{j}+\frac{1}{3}\beta \delta_{\hat{x}}\delta_{\hat {x}}\delta_{\hat{t}}v^{n}_{j}+ \frac{4}{3}\delta_{\hat{x}}\bigl(v^{n}_{j}\bigr)- \frac {1}{3}\delta_{\ddot{x}}\bigl(v^{n}_{j}\bigr) \\ &\phantom{Er^{n}_{j}=}{} +\frac{p}{p+1}\alpha\biggl[\frac{4}{3}\bigl\{ \bigl( \delta_{\hat{x}}\bar {v}^{n}_{j}\bigr) \bigl(v^{n}_{j}\bigr)^{p-1}+\delta_{\hat{x}} \bigl[\bigl(v^{n}_{j}\bigr)^{p-1}\bigl( \bar{v}^{n}_{j}\bigr)\bigr]\bigr\} \\ &\phantom{Er^{n}_{j}=}{} -\frac{1}{3}\bigl\{ \bigl(\delta_{\ddot{x}}\bar{v}^{n}_{j}\bigr) \bigl(v^{n}_{j}\bigr)^{p-1}+\delta_{\ddot {x}} \bigl[\bigl(v^{n}_{j}\bigr)^{p-1}\bigl( \bar{v}^{n}_{j}\bigr)\bigr]\bigr\} \biggr],\\ &\quad 1\le j\le J-1, 1\le n\le N-1, \end{aligned} \end{aligned}$$
(4.1)
$$\begin{aligned}& \begin{aligned}[b] s^{0}_{j}={}&v^{1}_{j}- \frac{4}{3}\beta\delta_{x}\delta_{\bar{x}}v^{1}_{j}+ \frac {1}{3}\beta\delta_{\hat{x}}\delta_{\hat{x}}v^{1}_{j}-u_{0}(x_{j})\\ &{} + \frac{d^{2} u_{0}}{dx^{2}}(x_{j})+\tau\frac{du_{0}}{dx}(x_{j})+\tau \alpha p u_{0}^{p-1}(x_{j})\frac{du_{0}}{dx}(x_{j}),\end{aligned} \end{aligned}$$
(4.2)
$$\begin{aligned}& v^{0}_{j}=u_{0}(x_{j}),\quad 1\le j\le J, \end{aligned}$$
(4.3)
$$\begin{aligned}& v^{n}_{0}=v^{n}_{J}=0,\quad 0\le n\le N. \end{aligned}$$
(4.4)

Using a Taylor expansion, we obtain \(\vert Er^{n}\vert +\vert s^{0}\vert =O(\tau^{2}+h^{4})\) holds if \(\tau,h\to0\).

Next, we shall discuss the convergence and stability of the scheme (2.1)-(2.4).

Lemma 4.1

(Discrete Gronwall inequality [37])

Suppose that the discrete mesh function \(\{w^{n}|n=1,2,\ldots,N; N\tau=T\}\) satisfies the recurrence formula

$$ w^{n}-w^{n-1}\le A\tau w^{n}+B\tau w^{n-1}+C_{n}\tau, $$

where A, B, and \(C_{n}\) (\(n=1,\ldots,N\)) are nonnegative constants. Then

$$ \bigl\Vert w^{n}\bigr\Vert _{\infty}\le\Biggl(w^{0}+ \tau\sum_{k=1}^{N}C_{k} \Biggr)e^{2(A+B)T}, $$

where Ï„ is small, such that \((A+B)\tau\le\frac{N-1}{2N}\) (\(N > 1\)).

Theorem 4.1

Assume that \(u_{0}\in H^{1}\), then the solution \(u^{n}\) of the scheme (2.1)-(2.4) converges to the solution of the initial-boundary problem (1.1)-(1.3) and the rate of convergence is \(O(\tau^{2}+h^{4})\) by the \(\Vert \cdot \Vert _{\infty}\) norm.

Proof

Let \(e^{n}_{j}=v^{n}_{j}-u^{n}_{j}\). From (4.1)-(4.4) and (2.1)-(2.4), we have

$$\begin{aligned}& \begin{aligned}[b] Er^{n}_{j}={}&\delta_{\hat{t}}e^{n}_{j}- \frac{4}{3}\beta\delta_{x}\delta_{\bar {x}} \delta_{\hat{t}}e^{n}_{j}+\frac{1}{3}\beta \delta_{\hat{x}}\delta_{\hat {x}}\delta_{\hat{t}}e^{n}_{j}+ \frac{4}{3}\delta_{\hat{x}}\bigl(e^{n}_{j}\bigr)- \frac {1}{3}\delta_{\ddot{x}}\bigl(e^{n}_{j}\bigr) \\ & {}+\frac{p}{p+1}\alpha\biggl[\frac{4}{3}\bigl\{ \bigl( \delta_{\hat{x}}\bar {v}^{n}_{j}\bigr) \bigl(v^{n}_{j}\bigr)^{p-1}+\delta_{\hat{x}} \bigl[\bigl(v^{n}_{j}\bigr)^{p-1}\bigl( \bar{v}^{n}_{j}\bigr)\bigr]\bigr\} \\ & {} -\frac{4}{3}\bigl\{ \bigl(\delta_{\hat{x}}\bar{u}^{n}_{j}\bigr) \bigl(u^{n}_{j}\bigr)^{p-1}+\delta_{\hat {x}} \bigl[\bigl(u^{n}_{j}\bigr)^{p-1}\bigl( \bar{u}^{n}_{j}\bigr)\bigr]\bigr\} \biggr] \\ & {}-\frac{p}{p+1}\alpha\biggl[\frac{1}{3}\bigl\{ \bigl( \delta_{\ddot{x}}\bar {v}^{n}_{j}\bigr) \bigl(v^{n}_{j}\bigr)^{p-1}+\delta_{\ddot{x}} \bigl[\bigl(v^{n}_{j}\bigr)^{p-1}\bigl( \bar{v}^{n}_{j}\bigr)\bigr]\bigr\} \\ & {}-\frac{1}{3}\bigl\{ \bigl(\delta_{\ddot{x}}\bar{u}^{n}_{j}\bigr) \bigl(u^{n}_{j}\bigr)^{p-1}+\delta_{\ddot {x}} \bigl[\bigl(u^{n}_{j}\bigr)^{p-1}\bigl( \bar{u}^{n}_{j}\bigr)\bigr]\bigr\} \biggr], \quad 1\le j\le J-1, 1\le n\le N-1, \end{aligned} \end{aligned}$$
(4.5)
$$\begin{aligned}& s^{0}_{j}=e^{1}_{j}- \frac{4}{3}\beta\delta_{x}\delta_{\bar{x}}e^{1}_{j}+ \frac {1}{3}\beta\delta_{\hat{x}}\delta_{\hat{x}}e^{1}_{j}, \end{aligned}$$
(4.6)
$$\begin{aligned}& e^{0}_{j}=0,\quad 1\le j\le J, \end{aligned}$$
(4.7)
$$\begin{aligned}& e^{n}_{0}=e^{n}_{J}=0,\quad 0\le n\le N. \end{aligned}$$
(4.8)

Taking in (4.5) the inner product with \(2\bar{e}^{n}\) (i.e. \(e^{n+1}+e^{n-1}\)), we obtain

$$\begin{aligned} \bigl(Er^{n},2\bar{e}^{n}\bigr)={}&\delta_{\hat{t}}\bigl\Vert e^{n}\bigr\Vert ^{2}+\frac{4}{3}\beta\delta _{\hat{t}}\bigl\Vert \delta_{x}e^{n}\bigr\Vert ^{2} -\frac{1}{3}\beta\delta_{\hat{t}}\bigl\Vert \delta_{\hat{x}}e^{n}\bigr\Vert ^{2}+ \frac {4}{3}h\sum_{j=1}^{J-1}\bigl( \delta_{x}e^{n}_{j}e^{n+1}_{j}-e^{n}_{j} \delta_{x}e^{n-1}_{j}\bigr) \\ & {}-\frac{1}{3}h\sum_{j=1}^{J-1}\bigl( \delta_{\ddot {x}}e^{n}_{j}e^{n+1}_{j}-e^{n}_{j} \delta_{\ddot {x}}e^{n-1}_{j}\bigr)+\bigl(P_{1}+P_{2}+Q_{1}+Q_{2},2 \bar{e}^{n}\bigr), \end{aligned}$$
(4.9)

where

$$\begin{aligned}& P_{1}=\frac{4p\alpha}{3(p+1)}\bigl[\bigl(\delta_{\hat{x}}\bar {v}^{n}\bigr) \bigl(v^{n}\bigr)^{p-1}-\bigl( \delta_{\hat{x}}\bar{u}^{n}\bigr) \bigl(u^{n} \bigr)^{p-1}\bigr],\\& P_{2}=\frac{4p\alpha}{3(p+1)}\bigl\{ \delta_{\hat{x}}\bigl[\bigl(v^{n}\bigr)^{p-1}\bar {v}^{n}\bigr]-\delta_{\hat{x}}\bigl[\bigl(u^{n} \bigr)^{p-1}\bar{u}^{n}\bigr]\bigr\} , \\& Q_{1}=-\frac{p\alpha}{3(p+1)}\bigl[\bigl(\delta_{\ddot{x}}\bar {v}^{n}\bigr) \bigl(v^{n}\bigr)^{p-1}-\bigl( \delta_{\ddot{x}}\bar{u}^{n}\bigr) \bigl(u^{n} \bigr)^{p-1}\bigr], \\& Q_{2}=-\frac{p\alpha}{3(p+1)}\bigl\{ \delta_{\ddot{x}}\bigl[\bigl(v^{n}\bigr)^{p-1}\bar {v}^{n}\bigr]-\delta_{\ddot{x}}\bigl[\bigl(u^{n} \bigr)^{p-1}\bar{u}^{n}\bigr]\bigr\} . \end{aligned}$$

Computing the sixth term on the right-hand side of (4.9) and using Lemma 3.1 and Theorem 3.1 yield

$$\begin{aligned} \bigl(P_{1},2\bar{e}^{n}\bigr)&=\frac{4p\alpha}{3(p+1)}\bigl( \bigl[\bigl(\delta_{\hat{x}}\bar {v}^{n}\bigr) \bigl(v^{n}\bigr)^{p-1}-\bigl(\delta_{\hat{x}} \bar{u}^{n}\bigr) \bigl(u^{n}\bigr)^{p-1}\bigr],2 \bar{e}^{n}\bigr) \\ &=\frac{8p\alpha}{3(p+1)}\bigl(\bigl[\bigl(\delta_{\hat{x}}\bar {e}^{n}\bigr) \bigl(v^{n}\bigr)^{p-1}+\bigl( \delta_{\hat{x}}\bar {u}^{n}\bigr) \bigl(\bigl(v^{n} \bigr)^{p-1}-\bigl(u^{n}\bigr)^{p-1}\bigr)\bigr], \bar{e}^{n}\bigr) \\ &=\frac{8p\alpha}{3(p+1)}h\Biggl\{ \sum_{j=1}^{J-1} \bigl(\delta_{\hat{x}}\bar {e}^{n}_{j}\bigr) \bigl(v^{n}_{j}\bigr)^{p-1}\bar{e}^{n}_{j} +\sum_{j=1}^{J-1}\bigl(\delta_{\hat{x}} \bar {u}^{n}_{j}\bigr)\bigl[\bigl(v^{n}_{j} \bigr)^{p-1}-\bigl(u^{n}_{j}\bigr)^{p-1} \bigr]\bar{e}^{n}_{j}\Biggr\} \\ &=\frac{8p\alpha}{3(p+1)}h\Biggl(\sum_{j=1}^{J-1} \bigl(\delta_{\hat{x}}\bar {e}^{n}_{j}\bigr) \bigl(v^{n}_{j}\bigr)^{p-1}\bar{e}^{n}_{j} +\sum_{j=1}^{J-1}\bigl(\delta_{\hat{x}} \bar{u}^{n}_{j}\bigr)\Biggl[e_{j}^{n} \sum_{k=0}^{p-2}\bigl(v^{n}_{j} \bigr)^{p-2-k}\bigl(u^{n}_{j}\bigr)^{k} \Biggr]\bar{e}^{n}_{j}\Biggr) \\ &\le C\bigl(\bigl\Vert \delta_{\hat{x}}\bar{e}^{n}\bigr\Vert ^{2}+\bigl\Vert e^{n}\bigr\Vert ^{2}+\bigl\Vert \bar {e}^{n}\bigr\Vert ^{2}\bigr) \\ &\le C\bigl(\bigl\Vert \delta_{x}e^{n+1}\bigr\Vert ^{2}+\bigl\Vert \delta _{x}e^{n-1}\bigr\Vert ^{2}+\bigl\Vert e^{n+1}\bigr\Vert ^{2}+\bigl\Vert e^{n}\bigr\Vert ^{2}+\bigl\Vert e^{n-1} \bigr\Vert ^{2}\bigr), \end{aligned}$$
(4.10)

where the Cauchy-Schwartz inequality and Lemma 3.1 are used.

Similarly, we can also obtain

$$\begin{aligned} &\bigl(P_{2},2\bar{e}^{n}\bigr) \le C\bigl(\bigl\Vert \delta_{x}e^{n+1}\bigr\Vert ^{2}+\bigl\Vert \delta _{x}e^{n-1}\bigr\Vert ^{2}+\bigl\Vert e^{n+1}\bigr\Vert ^{2}+\bigl\Vert e^{n}\bigr\Vert ^{2}+\bigl\Vert e^{n-1}\bigr\Vert ^{2} \bigr), \end{aligned}$$
(4.11)
$$\begin{aligned} &\bigl(Q_{1},2\bar{e}^{n}\bigr) \le C\bigl(\bigl\Vert \delta_{x}e^{n+1}\bigr\Vert ^{2}+\bigl\Vert \delta _{x}e^{n-1}\bigr\Vert ^{2}+\bigl\Vert e^{n+1}\bigr\Vert ^{2}+\bigl\Vert e^{n}\bigr\Vert ^{2}+\bigl\Vert e^{n-1}\bigr\Vert ^{2} \bigr), \end{aligned}$$
(4.12)
$$\begin{aligned} &\bigl(Q_{2},2\bar{e}^{n}\bigr) \le C\bigl(\bigl\Vert \delta_{x}e^{n+1}\bigr\Vert ^{2}+\bigl\Vert \delta _{x}e^{n-1}\bigr\Vert ^{2}+\bigl\Vert e^{n+1}\bigr\Vert ^{2}+\bigl\Vert e^{n}\bigr\Vert ^{2}+\bigl\Vert e^{n-1}\bigr\Vert ^{2} \bigr). \end{aligned}$$
(4.13)

In addition, it is obvious that

$$\begin{aligned}& \bigl(Er^{n},2\bar{e}^{n}\bigr)\le\bigl\Vert Er^{n}\bigr\Vert ^{2} +\frac{1}{2}\bigl(\bigl\Vert e^{n+1}\bigr\Vert ^{2}+\bigl\Vert e^{n-1}\bigr\Vert ^{2}\bigr), \end{aligned}$$
(4.14)
$$\begin{aligned}& \frac{4}{3}h\sum_{j=1}^{J-1}\bigl( \delta_{x}e^{n}_{j}e^{n+1}_{j}-e^{n}_{j} \delta _{x}e^{n-1}_{j}\bigr)\leq C\bigl(\bigl\Vert \delta_{x}e^{n}\bigr\Vert ^{2}+\bigl\Vert \delta _{x}e^{n-1}\bigr\Vert ^{2}+\bigl\Vert e^{n+1}\bigr\Vert ^{2}+\bigl\Vert e^{n}\bigr\Vert ^{2}\bigr), \end{aligned}$$
(4.15)
$$\begin{aligned}& -\frac{1}{3}h\sum_{j=1}^{J-1}\bigl( \delta_{\ddot {x}}e^{n}_{j}e^{n+1}_{j}-e^{n}_{j} \delta_{\ddot{x}}e^{n-1}_{j}\bigr)\leq C\bigl(\bigl\Vert \delta _{x}e^{n}\bigr\Vert ^{2}+\bigl\Vert \delta_{x}e^{n-1}\bigr\Vert ^{2}+\bigl\Vert e^{n+1}\bigr\Vert ^{2}+\bigl\Vert e^{n}\bigr\Vert ^{2}\bigr). \end{aligned}$$
(4.16)

It follows from (4.9)-(4.16) that

$$\begin{aligned} &\delta_{\hat{t}}\bigl\Vert e^{n}\bigr\Vert ^{2}+ \frac{4}{3}\beta\delta_{\hat{t}}\bigl\Vert \delta _{x}e^{n}\bigr\Vert ^{2}-\frac{1}{3} \beta\delta_{\hat{t}}\bigl\Vert \delta_{\hat {x}}e^{n}\bigr\Vert ^{2} \\ &\quad \le\bigl\Vert Er^{n}\bigr\Vert ^{2}+C\bigl(\bigl\Vert \delta_{x}e^{n+1}\bigr\Vert ^{2}+\bigl\Vert \delta_{x}e^{n}\bigr\Vert ^{2} \\ &\qquad {}+\bigl\Vert \delta _{x}e^{n-1}\bigr\Vert ^{2}+\bigl\Vert e^{n+1}\bigr\Vert ^{2}+\bigl\Vert e^{n}\bigr\Vert ^{2}+\bigl\Vert e^{n-1}\bigr\Vert ^{2} \bigr). \end{aligned}$$
(4.17)

Let \(B^{n}=\frac{1}{2}(\Vert e^{n+1}\Vert ^{2}+\Vert e^{n}\Vert ^{2})+\frac{\beta}{2}(\Vert \delta _{x}e^{n+1}\Vert ^{2}+\Vert \delta_{x}e^{n}\Vert ^{2})\). Using Lemma 3.1, (4.17) can be written as follows:

$$\begin{aligned} B^{n}-B^{n-1}\le\tau\bigl\Vert Er^{n}\bigr\Vert ^{2}+C\tau\bigl(B^{n}+B^{n-1}\bigr). \end{aligned}$$
(4.18)

According to Lemma 4.1, we can immediately obtain

$$\begin{aligned} B^{n}\le\Bigl(B^{0}+T\sup_{l\leq n\leq N}\bigl\Vert Er^{n}\bigr\Vert ^{2}\Bigr) e^{CT}. \end{aligned}$$
(4.19)

Taking the inner product of (4.6) with \(e^{1}\) yields

$$\begin{aligned} \bigl(s^{0},e^{1}\bigr)=\bigl\Vert e^{1}\bigr\Vert ^{2}+\frac{4}{3}\beta\bigl\Vert \delta_{x}e^{1} \bigr\Vert ^{2}-\frac{1}{3}\beta \bigl\Vert \delta_{\ddot{x}}e^{1}\bigr\Vert ^{2}. \end{aligned}$$
(4.20)

This, together with \((s^{0},e^{1})\leq\frac{1}{2}(\Vert s^{0}\Vert ^{2}+\Vert e^{1}\Vert ^{2})\), \(\vert s^{0}\vert =O(\tau^{2}+h^{4})\), and Lemma 3.1, gives

$$\begin{aligned} \bigl\Vert e^{1}\bigr\Vert \leq O\bigl(\tau^{2}+h^{4} \bigr), \qquad \bigl\Vert \delta_{x}e^{1}\bigr\Vert \leq O\bigl( \tau^{2}+h^{4}\bigr). \end{aligned}$$
(4.21)

From the discrete initial condition (4.7), we know that \(B^{0}=[O(\tau^{2}+h^{4})]^{2}\).

It follows from (4.19) that

$$\begin{aligned} B^{n}\le\bigl[O\bigl(\tau^{2}+h^{4}\bigr) \bigr]^{2}. \end{aligned}$$
(4.22)

Then we have

$$\begin{aligned} \bigl\Vert e^{n}\bigr\Vert \le O\bigl(\tau^{2}+h^{4} \bigr),\qquad \bigl\Vert \delta_{x}e^{n}\bigr\Vert \le O\bigl( \tau^{2}+h^{4}\bigr). \end{aligned}$$
(4.23)

This, together with Lemmas 3.2, gives

$$\begin{aligned} \bigl\Vert e^{n}\bigr\Vert _{\infty}\le O\bigl( \tau^{2}+h^{4}\bigr). \end{aligned}$$
(4.24)

This completes the proof of Theorem 4.1. □

Similarly, we can prove stability of the difference solution.

Theorem 4.2

Under the conditions of Theorem  4.1, the solution of the scheme (2.1)-(2.4) is unconditionally stable by the \(\Vert \cdot \Vert _{\infty}\) norm.

5 Numerical experiments

In this section, we give some numerical experiments to demonstrate our theoretical results obtained in the previous sections. We will measure the accuracy of the proposed scheme using the absolute error defined by \(e^{n}=\Vert v^{n}-u^{n}\Vert _{\infty}\).

Consider the initial-boundary value problem (1.1)-(1.3). In the numerical experiments, we take \(x_{l}=-40\), \(x_{r}=60\), \(T=10\), and choose three cases \(p=2,3,4\), respectively. In order to verify the accuracy \(O(\tau^{2}+h^{4})\), we take τ and h small enough to verify the fourth-order accuracy and second-order accuracy in the spatial and temporal directions, respectively. The convergence order figures of \(\log(e^{n})\mbox{-}\log(h)\) with h and the ones of \(\log (e^{n})\mbox{-}\log(\tau)\) with τ small enough are given in Figures 1-6 for various mesh steps h and τ at \(t=10\). From Figures 1-6, it is obvious that the scheme (2.1)-(2.4) is convergent in the maximum norm, and the convergence order is \(O(\tau^{2}+h^{4})\).

We show in Theorem 2.1 that the numerical solution \(u^{n}\) of the scheme (2.1) satisfies the conservation of discrete energy. In Tables 1-3, the values of \(E^{n}\) for the scheme (2.1) are presented for three cases \(p=2,3,4\) under various mesh steps h and τ, respectively. It is easy to see from Tables 1-3 that the scheme (2.1) preserves the discrete energy very well, which also shows the accuracy and efficiency of the scheme in this paper.

Case 5.1

Take \(p=2\). Consider the following initial-boundary problem of RLW equation:

$$\begin{aligned} &u_{t}-u_{xxt}+u_{x}+uu_{x}=0, \end{aligned}$$
(5.1)
$$\begin{aligned} &u(x,0)=u_{0}(x), \end{aligned}$$
(5.2)
$$\begin{aligned} &u(x_{l},t)=u(x_{r},t)=0. \end{aligned}$$
(5.3)

In computations, we choose the initial condition \(u(x,0)=\operatorname {sech}^{2}(\frac{1}{4}x)\) [10]. The convergence order figures and the values of \(E^{n}\) are shown in Figures 1-2 and Table 1, respectively.

Figure 1
figure 1

The spatial convergence order in maximal norm for \(\pmb{u^{n}}\) at \(\pmb{t=10}\) with different h and Ï„ computed by the scheme ( 2.1 )-( 2.4 ).

Figure 2
figure 2

The temporal convergence order in maximal norm for \(\pmb{u^{n}}\) at \(\pmb{t=10}\) with different h and Ï„ computed by the scheme ( 2.1 )-( 2.4 ).

Table 1 Discrete energy \(\pmb{E^{n}}\) of the scheme ( 2.1 ) at different time t when \(\pmb{h=0.1,0.05}\) and \(\pmb{p=2}\)

Case 5.2

Take \(p=3\). We consider the following initial-boundary problem of MRLW equation:

$$\begin{aligned} &u_{t}-u_{xxt}+u_{x}+\bigl(u^{3} \bigr)_{x}=0, \end{aligned}$$
(5.4)
$$\begin{aligned} &u(x,0)=u_{0}(x), \end{aligned}$$
(5.5)
$$\begin{aligned} &u(x_{l},t)=u(x_{r},t)=0. \end{aligned}$$
(5.6)

In experiments, we choose the initial condition \(u(x,0)=\frac{\sqrt {6}}{3}\operatorname{sech}(\frac{1}{2}x)\) [1]. The convergence order figures and the values of \(E^{n}\) are shown in Figures 3-4 and Table 2, respectively.

Figure 3
figure 3

The spatial convergence order in maximal norm for \(\pmb{u^{n}}\) at \(\pmb{t=10}\) with different h and Ï„ computed by the scheme ( 2.1 )-( 2.4 ).

Figure 4
figure 4

The temporal convergence order in maximal norm for \(\pmb{u^{n}}\) at \(\pmb{t=10}\) with different h and Ï„ computed by the scheme ( 2.1 )-( 2.4 ).

Table 2 Discrete energy \(\pmb{E^{n}}\) of the scheme ( 2.1 ) at different times t when \(\pmb{h=0.1,0.05}\) and \(\pmb{p=3}\)

Case 5.3

Take \(p=4\). We consider the initial-boundary problem (1.1)-(1.3) of GRLW equation:

$$\begin{aligned} &u_{t}-u_{xxt}+u_{x}+u^{3}u_{x}=0, \end{aligned}$$
(5.7)
$$\begin{aligned} &u(x,0)=u_{0}(x), \end{aligned}$$
(5.8)
$$\begin{aligned} &u(x_{l},t)=u(x_{r},t)=0. \end{aligned}$$
(5.9)

In the following experiments, we choose the initial condition \(u(x,0)=\operatorname{sech}^{\frac{2}{3}}(\frac{3\sqrt{11}}{22}x)\) [1]. The convergence order figures and the values of \(E^{n}\) are shown in Figures 5-6 and Table 3, respectively.

Figure 5
figure 5

The spatial convergence order in maximal norm for \(\pmb{u^{n}}\) at \(\pmb{t=10}\) with different h and Ï„ computed by the scheme ( 2.1 )-( 2.4 ).

Figure 6
figure 6

The temporal convergence order in maximal norm for \(\pmb{u^{n}}\) at \(\pmb{t=10}\) with different h and Ï„ computed by the scheme ( 2.1 )-( 2.4 ).

Table 3 Discrete energy \(\pmb{E^{n}}\) of the scheme ( 2.1 ) at different time t when \(\pmb{(h,\tau)=(0.1,0.1)}\) and (0.05,0.025) for \(\pmb{p=4}\)

References

  1. Zhang, L: A finite difference scheme for the generalized regularized long-wave equation. Appl. Math. Comput. 168, 962-972 (2005)

    Article  MathSciNet  Google Scholar 

  2. Peregrine, DH: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321-330 (1966)

    Article  Google Scholar 

  3. Benjamin, TB, Bona, JL, Mahony, JJ: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. A 272, 47-78 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  4. Albert, J: Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation. J. Differ. Equ. 63(1), 117-134 (1986)

    Article  Google Scholar 

  5. Ruggieri, M, Speciale, MP: Similarity reduction and closed form solutions for a model derived from two-layer fluids. Adv. Differ. Equ. 2013, 355 (2013)

    Article  MathSciNet  Google Scholar 

  6. Lmaco, J, Clark, HR, Medeiros, LA: Remarks on equations of Benjamin-Bona-Mahony type. J. Math. Anal. Appl. 328(2), 1117-1140 (2007)

    Article  MathSciNet  Google Scholar 

  7. Ruggieri, M, Speciale, MP: New exact solutions for a coupled KdV-like model. J. Phys. Conf. Ser. 482, 012036 (2014)

    Article  Google Scholar 

  8. Gear, JA, Grimshaw, R: Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 70, 235-258 (1984)

    MATH  MathSciNet  Google Scholar 

  9. Kutluay, S, Esen, A: A finite difference solution of the regularized long wave equation. Math. Probl. Eng. 2006, 1-14 (2006)

    Article  MathSciNet  Google Scholar 

  10. Zhang, L, Chang, Q: A new finite difference method for regularized long-wave equation. Chinese J. Numer. Methods Comput. Appl. 23, 58-66 (2001)

    MathSciNet  Google Scholar 

  11. Avilez-Valente, P, Seabra-Santos, FJ: A Petrov-Galerkin finite element scheme for the regularized long wave equation. Comput. Mech. 34, 256-270 (2004)

    Article  MathSciNet  Google Scholar 

  12. Esen, A, Kutluay, S: Application of a lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput. 174(2), 833-845 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Guo, L, Chen, H: \(H^{1}\)-Galerkin mixed finite element method for the regularized long wave equation. Computing 77, 205-221 (2006)

    Article  MathSciNet  Google Scholar 

  14. Gu, H, Chen, N: Least-squares mixed finite element methods for the RLW equations. Numer. Methods Partial Differ. Equ. 24, 749-758 (2008)

    Article  MathSciNet  Google Scholar 

  15. Saka, B, Dag, I: A numerical solution of the RLW equation by Galerkin method using quartic B-splines. Commun. Numer. Methods Eng. 24, 1339-1361 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dag, I: Least squares quadratic B-spline finite element method for the regularized long wave equation. Comput. Methods Appl. Mech. Eng. 182, 205-215 (2000)

    Article  MATH  Google Scholar 

  17. Dag, I, Ozer, MN: Approximation of the RLW equation by the least square cubic B-spline finite element method. Appl. Math. Model. 25, 221-231 (2001)

    Article  Google Scholar 

  18. Dag, I, Saka, B, Irk, D: Application of cubic B-splines for numerical solution of the RLW equation. Appl. Math. Comput. 159, 373-389 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Soliman, AA, Raslan, KR: Collocation method using quadratic B-spline for the RLW equation. Int. J. Comput. Math. 78, 399-412 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Soliman, AA, Hussien, MH: Collocation solution for RLW equation with septic spline. Appl. Math. Comput. 161, 623-636 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cohen, G: High-Order Numerical Methods for Transient Wave Equations. Springer, New York (2002)

    Book  Google Scholar 

  22. Dai, W: An improved compact finite difference scheme for solving a N-carrier system with Neumann boundary conditions. Numer. Methods Partial Differ. Equ. 27, 436-446 (2011)

    Article  MATH  Google Scholar 

  23. Dai, W, Tzou, DY: A fourth-order compact finite difference scheme for solving a N-carrier system with Neumann boundary conditions. Numer. Methods Partial Differ. Equ. 25, 274-289 (2010)

    MathSciNet  Google Scholar 

  24. Dehghan, M, Mohebbi, A, Asgari, Z: Fourth-order compact solution of the nonlinear Klein-Gordon equation. Numer. Algorithms 52, 523-540 (2009)

    Article  MathSciNet  Google Scholar 

  25. Mohebbia, A, Dehghan, M: High-order solution of one-dimensional Sine-Gordon equation using compact finite difference and DIRKN methods. Math. Comput. Model. 51, 537-549 (2010)

    Article  Google Scholar 

  26. Mohebbia, A, Dehghan, M: High-order compact solution of the one-dimensional heat and advection-diffusion equations. Appl. Math. Model. 34, 3071-3084 (2010)

    Article  MathSciNet  Google Scholar 

  27. Dehghan, M, Taleei, A: A compact split-step finite difference method for solving the nonlinear Schrodinger equations with constant and variable coefficients. Comput. Phys. Commun. 181, 43-51 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Xie, S, Li, G, Yi, S: Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation. Comput. Methods Appl. Mech. Eng. 198, 1052-1060 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wang, T, Guo, B: Unconditional convergence of two conservative compact difference schemes for non-linear Schrödinger equation in one dimension. Sci. Sin., Math. 41(3), 207-233 (2011) (in Chinese)

    MATH  Google Scholar 

  30. Pan, X, Zhang, L: High-order linear compact conservative method for the nonlinear Schrodinger equation coupled with the nonlinear Klein-Gordon equation. Nonlinear Anal. 92, 108-118 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Zheng, K, Hu, J: High-order conservative Crank-Nicolson scheme for regularized long wave equation. Adv. Differ. Equ. 2013, 287 (2013)

    Article  MathSciNet  Google Scholar 

  32. Alias, A, Grimshaw, RHJ, Khusnutdinova, KR: On strongly interacting internal waves in a rotating ocean and coupled Ostrovsky equations. Chaos 23(2), 023121 (2013)

    Article  Google Scholar 

  33. Ruggieri, M, Speciale, MP: KdV-like equations for fluid dynamics. AIP Conf. Proc. 1637, 918-924 (2014)

    Article  Google Scholar 

  34. Alias, A, Grimshaw, RHJ, Khusnutdinova, KR: Coupled Ostrovsky equations for internal waves in a shear flow. Phys. Fluids 26, 126603 (2014)

    Article  Google Scholar 

  35. Ruggieri, M, Speciale, MP: Quasi self-adjoint coupled KdV-like equations. In: 11th International Conference of Numerical Analysis and Applied Mathematics, vol. 1558, pp. 1220-1223. AIP, New York (2013)

    Google Scholar 

  36. Ruggieri, M, Speciale, MP: On a hierarchy of traveling wave solutions in a shallow stratified fluid. In: 11th International Conference of Numerical Analysis and Applied Mathematics, vol. 1558, pp. 1793-1796. AIP, New York (2013)

    Google Scholar 

  37. Zhou, Y: Application of Discrete Functional Analysis to the Finite Difference Method. Inter. Acad. Publishers, Beijing (1990)

    Google Scholar 

Download references

Acknowledgements

This work is supported by the Natural Science Foundation of China (No. 11201343, 11401438), Natural Science Foundation of Shandong Province (ZR2012AM017, ZR2013FL032), a Project of Shandong Province Higher Educational Science and Technology Program (No. J14LI52, J15LI56), the Youth Research Foundation of WFU (No. 2013Z11) and the Project of Science and Technology Program of Weifang (Grant no. 201301006).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xintian Pan.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The article was carried out in collaboration between all authors. The three authors have contributed to, read, and approved the manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, X., Che, H. & Wang, Y. A high-accuracy compact conservative scheme for generalized regularized long-wave equation. Bound Value Probl 2015, 141 (2015). https://doi.org/10.1186/s13661-015-0404-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-015-0404-7

Keywords