On the Conservative Finite Difference Scheme for the Generalized Novikov Equation

Wenxia Chen^*, Qianqian Zhu, Ping Yang

Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, China.

DOI: 10.4236/jamp.2017.59150                

Abstract

In this paper, we investigate a numerical method for the generalized Novikov equation. We propose a conservative finite difference scheme and use Brouwer fixed point theorem to obtain the existence of the solution of the corresponding difference equation. We also prove the convergence and stability of the solution by using the discrete energy method. Moreover, we obtain the truncation error of the difference scheme which is .

Keywords

Generalized Novikov Equation, Finite Difference Scheme, Conservation Law, Stability, Convergence

1. Introduction

The Camassa-Holm (CH) equation

$m_t+2u_{x}m+u{m}_x=0$ (1.1)

was derived by Camassa and Holm [1] as a model for the unidirectional propagation of the shallow water waves over a flat bottom [2] [3] , where $m=u-u_{xx}$ . It was found earlier by Fuchssteiner and Fokas [4] using the recursion operator of the KdV equation. More interestingly, the CH equation could also be obtained by the tri-Hamiltonian duality approach from the bi-Hamiltonian structure of the KdV equation [5] [6] . So the CH equation could be regarded as a dual system of the KdV equation. The CH equation had attracted much attention because of its nice features: existence of peaked solitons [1] [7] (which has a discontinuous first derivative at the spike), complete integrability [1] [4] , nice geometric formulations [8] [9] [10] and wave breaking phenomena [11] [12] [13] [14] etc. It was noticed that the peaked solitons were not allowed by the well-known integrable equations such as the KdV equation, the mKdV equation, and the Schrodinger equation etc. The stability of peakons of the CH Equation (1.1) was proved in [15] [16] . A similar integrable equation with quadratic nonlinearities was the Degasperis-Procesi (DP) equation [17] , which took the form

$m_t+3u_{x}m+u{m}_x=0$ (1.2)

It was regarded as a model for nonlinear shallow water dynamics and could be derived from the governing equation for water waves [2] . Analogous to the CH equation, the DP equation also possessed an infinite number of conservation laws, bi-Hamiltonian structure, peaked solitons etc. Moreover, it admitted shocked peaked solitons [18] . Its integrability, existence of peaked solitons, stability of peaked solitons, and wave breaking phenomena were studied extensively in [19] [20] [21] [22] .

Notice that both the CH and DP equations had the quadratic nonlinearities and $H^1$ conservation laws. So it was of great interest to search for such type of equations with higher-order nonlinearities. To the best of our knowledge, two integrable equations with cubic nonlinearities had been proposed. One was a modified CH equation

$m_t+{\left(\left(u^2-u_x^2\right)m\right)}_x=0$ (1.3)

which was obtained by the tri-Hamiltonian duality approach from the bi-Hamiltonian structure of the mKdV equation [5] [6] . Its well-posedness, blow-up, wave breaking, peaked solitons, and their stability were studied in recent works [23] [24] [25] [26] . The mCH Equation (1.3) exhibits new features of peaked solitons, wave-breaking mechanism, and blow-up criteria. The mCH equation could also be obtained from an invariant non-stretching planar curve flow in Euclidean geometry [24] . So it was regarded as an Euclidean version of the CH equation in this sense. The other one was the so-called Novikov equation

$m_t+3u{u}_{x}m+u^2{m}_x=0$ (1.4)

which was obtained by Novikov [27] in the symmetry classification of such type equations. The integrability, existence of peaked solitons and their stability, global well-posedness and wave breaking phenomena of Equation (1.4) were discussed in [28] [29] [30] [31] . Applying the tri-Hamiltonian duality approach to the Gardner equation

$u_t+u_{xxx}+a_1{u}^2{u}_x+a_{2}u{u}_x=0$

they deduced the following generalized CH equation with the cubic and quadratic nonlinearities [32]

$m_t+a_1{\left(\left(u^2-u_x^2\right)m\right)}_x+a_2\left(2u_{x}m+u{m}_x\right)=0$ (1.5)

This equation admitted Lax-pair and peaked solitons [33] .

There are few studies on numerical solutions of the generalized Novikov equation. Therefore in this paper, we will construct a finite difference scheme for the initial boundary problem of equations as follows,

$\begin{array}{l}{m}_t+a_1\left(3u{u}_{x}m+u^2{m}_x\right)+a_2{u}_{x}m+a_{3}u{m}_x=0,\\ m=u-u_{xx},0<t\le T,0\le x\le L\end{array}$ (1.6)

$u\left(0,t\right)=u\left(L,t\right),0<t\le T$ (1.7)

$u\left(0,x\right)=u_0\left(x\right),0<x\le L$ (1.8)

where $u\left(x,t\right)$ is a function of time $t$ and a single spatial variable $x$ , and $a_1$ , $a_2$ , $a_3$ are constants. It is clear that Equation (1.6) is reduced respectively to the Novikov Equation (1.4), the CH Equation (1.1) and the DP Equation (1.2), when $a_1=1$ , $a_2=0$ , $a_3=0$ ; $a_1=0$ , $a_2=2$ , $a_3=1$ ; $a_1=0$ , $a_2=3$ , $a_3=1$ . Clearly, Equation (1.6) is a linear combination of the Novikov equation, the CH equation, and the DP equation.

We shall give an energy conservative finite scheme for Equations (1.6)-(1.8) and obtain the module estimation. Moreover, we prove the convergence and stability of the finite scheme by use of discrete energy method.

2. Preliminaries

For convenience, we denote $\Omega =\left\{\left(x,t\right)|0\le x\le L,0<t\le T\right\}$ in the following section. Let $N$ , $J$ be any positive integers and $h=\frac{L}{J+1}$ , $\tau =\frac{T}{N}$ , $x_j=jh$

for $j=0,1,\cdots ,J+1$ . Denote $t^n=n\tau$ , $u_j^n=u\left(x_j,t^n\right)$ , $u^n=\left(u_0^n,\cdots ,u_{J+1}^n\right)$ for $n=0,\cdots ,N$ and $Z_h^0=\left\{u=\left(u_0,\cdots ,u_{J+1}\right)|u_0=u_{J+1}=0\right\}$ .

For simplicity, we introduce some notations as follows:

$u_x^n=\frac{u_{j+1}^n-u_j^n}{h}$ , $u_{\bar{x}}^n=\frac{u_j^n-u_{j-1}^n}{h}$ , $u_{\hat{x}}^n=\frac{u_{j+1}^n-u_{j-1}^n}{2h}$ ,

$u_t^n=\frac{u_j^{n+1}-u_j^n}{\tau }$ , $u_{\bar{t}}^n=\frac{u_j^n-u_j^{n-1}}{\tau }$ , $u_{\hat{t}}^n=\frac{u_j^{n+1}-u_j^{n-1}}{2\tau }$ ,

$\bar{u}=\frac{u_j^{n+1}+u_j^{n-1}}{2}$ , $u^n=u_j^n$ (2.1)

For $u,v\in Z_h^0$ , we define a discrete inner product and the discrete $L^2$ -norm ${\Vert \cdot \Vert }_h$ as

${\left(u,v\right)}_h=h{\displaystyle \underset{j=1}{\overset{J}{\sum }}{u}_j{v}_j}$ , ${\Vert u\Vert }_h={\left(h{\displaystyle \underset{j=1}{\overset{J}{\sum }}{u}_j^2}\right)}^{\frac{1}{2}}$ (2.2)

In order to obtain the module estimation, investigate the convergence and stability of the finite difference scheme, we need introduce two lemmas as follows,

Lemma 2.1. (Discrete Sobolev inequality [34] ) There exist constants $c_1$ , $c_2$ satisfying

${\Vert u^n\Vert }_{\infty }\le c_1\Vert u^n\Vert +c_2\Vert u_x^n\Vert$ (2.3)

Lemma 2.2. (Discrete Gronwall inequality [34] ) Suppose that there exist negative functions $\omega \left(k\right)$ , $\rho \left(k\right)$ , where $\rho \left(k\right)$ is decreasing. For any $k$ and $c>0$ , if

$\omega \left(k\right)\le \rho \left(k\right)+c\tau {\displaystyle \underset{l=0}{\overset{k-1}{\sum }}w\left(l\right)}$ (2.4)

then

$\omega \left(k\right)\le \rho \left(k\right)c^{c\tau k}$ (2.5)

3. A Energy Conservative Finite Scheme

The Equation (1.6) can be rewritten in the following form:

$u_t-u_{xxt}+a_1\left(3u{u}_x\left(u-u_{xx}\right)+u^2\left(u_x-u_{xxx}\right)\right)+a_2{u}_x\left(u-u_{xx}\right)+a_{3}u\left(u_x-u_{xxx}\right)=0$

which can be expressed as

$u_t-u_{xxt}+4a_1{u}^2{u}_x+\left(a_2+a_3\right)u{u}_x=3a_{1}u{u}_x{u}_{xx}+a_1{u}^2{u}_{xxx}+a_2{u}_x{u}_{xx}+a_{3}u{u}_{xxx}$

Firstly, we construct an energy conservative finite scheme for the problem (1.6)-(1.8) as follows:

$\begin{array}{l}{u}_t^n-u_{tx\bar{x}}^n+a_1\left({\left({\left(u^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}+{\left(u^{n+\frac{1}{2}}\right)}^2{u}_x^{n+\frac{1}{2}}\right)+\frac{a_2+a_3}{3}\left({\left({\left(u^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}+u^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}\right)\\ =a_1\left({\left({\left(u^{n+\frac{1}{2}}\right)}^2{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+u^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)+a_3\left({\left(u^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+u_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)\\ +\left(a_2-2a_3\right)\left({\left({\left(u_x^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}-u_x^{n+\frac{1}{2}}{u}_{x\bar{x}}^{n+\frac{1}{2}}\right)\end{array}$ (3.1)

$u_j^0=u_0\left(x_j\right),0\le j\le J+1$ (3.2)

and

$u_0^n=u_{J+1}^n=0,0\le n\le N$ (3.3)

Lemma 3.1. The difference scheme satisfies discrete conservative law as follows:

$E^n={\Vert u^n\Vert }^2+{\Vert u_x^n\Vert }^2=E^{n-1}=\cdots =E^0$ (3.4)

Proof. Computing the inner product of the difference scheme with $2u^{n+\frac{1}{2}}$ , we have

$\begin{array}{c}\left(u_t,2u^{n+\frac{1}{2}}\right)=h{\displaystyle \underset{j=1}{\overset{J}{\sum }}{u}_t}\cdot \left(2u^{n+\frac{1}{2}}\right)=\frac{h}{\tau }{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left(u_j^{n+1}-u_j^n\right)\left(u_j^{n+1}+u_j^n\right)}\\ =\frac{1}{\tau }\left({\Vert u^{n+1}\Vert }^2-{\Vert u^n\Vert }^2\right)\end{array}$

$\begin{array}{c}\left(u_{tx\bar{x}},2u^{n+\frac{1}{2}}\right)=h{\displaystyle \underset{j=1}{\overset{J}{\sum }}{u}_{tx\bar{x}}\cdot \left(2u^{n+\frac{1}{2}}\right)}=-\frac{h}{\tau }{\displaystyle \underset{j=1}{\overset{J}{\sum }}{\left(u_j^{n+1}-u_j^n\right)}_x}{\left(u_j^{n+1}+u_j^n\right)}_x\\ =-\frac{1}{\tau }\left({\Vert u_x^{n+1}\Vert }^2-{\Vert u_x^n\Vert }^2\right)\end{array}$

$\begin{array}{l}\left(a_1\left({\left({\left(u^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}+{\left(u^{n+\frac{1}{2}}\right)}^2{u}_x^{n+\frac{1}{2}}\right),2u^{n+\frac{1}{2}}\right)\\ =2a_{1}h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left({\left(u_j^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}+{\left(u_j^{n+\frac{1}{2}}\right)}^2{u}_{xj}^{n+\frac{1}{2}}\cdot u_j^{n+\frac{1}{2}}\right)}\\ =2a_{1}h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left({\left(u_j^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}-{\left({\left(u_j^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}{u}_j^{n+\frac{1}{2}}\right)}\\ =0\end{array}$

$\begin{array}{l}\left(\frac{a_2+a_3}{3}\left({\left({\left(u^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}+u^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}\right),2u^{n+\frac{1}{2}}\right)\\ =\frac{2\left(a_2+a_3\right)}{3}h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left({\left(u_j^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}+u_j^{n+\frac{1}{2}}{u}_{xj}^{n+\frac{1}{2}}\cdot u_j^{n+\frac{1}{2}}\right)}\\ =\frac{2\left(a_2+a_3\right)}{3}h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left({\left(u_j^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}-{\left({\left(u_j^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}{u}_j^{n+\frac{1}{2}}\right)}\\ =0\end{array}$

$\begin{array}{l}\left(a_1\left({\left({\left(u^{n+\frac{1}{2}}\right)}^2{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+u^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right),2u^{n+\frac{1}{2}}\right)\\ =2a_{1}h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left({\left(u_j^{n+\frac{1}{2}}\right)}^2{u}_{xxj}^{n+\frac{1}{2}}\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}+u_j^{n+\frac{1}{2}}{u}_{xj}^{n+\frac{1}{2}}{u}_{xxj}^{n+\frac{1}{2}}\cdot u_j^{n+\frac{1}{2}}\right)}\\ =2a_{1}h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left({\left(u_j^{n+\frac{1}{2}}\right)}^2{u}_{xxj}^{n+\frac{1}{2}}\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}-{\left({\left(u_j^{n+\frac{1}{2}}\right)}^2{u}_{xxj}^{n+\frac{1}{2}}\right)}_{\bar{x}}{u}_j^{n+\frac{1}{2}}\right)}\\ =0\end{array}$

$\begin{array}{l}\left(a_3\left({\left(u^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+u_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right),2u^{n+\frac{1}{2}}\right)\\ =2a_{3}h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left(u_j^{n+\frac{1}{2}}{u}_{xxj}^{n+\frac{1}{2}}\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}+u_{xj}^{n+\frac{1}{2}}{u}_{xxj}^{n+\frac{1}{2}}\cdot u_j^{n+\frac{1}{2}}\right)}\\ =2a_{3}h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left(u_j^{n+\frac{1}{2}}{u}_{xxj}^{n+\frac{1}{2}}\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}-u_j^{n+\frac{1}{2}}{\left(u_j^{n+\frac{1}{2}}{u}_{xxj}^{n+\frac{1}{2}}\right)}_{\bar{x}}\right)}\\ =0\end{array}$

$\begin{array}{l}\left(\left(a_2-2a_3\right)\left({\left({\left(u_x^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}-u_x^{n+\frac{1}{2}}{u}_{x\bar{x}}^{n+\frac{1}{2}}\right),2u^{n+\frac{1}{2}}\right)\\ =2\left(a_2-2a_3\right)h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left({\left(u_{xj}^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}\cdot u_j^{n+\frac{1}{2}}-u_{xj}^{n+\frac{1}{2}}{u}_{x\bar{x}j}^{n+\frac{1}{2}}\cdot u_j^{n+\frac{1}{2}}\right)}\\ =2\left(a_2-2a_3\right)h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left(-{\left(u_{xj}^{n+\frac{1}{2}}\right)}^2\cdot u_{xj}^{n+\frac{1}{2}}+u_{xj}^{n+\frac{1}{2}}{u}_{xj}^{n+\frac{1}{2}}\cdot u_{xj}^{n+\frac{1}{2}}\right)}\\ =0\end{array}$

From the above formulas, we can simplify formula as

$\frac{1}{\tau }\left({\Vert u^{n+1}\Vert }^2-{\Vert u^n\Vert }^2\right)+\frac{1}{\tau }\left({\Vert u_x^{n+1}\Vert }^2-{\Vert u_x^n\Vert }^2\right)=0$ (3.5)

then

${\Vert u^{n+1}\Vert }^2+{\Vert u_x^{n+1}\Vert }^2={\Vert u^n\Vert }^2+{\Vert u_x^n\Vert }^2$ (3.6)

So the difference scheme satisfies the discrete conservative law as follows

$E^n={\Vert u^n\Vert }^2+{\Vert u_x^n\Vert }^2=E^{n-1}=\cdots =E^0$ (3.7)

This completes the proof of Lemma 3.1.

Theorem 3.2. The solution of the difference scheme (3.1)-(3.3) satisfies:

$\Vert u^n\Vert \le C,\Vert u_x^n\Vert \le C,{\Vert u^n\Vert }_{\infty }\le C$ (3.8)

Proof. From Lemma 3.1, we know $\Vert u^n\Vert \le C$ and $\Vert u_x^n\Vert \le C$ . Then by Lemma 2.1, we obtain ${\Vert u^n\Vert }_{\infty }\le C$ . This completes the proof of Theorem 3.2.

Theorem 3.3. There exists $u^n\in Z_h^0$ satisfying the difference scheme (3.1)-(3.3).

Proof. We shall use Mathematical Induction to prove Theorem 3.3.

When $n=1$ , it is known that there exists $u^1$ satisfying the difference scheme from initial condition. Next, we need to prove the case of $n>1$ .

Assume that there exists $u^n$ satisfies the difference scheme when $n<N$ , then we need to prove that there exists $u^{n+1}$ satisfies the difference scheme.

Define a operator $\omega \left(v\right)$ in $Z_h^0$ as follows:

$\begin{array}{c}\omega \left(v\right)=2v-2u^n-\left(2v_{x\bar{x}}-2u_{x\bar{x}}^n\right)+a_1\tau \left({\left(v^3\right)}_{\bar{x}}+v^2{v}_x\right)\\ +\frac{a_2+a_3}{3}\tau \left({\left(v^2\right)}_{\bar{x}}+v{v}_x\right)-a_1\tau \left({\left(v^2{v}_{xx}\right)}_{\bar{x}}+v{v}_x{v}_{xx}\right)\\ -a_3\tau \left({\left(v{v}_{xx}\right)}_{\bar{x}}+v_x{v}_{xx}\right)-\left(a_2-2a_3\right)\tau \left({\left({\left(v_x\right)}^2\right)}_{\bar{x}}-v_x{v}_{x\bar{x}}\right)\end{array}$ (3.9)

It is clear that $\omega$ is continuous.

Computing the inner product of the operator $\omega \left(v\right)$ , we obtain

$\left(2v,v\right)=2{\Vert v\Vert }^2$

$\left(2v_{x\bar{x}},v\right)=-2\left(v_x,v_x\right)=-2{\Vert v_x\Vert }^2$

$\left({\left(v^3\right)}_{\bar{x}}+v^2{v}_x,v\right)=h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left(v^3\right)}_{\bar{x}}\cdot v+v^2{v}_x\cdot v\right)}=0$

$\left({\left(v^2\right)}_{\bar{x}}+v{v}_x,v\right)=h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left(v^2\right)}_{\bar{x}}\cdot v+v{v}_x\cdot v\right)}=0$

$\left({\left(v^2{v}_{xx}\right)}_{\bar{x}}+v{v}_x{v}_{xx},v\right)=h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left(v^2{v}_{xx}\right)}_{\bar{x}}\cdot v+v{v}_x{v}_{xx}\cdot v\right)}=0$

$\left({\left(v{v}_{xx}\right)}_{\bar{x}}+v_x{v}_{xx},v\right)=h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left({\left(v{v}_{xx}\right)}_{\bar{x}}\cdot v+v_x{v}_{xx}\cdot v\right)}=0$

$\left({\left({\left(v_x\right)}^2\right)}_{\bar{x}}-v_x{v}_{x\bar{x}},v\right)=h{\displaystyle \underset{j=1}{\overset{J}{\sum }}\left(-{\left(v_x\right)}^2\cdot v_x+v_x{v}_x\cdot v_x\right)}=0$

By using Cauchy-Schwartz inequality, we obtain

$\left(2u^n,v\right)\le 2\Vert u^n\Vert \cdot \Vert v\Vert$ (3.10)

$\left(2u_{x\bar{x}}^n,v\right)=-2\left(u_x^n,v_x\right)\ge -2\Vert u_x^n\Vert \cdot \Vert v_x\Vert$ (3.11)

From above discussions, we get

$\begin{array}{c}\left(\omega \left(v\right),v\right)\ge 2{\Vert v\Vert }^2-2\Vert u^n\Vert \cdot \Vert v\Vert +2{\Vert v_x\Vert }^2-2\Vert u_x^n\Vert \cdot \Vert v_x\Vert \\ \ge {\Vert v\Vert }^2-{\Vert u^n\Vert }^2-{\Vert u_x^n\Vert }^2\end{array}$ (3.12)

For $\forall v\in Z_h^0$ , then

${\Vert v\Vert }^2={\Vert u^n\Vert }^2+{\Vert u_x^n\Vert }^2+1$ (3.13)

So we get $\left(\omega \left(v\right),v\right)>0$ . By Brouwer fixed point theorem, there exists $v^{\ast }\in H$ such that $\omega \left(v^{\ast }\right)=0$ and $\Vert v^{\ast }\Vert \le \partial$ .

Let $u^{n+1}=2v^{\ast }-u^n$ , it is easy to verify that $u^{n+1}$ satisfies the difference scheme for the problem.

So we complete the proof of Theorem 3.3.

4. Convergence and Stability of the Difference Scheme

In order to investigate the convergence and stability of the difference scheme, we need to obtain the truncation error of the scheme.

Theorem 4.1. If the solution $u\left(x,t\right)$ of equation is sufficiently regular, then the truncation error of the difference scheme is $R_j^n=O\left({\tau }^2+h^2\right)$ .

Proof. Firstly, we can use the Taylor expansion of $u_j^{n+1}$ , $u_j^n$ , $u_{j+1}^{n+1}$ , $u_{j+1}^n$ , $u_{j-1}^{n+1}$ , $u_{j-1}^n$ at the point $\left(x_j,t_{n+\frac{1}{2}}\right)$ . Secondly, from the above Taylor expansions we reorganize difference equation at point $\left(x_j,t_{n+\frac{1}{2}}\right)$ , then

$\begin{array}{c}\left(x_j,t_{n+\frac{1}{2}}\right)u_t^n=\frac{u_j^{n+1}-u_j^n}{\tau }\\ =\frac{1}{\tau }(\left(u_j^{n+\frac{1}{2}}+\frac{\tau }{2}{\left(\frac{\partial u}{\partial t}\right)}_j^{n+\frac{1}{2}}+\frac{{\tau }^2}{8}{\left(\frac{{\partial }^{2}u}{\partial t^2}\right)}_j^{n+\frac{1}{2}}+\frac{{\tau }^3}{48}{\left(\frac{{\partial }^{3}u}{\partial t^3}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\\ -\left(u_j^{n+\frac{1}{2}}-\frac{\tau }{2}{\left(\frac{\partial u}{\partial t}\right)}_j^{n+\frac{1}{2}}+\frac{{\tau }^2}{8}{\left(\frac{{\partial }^{2}u}{\partial t^2}\right)}_j^{n+\frac{1}{2}}+\cdots \right))\\ ={\left(\frac{\partial u}{\partial t}\right)}_j^{n+\frac{1}{2}}+\frac{{\tau }^3}{24}{\left(\frac{{\partial }^{3}u}{\partial t^3}\right)}_j^{n+\frac{1}{2}}+\cdots \end{array}$

$\begin{array}{c}{u}_{tx\bar{x}}^n=\frac{{\left(u_{x\bar{x}}\right)}_j^{n+1}-{\left(u_{x\bar{x}}\right)}_j^n}{\tau }=\frac{\left({\left(u_{\bar{x}}\right)}_{j+1}^{n+1}-{\left(u_{\bar{x}}\right)}_j^{n+1}\right)-\left({\left(u_{\bar{x}}\right)}_{j+1}^n-{\left(u_{\bar{x}}\right)}_j^n\right)}{\tau h}\\ =\frac{1}{h^2}(\left({\left(\frac{\partial u}{\partial t}\right)}_{j+1}^{n+\frac{1}{2}}-2{\left(\frac{\partial u}{\partial t}\right)}_j^{n+\frac{1}{2}}+{\left(\frac{\partial u}{\partial t}\right)}_{j-1}^{n+\frac{1}{2}}\right)\\ +\left(\frac{{\tau }^2}{24}{\left(\frac{{\partial }^{3}u}{\partial t^3}\right)}_{j+1}^{n+\frac{1}{2}}-\frac{{\tau }^2}{12}{\left(\frac{{\partial }^{3}u}{\partial t^3}\right)}_j^{n+\frac{1}{2}}+\frac{{\tau }^2}{24}{\left(\frac{{\partial }^{3}u}{\partial t^3}\right)}_{j-1}^{n+\frac{1}{2}}\right)+\cdots )\\ ={\left(\frac{{\partial }^{3}u}{\partial t\partial x^2}\right)}_j^{n+\frac{1}{2}}+\cdots \end{array}$

$\begin{array}{l}{\left({\left(u^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}=\frac{{\left({\left(u^{n+\frac{1}{2}}\right)}^3\right)}_j-{\left({\left(u^{n+\frac{1}{2}}\right)}^3\right)}_{j-1}}{h}\\ =3{\left(u^{n+\frac{1}{2}}\right)}^2{\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}-\frac{h}{2}\left(3{\left(u^{n+\frac{1}{2}}\right)}^2{\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+6u^{n+\frac{1}{2}}{\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\end{array}$

$\begin{array}{c}{\left(u^{n+\frac{1}{2}}\right)}^2{u}_x^{n+\frac{1}{2}}={\left(u^{n+\frac{1}{2}}\right)}^2\cdot \frac{u_{j+1}^{n+\frac{1}{2}}-u_j^{n+\frac{1}{2}}}{h}\\ ={\left(u^{n+\frac{1}{2}}\right)}^2\cdot \left({\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+\frac{h}{2}{\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+\frac{h^2}{6}{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\end{array}$

$\begin{array}{c}{\left({\left(u^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}=\frac{{\left({\left(u^{n+\frac{1}{2}}\right)}^2\right)}_j-{\left({\left(u^{n+\frac{1}{2}}\right)}^2\right)}_{j-1}}{h}\\ =2u^{n+\frac{1}{2}}{\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}-\frac{h}{2}\left(2u^{n+\frac{1}{2}}{\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+2{\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\end{array}$

$\begin{array}{c}{u}^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}=u^{n+\frac{1}{2}}\cdot \frac{u_{j+1}^{n+\frac{1}{2}}-u_j^{n+\frac{1}{2}}}{h}\\ =u^{n+\frac{1}{2}}\cdot \left({\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+\frac{h}{2}{\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+\frac{h^2}{6}{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\end{array}$

$\begin{array}{l}{u}^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}=u^{n+\frac{1}{2}}\cdot \frac{u_{j+1}^{n+\frac{1}{2}}-u_j^{n+\frac{1}{2}}}{h}\cdot \frac{u_{j+2}^{n+\frac{1}{2}}-2u_{j+1}^{n+\frac{1}{2}}+u_j^{n+\frac{1}{2}}}{h^2}\\ =u^{n+\frac{1}{2}}\cdot \left({\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+\frac{h}{2}{\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\cdot \left({\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+h{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\end{array}$

$\begin{array}{l}{\left({\left(u^{n+\frac{1}{2}}\right)}^2{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}=\frac{{\left(u_j^{n+\frac{1}{2}}\right)}^2{u}_{xxj}^{n+\frac{1}{2}}-{\left(u_{j-1}^{n+\frac{1}{2}}\right)}^2{u}_{xx\left(j-1\right)}^{n+\frac{1}{2}}}{h}\\ =2u^{n+\frac{1}{2}}\cdot \left({\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}\cdot {\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+h{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\end{array}$

$\begin{array}{l}{u}_x^{x+\frac{1}{2}}{u}_{xx}^{x+\frac{1}{2}}=\frac{u_{j+1}^{x+\frac{1}{2}}-u_j^{x+\frac{1}{2}}}{h}\cdot \frac{u_{j+2}^{x+\frac{1}{2}}-2u_{j+1}^{x+\frac{1}{2}}+u_j^{x+\frac{1}{2}}}{h^2}\\ =\left({\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+\frac{h}{2}{\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\cdot \left({\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+h{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\end{array}$

$\begin{array}{c}{\left(u^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}=\frac{u_j^{n+\frac{1}{2}}{u}_{xxj}^{n+\frac{1}{2}}-u_{j-1}^{n+\frac{1}{2}}{u}_{xx\left(j-1\right)}^{n+\frac{1}{2}}}{h}\\ ={\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}\cdot {\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+h{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+\cdots \end{array}$

$\begin{array}{c}{\left({\left(u_x^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}=\frac{u_{xj}^{n+\frac{1}{2}}{u}_{xj}^{n+\frac{1}{2}}-u_{x\left(j-1\right)}^{n+\frac{1}{2}}{u}_{x\left(j-1\right)}^{n+\frac{1}{2}}}{h}\\ ={\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}\cdot {\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+h{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+\cdots \end{array}$

$\begin{array}{l}{u}_x^{n+\frac{1}{2}}{u}_{x\bar{x}}^{n+\frac{1}{2}}=\frac{u_{j+1}^{n+\frac{1}{2}}-u_j^{n+\frac{1}{2}}}{h}\cdot \frac{u_{j+1}^{n+\frac{1}{2}}-2u_j^{n+\frac{1}{2}}+u_{j-1}^{n+\frac{1}{2}}}{h^2}\\ =\left({\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+\frac{h}{2}{\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\cdot \left({\left(\frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+h{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}+\cdots \right)\end{array}$

From the above expansions, we obtain the linear part of Equation (3.1) $u_t^n-u_{tx\bar{x}}^n$ at the point $\left(x_j,t_{n+\frac{1}{2}}\right)$ satisfying

$u_t^n-u_{tx\bar{x}}^n={\left(\frac{\partial u}{\partial t}-\frac{{\partial }^{3}u}{\partial t\partial x^2}\right)}_{\left(x_j,t_{n+\frac{1}{2}}\right)}+O\left({\tau }^2+h^2\right)$

On the other hand, the nonlinear parts at the point $\left(x_j,t_{n+\frac{1}{2}}\right)$ satisfying

$a_1\left({\left({\left(u^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}+{\left(u^{n+\frac{1}{2}}\right)}^2{u}_x^{n+\frac{1}{2}}\right)=4a_1{u}^2{\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+O\left({\tau }^2+h^2\right)$

$\frac{a_2+a_3}{3}\left({\left({\left(u^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}+u^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}\right)=\left(a_2+a_3\right)u{\left(\frac{\partial u}{\partial x}\right)}_j^{n+\frac{1}{2}}+O\left({\tau }^2+h^2\right)$

$\begin{array}{l}{a}_1\left({\left({\left(u^{n+\frac{1}{2}}\right)}^2{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+u^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)\\ =a_1\left(3u{\left(\frac{\partial u}{\partial x}\cdot \frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+2u{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}\right)+O\left({\tau }^2+h^2\right)\end{array}$

$a_3\left({\left(u^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+u_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)=a_3\left(2{\left(\frac{\partial u}{\partial x}\cdot \frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}\right)+O\left({\tau }^2+h^2\right)$

$\begin{array}{l}\left(a_2-2a_3\right)\left({\left({\left(u_x^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}-u_x^{n+\frac{1}{2}}{u}_{x\bar{x}}^{n+\frac{1}{2}}\right)\\ =\left(a_2-2a_3\right)\left(2{\left(\frac{\partial u}{\partial x}\cdot \frac{{\partial }^{2}u}{\partial x^2}\right)}_j^{n+\frac{1}{2}}+{\left(\frac{{\partial }^{3}u}{\partial x^3}\right)}_j^{n+\frac{1}{2}}\right)+O\left({\tau }^2+h^2\right)\end{array}$

Then we can easily obtain that truncation error of scheme (3.1) is $R_j^n=O\left({\tau }^2+h^2\right)$ .

Theorem 4.2. The solution of the difference scheme (3.1)-(3.3) approaches to the solution of the original differential Equations (1.6)-(1.8) in $L_{\infty }$ , and the corresponding truncation error is $O\left({\tau }^2+h^2\right)$ .

Proof. Assume that $u_j^n$ is the solution of the difference scheme (3.1)-(3.3), $U_j^n$ is the solution of the original differential Equations (1.6)-(1.8). We can easily have $e_j^n=U_j^n-u_j^n$ .

For the difference scheme (3.1),

$\begin{array}{c}{R}^n=U_t^n-U_{tx\bar{x}}^n+a_1\left({\left({\left(U^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}+{\left(U^{n+\frac{1}{2}}\right)}^2{U}_x^{n+\frac{1}{2}}\right)\\ +\frac{a_2+a_3}{3}\left({\left({\left(U^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}+U^{n+\frac{1}{2}}{U}_x^{n+\frac{1}{2}}\right)-a_1\left({\left({\left(U^{n+\frac{1}{2}}\right)}^2{U}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+U^{n+\frac{1}{2}}{U}_x^{n+\frac{1}{2}}{U}_{xx}^{n+\frac{1}{2}}\right)\\ -a_3\left({\left(U^{n+\frac{1}{2}}{U}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+U_x^{n+\frac{1}{2}}{U}_{xx}^{n+\frac{1}{2}}\right)-\left(a_2-2a_3\right){\left({\left(U_x^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}-U_x^{n+\frac{1}{2}}{U}_{x\bar{x}}^{n+\frac{1}{2}}\end{array}$ (4.1)

then (4,1) minus (3.1), yields

$R^n=e_t^n-e_{tx\bar{x}}^n+I_1+I_2+I_3+I_4+I_5$ (4.2)

where

$I_1=a_1\left({\left({\left(U^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}+{\left(U^{n+\frac{1}{2}}\right)}^2{U}_x^{n+\frac{1}{2}}-{\left({\left(u^{n+\frac{1}{2}}\right)}^3\right)}_{\bar{x}}-{\left(u^{n+\frac{1}{2}}\right)}^2{u}_x^{n+\frac{1}{2}}\right)$

$I_2=\frac{a_2+a_3}{3}\left({\left({\left(U^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}+U^{n+\frac{1}{2}}{U}_x^{n+\frac{1}{2}}-{\left({\left(u^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}-u^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}\right)$

$I_3=a_1\left({\left({\left(u^{n+\frac{1}{2}}\right)}^2{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+u^{n+\frac{1}{2}}{u}_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}-{\left({\left(U^{n+\frac{1}{2}}\right)}^2{U}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}-U^{n+\frac{1}{2}}{U}_x^{n+\frac{1}{2}}{U}_{xx}^{n+\frac{1}{2}}\right)$

$I_4=a_3\left({\left(u^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}+u_x^{n+\frac{1}{2}}{u}_{xx}^{n+\frac{1}{2}}-{\left(U^{n+\frac{1}{2}}{U}_{xx}^{n+\frac{1}{2}}\right)}_{\bar{x}}-U_x^{n+\frac{1}{2}}{U}_{xx}^{n+\frac{1}{2}}\right)$

$I_5=\left(a_2-2a_3\right)\left({\left({\left(u_x^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}-u_x^{n+\frac{1}{2}}{u}_{x\bar{x}}^{n+\frac{1}{2}}-{\left({\left(U_x^{n+\frac{1}{2}}\right)}^2\right)}_{\bar{x}}+U_x^{n+\frac{1}{2}}{U}_{x\bar{x}}^{n+\frac{1}{2}}\right)$

Taking the inner product of (4.2) with $2e^{n+\frac{1}{2}}$ , we obtain

$\begin{array}{c}\left(R^n,2e^{n+\frac{1}{2}}\right)=\left(e_t^n,2e^{n+\frac{1}{2}}\right)-\left(e_{tx\bar{x}}^n,2e^{n+\frac{1}{2}}\right)+\left(I_1,2e^{n+\frac{1}{2}}\right)+\left(I_2,2e^{n+\frac{1}{2}}\right)\\ +\left(I_3,2e^{n+\frac{1}{2}}\right)+\left(I_4,2e^{n+\frac{1}{2}}\right)+\left(I_5,2e^{n+\frac{1}{2}}\right)\end{array}$

By computing $\left(e_t^n,2e^{n+\frac{1}{2}}\right)$ , $\left(e_{tx\bar{x}}^n,2e^{n+\frac{1}{2}}\right)$ , $\left(I_1,2e^{n+\frac{1}{2}}\right)$ , $\left(I_2,2e^{n+\frac{1}{2}}\right)$ , $\left(I_3,2e^{n+\frac{1}{2}}\right)$ , $\left(I_4,2e^{n+\frac{1}{2}}\right)$ and $\left(I_5,2e^{n+\frac{1}{2}}\right)$ respectively, and from Lemma 2.3 and Cauchy-Schwartz inequality, we have

$\left(e_t^n,2e^{n+\frac{1}{2}}\right)=h{\displaystyle \underset{j=0}{\overset{J}{\sum }}{e}_{tj}}\cdot 2e_j^{n+\frac{1}{2}}=\frac{1}{\tau }\left({\Vert e^{n+1}\Vert }^2-{\Vert e^n\Vert }^2\right)$

$\left(e_{tx\bar{x}}^n,2e^{n+\frac{1}{2}}\right)=h{\displaystyle \underset{j=0}{\overset{J}{\sum }}{e}_{tx\bar{x}j}}\cdot 2e_j^{n+\frac{1}{2}}=\frac{1}{\tau }\left({\Vert e_x^{n+1}\Vert }^2-{\Vert e_x^n\Vert }^2\right)$

$\left(I_1,2e^{n+\frac{1}{2}}\right)\le C\left({\Vert e_x^{n+1}\Vert }^2+{\Vert e_x^n\Vert }^2+{\Vert e^{n+1}\Vert }^2+{\Vert e^n\Vert }^2\right)$

$\left(I_2,2e^{n+\frac{1}{2}}\right)\le C\left({\Vert e_x^{n+1}\Vert }^2+{\Vert e_x^n\Vert }^2+{\Vert e^{n+1}\Vert }^2+{\Vert e^n\Vert }^2\right)$

$\left(I_3,2e^{n+\frac{1}{2}}\right)\le C\left({\Vert e_x^{n+1}\Vert }^2+{\Vert e_x^n\Vert }^2+{\Vert e^{n+1}\Vert }^2+{\Vert e^n\Vert }^2\right)$

$\left(I_4,2e^{n+\frac{1}{2}}\right)\le C\left({\Vert e_x^{n+1}\Vert }^2+{\Vert e_x^n\Vert }^2+{\Vert e^{n+1}\Vert }^2+{\Vert e^n\Vert }^2\right)$

$\left(I_5,2e^{n+\frac{1}{2}}\right)\le C\left({\Vert e_x^{n+1}\Vert }^2+{\Vert e_x^n\Vert }^2+{\Vert e^{n+1}\Vert }^2+{\Vert e^n\Vert }^2\right)$

Therefore,

$\begin{array}{c}\left(R^n,2e^{n+\frac{1}{2}}\right)=\left(e_t^n,2e^{n+\frac{1}{2}}\right)-\left(e_{tx\bar{x}}^n,2e^{n+\frac{1}{2}}\right)+\left(I_1,2e^{n+\frac{1}{2}}\right)+\left(I_2,2e^{n+\frac{1}{2}}\right)\\ +\left(I_3,2e^{n+\frac{1}{2}}\right)+\left(I_4,2e^{n+\frac{1}{2}}\right)+\left(I_5,2e^{n+\frac{1}{2}}\right)\\ =\frac{1}{\tau }\left({\Vert e^{n+1}\Vert }^2-{\Vert e^n\Vert }^2\right)-\frac{1}{\tau }\left({\Vert e_x^{n+1}\Vert }^2-{\Vert e_x^n\Vert }^2\right)+\left(I_1,2e^{n+\frac{1}{2}}\right)+\left(I_2,2e^{n+\frac{1}{2}}\right)\\ +\left(I_3,2e^{n+\frac{1}{2}}\right)+\left(I_4,2e^{n+\frac{1}{2}}\right)+\left(I_5,2e^{n+\frac{1}{2}}\right)\end{array}$

Taking Schwartz inequality, it follows that

$\left(R^n,2e^{n+\frac{1}{2}}\right)\le 2\Vert R^n\Vert \cdot \Vert e^{n+\frac{1}{2}}\Vert \le {\Vert R^n\Vert }^2+2\left({\Vert e^{n+1}\Vert }^2+{\Vert e^n\Vert }^2\right)$

From the above discussion, we have

$\begin{array}{l}{\Vert e^{n+1}\Vert }^2-{\Vert e^n\Vert }^2+{\Vert e_x^{n+1}\Vert }^2-{\Vert e_x^n\Vert }^2\\ \le C\tau {\Vert R^n\Vert }^2+C\tau \left({\Vert e^{n+1}\Vert }^2+{\Vert e^n\Vert }^2+{\Vert e_x^{n+1}\Vert }^2+{\Vert e_x^n\Vert }^2\right)\end{array}$ (4.3)

Let ${\phi }_n={\Vert e^n\Vert }^2+{\Vert e_x^n\Vert }^2$ , then (4.3) becomes to

${\phi }_{n+1}-{\phi }_n\le C\tau \left({\phi }_{n+1}+{\phi }_n\right)+C\tau {\Vert R^n\Vert }^2$

Hence we have

${\phi }_n-{\phi }_{n-1}\le C\tau \left({\phi }_n+{\phi }_{n-1}\right)+C\tau {\Vert R^{n-1}\Vert }^2$

${\phi }_{n-1}-{\phi }_{n-2}\le C\tau \left({\phi }_{n-1}+{\phi }_{n-2}\right)+C\tau {\Vert R^{n-2}\Vert }^2$

$⋮$

${\phi }_1-{\phi }_0\le C\tau \left({\phi }_1+{\phi }_0\right)+C\tau {\Vert R^0\Vert }^2$

From the above inequalities, one has

${\phi }_n\le {\phi }_0+C\tau {\displaystyle \underset{i=0}{\overset{n-1}{\sum }}{\phi }_i}+C\tau {\displaystyle \underset{i=0}{\overset{n-1}{\sum }}{\Vert R^i\Vert }^2}$

where $k{\displaystyle {\sum }_{i=0}^{n-1}{\Vert R^i\Vert }^2}\le n\tau \underset{0\le i\le n-1}{\max }{\Vert R^i\Vert }^2\le TO{\left({\tau }^2+h^2\right)}^2$ .

Since ${\phi }_0=0$ , then ${\phi }_n\le C\tau {\displaystyle {\sum }_{i=0}^{n-1}{\phi }_i}+CTO{\left({\tau }^2+h^2\right)}^2$ . From discrete Gronwall inequality, ${\phi }_n\le O{\left({\tau }^2+h^2\right)}^2$ , that is

${\Vert e_x^n\Vert }^2+{\Vert e^n\Vert }^2\le O{\left({\tau }^2+h^2\right)}^2$

Then we have

$\Vert e^n\Vert \le O\left({\tau }^2+h^2\right),\Vert e_x^n\Vert \le O\left({\tau }^2+h^2\right)$

From Theorem 3.2, we assert

${\Vert e_x^n\Vert }_{\infty }\le O\left({\tau }^2+h^2\right)$

We complete the proof of Theorem 4.2.

Theorem 4.3. The solution of the difference scheme (3.1)-(3.3) is stable in $L_{\infty }$ .

Proof. Assume that $u_j^n$ is the solution of the difference scheme (3.1)-(3.3), $U_j^n$ is the solution of original differential Equations (1.6)-(1.8). We can easily obtain that $e_j^n=U_j^n-u_j^n$ .

Then from Theorem 4.2, the following inequality holds true

${\Vert e^n\Vert }^2\le C{\Vert U_0-u_0\Vert }^2$

Thus, we complete the proof of Theorem 4.3.

5. Conclusion

In this paper, we give a difference scheme for the generalized Novikov equation. In Section 2, we give some preparation knowledge. In Section 3, we propose a conservative finite difference scheme for the generalized Novikov equation and use Brouwer fixed point theorem to obtain the existence of the solution for the corresponding difference equation. In Section 4, we prove the convergence and stability of the solution by using the discrete energy method.

Acknowledgements

This research is supported by the National Nature Science Foundation of China (Grant Nos. 11501253, 11571141 and 11571140), and the Nature Science Foundation of Jiangsu Province (Grant No. BK20140525), and the Advanced talent of Jiangsu University (Grant Nos. 14JDG070, 15JDG079).

Conflicts of Interest

The authors declare no conflicts of interest.

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