C-N Difference Schemes for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

Abstract

We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term. Crank-Nicolson nonlinear-implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify the theoretical analysis.

1. Introduction

A symmetric version of regularized long wave equation (SRLWE),𝑢_𝑥𝑥𝑡−𝜌_𝑡𝑢_𝑡=𝜌_𝑥+𝑢𝑢_𝑥,+𝑢_𝑥=0,(1.1) has been proposed to model the propagation of weakly nonlinear ion acoustic and space charge waves [1]. The sech² solitary wave solutions are 𝑢(𝑥,𝑡)=3𝑣²−1𝑣sech²12𝑣²−1𝑣²𝜌(𝑥,𝑡)=3𝑣₂(𝑥−𝑣𝑡),−1𝑣²sech²12𝑣²−1𝑣²(𝑥−𝑣𝑡).(1.2) The four invariants and some numerical results have been obtained in [1], where 𝑣 is the velocity, 𝑣²>1. Obviously, eliminating 𝜌 from (1.1), we get a class of SRLWE𝑢_𝑡𝑡−𝑢_𝑥𝑥+12𝑢²_𝑥𝑡−𝑢_{𝑥𝑥𝑡𝑡}=0.(1.3) Equation (1.3) is explicitly symmetric in the 𝑥 and 𝑡 derivatives and is very similar to the regularized long wave equation that describes shallow water waves and plasma drift waves [2, 3]. The SRLW equation also arises in many other areas of mathematical physics [4–6]. Numerical investigation indicates that interactions of solitary waves are inelastic [7] thus, the solitary wave of the SRLWE is not a solution. Research on the well-posedness for its solution and numerical methods has aroused more and more interest. In [8], Guo studied the existence, uniqueness and regularity of the numerical solutions for the periodic initial value problem of generalized SRLW by the spectral method. In [9], Zheng et al. presented a Fourier pseudospectral method with a restraint operator for the SRLWEs, and proved its stability and obtained the optimum error estimates. There are other methods such as pseudospectral method, finite difference method for the initial-boundary value problem of SRLWEs (see [10–15]).

Because of gravity and resistance of propagation medium and air, the principle of dissipation must be considered when studying the move of nonlinear wave. In applications, the viscous damping effect is inevitable and it plays the same important role as the dispersive effect. Therefore, it is more significant to study the dissipative symmetric regularized long wave equations with the damping term𝑢_𝑥𝑥𝑡−𝜌_𝑡𝑢_𝑡+𝜐𝑢_𝑥𝑥=𝜌_𝑥+𝑢𝑢_𝑥,+𝑢_𝑥+𝛾𝜌=0,(1.4) where 𝜐,𝛾 are positive, 𝜐>0 is the dissipative coefficient, 𝛾>0 is the damping coefficient. Equation (1.4) are a reasonable model to render essential phenomena of nonlinear ion acoustic wave motion when dissipation is considered (see [16–20]). Existence, uniqueness and well-posedness of global solutions to (1.4) are presented (see [16–20]). But it is difficult to find the analytical solution to (1.4), which makes numerical solution important.

In this paper, we study (1.4) with𝑢(𝑥,0)=𝑢₀(𝑥),𝜌(𝑥,0)=𝜌₀(𝑥),𝑥∈𝑥_𝐿,𝑥_𝑅(1.5) and the boundary conditions𝑢𝑥_𝐿,𝑡=𝑢𝑥_𝑅,𝑡=0,𝜌𝑥_𝐿,𝑡=𝜌𝑥_𝑅,𝑡=0,𝑡∈[0,𝑇].(1.6) In [21] we proposed a three-level implicit finite difference scheme to (1.4)–(1.6) with second-order convergence. But the three-level implicit finite difference scheme can not start by itself, we need to select other two-level schemes (such as the C-N Scheme) to get 𝑢¹,𝜌¹. Then, reusing initial vale 𝑢⁰,𝜌⁰, we can work out 𝑢²,𝜌²,𝑢³,𝜌³,…. Since the form of SRLW equations is similar ro the Rosenau equation and Rosenau-Burgers equation, the established difference schemes in [22, 23] for solving Rosenau equation and Rosenau-Burgers equation are helpful to investigate the SRLWEs. We propose the Crank-Nicolson finite difference scheme for (1.4)–(1.6) which can start by itself. We will show that this difference scheme is uniquely solvable, convergent and stable in both theoretical and numerical senses.

2. Finite Difference Scheme and Its Error Estimation

Let ℎ and 𝜏 be the uniform step size in the spatial and temporal direction, respectively. Denote 𝑥_𝑗=𝑥_𝐿+𝑗ℎ(𝑗=0,1,2,…,𝐽), 𝑡_𝑛=𝑛𝜏(𝑛=0,1,2,…,𝑁), 𝑁=[𝑇/𝜏], 𝑢^𝑛𝑗≈𝑢(𝑥_𝑗,𝑡_𝑛), 𝜌^𝑛𝑗≈𝜌(𝑥_𝑗,𝑡_𝑛) and 𝑍^0ℎ={𝑢=(𝑢_𝑗)∣𝑢₀=𝑢_𝐽=0, 𝑗=0,1,2,…,𝐽}. Throughout this paper, we will denote 𝐶 as a generic constant independent of ℎ and 𝜏 that varies in the context. We define the difference operators, inner product, and norms that will be used in this paper as follows: 𝑢^𝑛𝑗_𝑥=𝑢^𝑛𝑗+1−𝑢^𝑛𝑗ℎ,𝑢^𝑛𝑗_𝑥=𝑢^𝑛𝑗−𝑢^𝑛𝑗−1ℎ,𝑢^𝑛𝑗𝑥̂=𝑢^𝑛𝑗+1−𝑢^𝑛𝑗−12ℎ,𝑢^𝑛𝑗_𝑡=𝑢^𝑗𝑛+1−𝑢^𝑛𝑗𝑢_𝑗𝑛+1/2𝜏,=𝑢^𝑗𝑛+1+𝑢^𝑛𝑗2,⟨𝑢^𝑛,𝑣^𝑛⟩=ℎ^𝐽−1_𝑗=1𝑢^𝑛𝑗𝑣^𝑛𝑗,‖𝑢^𝑛‖²=⟨𝑢^𝑛,𝑢^𝑛⟩,‖𝑢^𝑛‖_∞=max_{1≤𝑗≤𝐽−1}||𝑢^𝑛𝑗||.(2.1)

Then, the Crank-Nicolson finite difference scheme for the solution of (1.4)–(1.6) is as follows:𝑢^𝑛𝑗_𝑡−𝑢^𝑛𝑗_𝑥𝑥𝑡+𝜌^𝑗𝑛+1/2𝑥̂−𝜌𝜐𝑢^𝑗𝑛+1/2_𝑥𝑥+13𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1}𝑢^𝑗𝑛+1/2𝑥̂=0,(2.2)_𝑛𝑗𝑢_𝑡+𝑢^𝑗𝑛+1/2𝑥̂+𝛾𝜌^𝑗𝑛+1/2=0,(2.3)_0𝑗𝑢=𝑢₀𝑥_𝑗,𝜌^0𝑗=𝜌₀𝑥_𝑗,1≤𝑗≤𝐽−1,(2.4)_𝑛0=𝑢^𝑛𝐽=0,𝜌^𝑛0=𝜌^𝑛𝐽=0,1≤𝑛≤𝑁.(2.5)

Lemma 2.1. It follows summation by parts [12, 23] that for any two discrete functions 𝑢,𝑣∈𝑍^0ℎ, ⟨𝑢_𝑥,𝑣⟩=−⟨𝑢,𝑣_𝑥⟩,⟨𝑢_𝑥𝑥,𝑣⟩=−⟨𝑢_𝑥,𝑣_𝑥⟩.(2.6)

Lemma 2.2 (Discrete Sobolev's inequality [12, 23]). There exist two constants 𝐶₁ and 𝐶₂ such that ‖𝑢^𝑛‖_∞≤𝐶₁‖𝑢^𝑛‖+𝐶₂‖‖𝑢^𝑛𝑥‖‖.(2.7)

Lemma 2.3 (Discrete Gronwall’s inequality [12, 23]). Suppose 𝑤(𝑘), 𝜌(𝑘) are nonnegative function and 𝜌(𝑘) is nondecreasing. If 𝐶>0 and 𝑤(𝑘)≤𝜌(𝑘)+𝐶𝜏∑^{𝑘−1𝑙=0}𝑤(𝑙), then 𝑤(𝑘)≤𝜌(𝑘)𝑒^𝐶𝜏𝑘.(2.8)

Theorem 2.4. If 𝑢₀∈𝐻¹, 𝜌₀∈𝐿₂, then the solution of (2.2)–(2.5) satisfies ‖𝑢^𝑛‖≤𝐶,‖‖𝑢^𝑛𝑥‖‖≤𝐶,‖𝜌^𝑛‖≤𝐶,‖𝑢^𝑛‖_∞≤𝐶,(𝑛=1,2,…,𝑁).(2.9)

Proof. Taking an inner product of (2.2) with 2𝑢^𝑛+1/2 (i.e., 𝑢^𝑛+1+𝑢^𝑛) and considering the boundary condition (2.5) and Lemma 2.1, we obtain 1𝜏‖‖𝑢^𝑛+1‖‖²−‖𝑢^𝑛‖²+𝜏1‖‖𝑢^𝑥𝑛+1‖‖²−‖‖𝑢^𝑛𝑥‖‖²−𝜐𝑢_{𝑥𝑛+1/2𝑥}+𝜌𝑥̂^𝑛+1/2,2𝑢^𝑛+1/2,2𝑢^𝑛+1/2+𝑃^𝑛+1/2,2𝑢^𝑛+1/2=0,(2.10) where 𝑃^𝑗𝑛+1/2=(1/3)(𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1})⋅(𝑢^𝑗𝑛+1/2)𝑥̂.
Since𝜌𝑥̂𝑢^𝑛+1/2_{𝑥𝑛+1/2𝑥},2𝑢^𝑛+1/2=−2𝑢𝑥̂^𝑛+1/2,𝜌^𝑛+1/2,𝑃,2𝑢^𝑛+1/2=−2‖‖𝑢^𝑥𝑛+1/2‖‖²,(2.11)_𝑛+1/2,2𝑢^𝑛+1/2=2ℎ^𝐽−1_𝑗=1𝑃^𝑗𝑛+1/2𝑢^𝑗𝑛+1/2=23ℎ^𝐽−1_𝑗=1𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1}⋅𝑢^𝑗𝑛+1/2𝑥̂𝑢^𝑗𝑛+1/2=13^𝐽−1_𝑗=1𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1}⋅𝑢^{𝑛+1/2𝑗+1}−𝑢^{𝑛+1/2𝑗−1}𝑢^𝑗𝑛+1/2=13^𝐽−1_𝑗=1𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2𝑢^{𝑛+1/2𝑗+1}𝑢^𝑗𝑛+1/2−13^𝐽−1_𝑗=1𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1}𝑢^{𝑛+1/2𝑗−1}𝑢^𝑗𝑛+1/2=0,(2.12) we obtain 1𝜏‖‖𝑢^𝑛+1‖‖²−‖𝑢^𝑛‖²+𝜏1‖‖𝑢^𝑥𝑛+1‖‖²−‖‖𝑢^𝑛𝑥‖‖²−2𝑢𝑥̂^𝑛+1/2,𝜌^𝑛+1/2+2𝜐‖‖𝑢^𝑥𝑛+1/2‖‖²=0.(2.13) Taking an inner product of (2.3) with 2𝜌^𝑛+1/2 (i.e., 𝜌^𝑛+1+𝜌^𝑛) and considering the boundary condition (2.5) and Lemma 2.1, we obtain 1𝜏‖‖𝜌^𝑛+1‖‖²−‖𝜌^𝑛‖²+𝑢𝑥̂^𝑛+1/2,2𝜌^𝑛+1/2+2𝛾‖‖𝜌^𝑛+1/2‖‖²=0.(2.14) Adding (2.13) to (2.14), we have ‖‖𝑢^𝑛+1‖‖²−‖𝑢^𝑛‖²+‖‖𝑢^𝑥𝑛+1‖‖²−‖‖𝑢^𝑛𝑥‖‖²+‖‖𝜌^𝑛+1‖‖²−‖𝜌^𝑛‖²=−2𝜏𝜐‖‖𝑢_𝑥𝑛+1/2‖‖²+𝛾‖‖𝜌^𝑛+1/2‖‖²≤0,(2.15) which implies ‖𝑢^𝑛‖²+‖‖𝑢^𝑛𝑥‖‖²+‖𝜌^𝑛‖²≤‖‖𝑢^𝑛−1‖‖²+‖‖𝑢^𝑥𝑛−1‖‖²+‖‖𝜌^𝑛−1‖‖²≤⋯≤‖‖𝑢⁰‖‖²+‖‖𝑢^0𝑥‖‖²+‖‖𝜌⁰‖‖²=𝐶.(2.16) Then, it holds ‖𝑢^𝑛‖≤𝐶,‖‖𝑢^𝑛𝑥‖‖≤𝐶,‖𝜌^𝑛‖≤𝐶.(2.17) By Lemma 2.2, we obtain ‖𝑢^𝑛‖_∞≤𝐶.

Theorem 2.5. Assume that 𝑢⁰∈𝐻², 𝜌⁰∈𝐻¹, the solution of difference scheme (2.2)–(2.5) satisfies: ‖‖𝜌^𝑛𝑥‖‖≤𝐶,‖‖𝑢^𝑛𝑥𝑥‖‖≤𝐶,‖‖𝑢^𝑛𝑥‖‖≤𝐶,‖𝜌^𝑛‖_∞≤𝐶,(𝑛=1,2,…,𝑁).(2.18)

Proof. Differentiating backward (2.2)–(2.5) with respect to 𝑥, we obtain 𝑢^𝑛𝑗_𝑥𝑡−𝑢^𝑛𝑗_{𝑥𝑥𝑥𝑡}+𝜌^𝑗𝑛+1/2_𝑥𝑥̂−𝜐𝑢^𝑗𝑛+1/2_𝑥𝑥𝑥𝜌+𝑃^𝑗𝑛+1/2_𝑥=0,(2.19)_𝑛𝑗_𝑥𝑡+𝑢^𝑗𝑛+1/2_𝑥𝑥̂+𝛾𝜌^𝑗𝑛+1/2_𝑥𝑢=0,(2.20)_0𝑗𝑢_𝑥=𝑢_0,𝑥𝑥_𝑗,𝜌^0𝑗_𝑥=𝜌_0,𝑥𝑥_𝑗,1≤𝑗≤𝐽−1,(2.21)_𝑛0_𝑥=𝑢^𝑛𝐽_𝑥=0,𝜌^𝑛0_𝑥=𝜌^𝑛𝐽_𝑥=0,1≤𝑛≤𝑁,(2.22) where (𝑃^𝑗𝑛+1/2)_𝑥=(1/3)[(𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1})(𝑢^𝑗𝑛+1/2)𝑥̂]_𝑥.
Computing the inner product of (2.19) with 2𝑢^𝑥𝑛+1/2 (i.e., 𝑢^𝑥𝑛+1+𝑢^𝑛𝑥) and considering (2.22) and Lemma 2.1, we obtain1𝜏‖‖𝑢^𝑥𝑛+1‖‖²−‖‖𝑢^𝑛𝑥‖‖²+𝜏1‖‖𝑢^{𝑛+1𝑥𝑥}‖‖²−‖‖𝑢^𝑛𝑥𝑥‖‖²+𝜌_𝑥𝑛+1/2𝑥̂,2𝑢^𝑥𝑛+1/2−𝜐𝑢^{𝑛+1/2𝑥𝑥}_𝑥,2𝑢^𝑥𝑛+1/2+𝑃^𝑥𝑛+1/2,2𝑢^𝑥𝑛+1/2=0.(2.23) It follows from Theorem 2.4 that |||𝑢_{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1}|||≤𝐶,(𝑗=0,1,2,…,𝐽).(2.24) By the Schwarz inequality and Lemma 2.1, we get 𝑃^𝑥𝑛+1/2,2𝑢^𝑥𝑛+1/2==−−22𝑃^𝑛+1/2,𝑢^𝑥𝑛+1/2_𝑥3ℎ^𝐽−1_𝑗=1𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1}⋅𝑢^𝑗𝑛+1/2𝑥̂𝑢^𝑗𝑛+1/2_𝑥𝑥≤𝐶ℎ^𝐽−1_𝑗=1|||𝑢_𝑗𝑛+1/2𝑥̂|||⋅|||𝑢^𝑗𝑛+1/2_𝑥𝑥|||≤𝐶‖‖𝑢_𝑥𝑛+1/2‖‖²≤𝐶‖‖𝑢_𝑥𝑛+1+‖‖𝑢^{𝑛+1/2𝑥𝑥}‖‖²‖‖²+‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢^{𝑛+1𝑥𝑥}‖‖²+‖‖𝑢^𝑛𝑥𝑥‖‖².(2.25) By 𝜌𝑥𝑢̂^𝑥𝑛+1/2,_{𝑛+1/2𝑥𝑥𝑥}2𝑢^𝑥𝑛+1/2=−2𝑢𝑥̂^𝑥𝑛+1/2,𝜌^𝑥𝑛+1/2,,2𝑢^𝑥𝑛+1/2=−2‖‖𝑢^{𝑛+1/2𝑥𝑥}‖‖²,(2.26) it follows from (2.23) that ‖‖𝑢^𝑥𝑛+1‖‖²−‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢^{𝑛+1𝑥𝑥}‖‖²−‖‖𝑢^𝑛𝑥𝑥‖‖²≤−2𝜐𝜏‖‖𝑢_{𝑛+1/2𝑥𝑥}−2𝜏𝑢𝑥̂^𝑥𝑛+1/2,𝜌^𝑥𝑛+1/2‖‖²+𝐶𝜏‖‖𝑢^𝑥𝑛+1‖‖²+‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢^{𝑛+1𝑥𝑥}‖‖²+‖‖𝑢^𝑛𝑥𝑥‖‖².(2.27) Computing the inner product of (2.20) with 2𝜌^𝑥𝑛+1/2 (i.e., 𝜌^𝑥𝑛+1+𝜌^𝑛𝑥) and considering (2.22) and Lemma 2.1, we obtain ‖‖𝜌^𝑥𝑛+1‖‖²−‖‖𝜌^𝑛𝑥‖‖²+2𝜏𝑢𝑥̂^𝑥𝑛+1/2,𝜌^𝑥𝑛+1/2+2𝛾𝜏‖‖𝜌^𝑥𝑛+1/2‖‖²=0.(2.28) Adding (2.28) to (2.27),we have ‖‖𝑢^𝑥𝑛+1‖‖²−‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢^{𝑛+1𝑥𝑥}‖‖²−‖‖𝑢^𝑛𝑥𝑥‖‖²+‖‖𝜌^𝑥𝑛+1‖‖²−‖‖𝜌^𝑛𝑥‖‖²≤−2𝜐𝜏‖‖𝑢_{𝑛+1/2𝑥𝑥}‖‖²−2𝛾𝜏‖‖𝜌^𝑥𝑛+1/2‖‖²+𝐶𝜏‖‖𝑢^𝑥𝑛+1‖‖²+‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢^{𝑛+1𝑥𝑥}‖‖²+‖‖𝑢^𝑛𝑥𝑥‖‖²≤𝐶𝜏‖‖𝑢_𝑥𝑛+1‖‖²+‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢^{𝑛+1𝑥𝑥}‖‖²+‖‖𝑢^𝑛𝑥𝑥‖‖²+‖‖𝜌^𝑥𝑛+1‖‖²+‖‖𝜌^𝑛𝑥‖‖².(2.29) Let 𝐴^𝑛=‖𝑢^𝑛𝑥‖²+‖𝑢^𝑛𝑥𝑥‖²+‖𝜌^𝑛𝑥‖², we obtain 𝐴^𝑛+1−𝐴^𝑛≤𝐶𝜏(𝐴^𝑛+1+𝐴^𝑛).
Choosing suitable 𝜏 which is small enough to satisfy 1−𝐶𝜏>0, we get𝐴^𝑛+1−𝐴^𝑛≤𝐶𝜏𝐴^𝑛.(2.30) Summing up (2.30) from 0 to 𝑛−1, we have 𝐴^𝑛≤𝐴⁰+𝐶𝜏^𝑛−1_𝑙=0𝐴^𝑙.(2.31) By Lemma 2.3, we get 𝐴^𝑛≤𝐶, which implies ‖𝜌^𝑛𝑥‖≤𝐶, ‖𝑢^𝑛𝑥𝑥‖≤𝐶. It follows from Theorem 2.4 and Lemma 2.2 that ‖𝑢^𝑛𝑥‖_∞≤𝐶, ‖𝜌^𝑛‖_∞≤𝐶.

3. Solvability, Convergence, and Stability

The following Brouwer fixed point theorem will be needed in order to show the existence of solution for (2.2)–(2.5). For the proof, see [24].

Lemma 3.1 (Brouwer fixed point theorem). Let 𝐻 be a finite dimensional inner product space, suppose that 𝑔∶𝐻→𝐻 is continuous and there exists an 𝛼>0 such that <𝑔(𝑥),𝑥>0 for all 𝑥∈𝐻 with ‖𝑥‖=𝛼. Then there exists 𝑥^∗∈𝐻 such that 𝑔(𝑥^∗)=0 and ‖𝑥^∗‖≤𝛼.

Let 𝑍_Δ={𝑣=(𝑣₁,𝑣₂)=(𝑣_1,𝑗,𝑣_2,𝑗)∣𝑣_1,0=𝑣_1,𝐽=𝑣_2,0=𝑣_2,𝐽=0, 𝑗=0,1,2,…,𝐽}, equipped with the inner product ⟨𝑣,𝑣^⟩=⟨(𝑣₁,𝑣₂),(𝑣^1,𝑣^2)⟩=⟨𝑣₁,𝑣^1⟩+⟨𝑣₂,𝑣^2⟩ and the norm ‖𝑣‖²=‖𝑣₁‖²+‖𝑣₂‖².

Theorem 3.2. There exists (𝑢^𝑛,𝜌^𝑛)∈𝑍_Δ which satisfies the difference scheme (2.2)–(2.5).

Proof. In order to prove the theorem by the mathematical induction, we assume that (𝑢⁰,𝜌⁰),(𝑢¹,𝜌¹),…,(𝑢^𝑛,𝜌^𝑛)∈𝑍_Δ satisfying (2.2)–(2.5). Next prove there exists (𝑢^𝑛+1,𝜌^𝑛+1) which satisfies (2.2)–(2.5).
Let 𝑔=(𝑔₁,𝑔₂) be a operator on 𝑍_Δ defined by𝑔𝑔₁(𝑣)=2𝑣₁−2𝑢^𝑛−2𝑣_1𝑥𝑥+2𝑢^𝑛𝑥_𝑥+𝜏𝑣₂𝑥̂−𝜐𝜏𝑣^1𝑥𝑥+𝜏𝑊,₂(𝑣)=2𝑣₂−2𝜌^𝑛+𝜏𝑣₁𝑥̂+𝛾𝜏𝑣²,∀𝑣=𝑣₁,𝑣₂∈𝑍_Δ,(3.1) where 𝑊_𝑗=(1/3)(𝑣_1,𝑗+1+𝑣_1,𝑗+𝑣_1,𝑗−1)(𝑣_1,𝑗)𝑥̂. Computing the inner product of (3.1) with 𝑣=(𝑣₁,𝑣₂), similarly to (2.11) and (2.12), we obtain 𝑣₂𝑥̂,𝑣¹=−𝑣₁𝑥̂,𝑣²,⟨𝑊,𝑣₁⟩=0.(3.2) By (2.5) and the Schwarz inequality, we obtain ⟨𝑔(𝑣),𝑣⟩==⟨2𝑔‖‖₁𝑣(₁𝑣),𝑣₁⟩+⟨𝑔₂(𝑣),𝑣₂⟩‖‖²−2⟨𝑢^𝑛,𝑣₁⟩+2‖‖𝑣_1𝑥‖‖²−2⟨𝑢^𝑛𝑥,𝑣_1𝑥⟩+𝜐𝜏‖‖𝑣_1𝑥‖‖²+2‖‖𝑣₂‖‖²−2⟨𝜌^𝑛,𝑣₂⟩+𝛾𝜏‖‖𝑣₂‖‖²≥2‖‖𝑣₁‖‖²−‖𝑢^𝑛‖²+‖‖𝑣₁‖‖²+2‖‖𝑣^1𝑥‖‖²−‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑣_1𝑥‖‖²+𝜐𝜏‖‖𝑣^1𝑥‖‖²+2‖‖𝑣₂‖‖²−‖𝜌^𝑛‖²+‖‖𝑣₂‖‖²+𝛾𝜏‖‖𝑣²‖‖²=‖‖𝑣₁‖‖²+‖‖𝑣₂‖‖²+‖‖𝑣_1𝑥‖‖²−‖𝑢^𝑛‖²−‖‖𝑢^𝑛𝑥‖‖²−‖𝜌^𝑛‖²+𝜐𝜏‖‖𝑣_1𝑥‖‖²+𝛾𝜏‖‖𝑣₂‖‖²≥‖𝑣‖²−‖𝑢^𝑛‖²+‖‖𝑢^𝑛𝑥‖‖²+‖𝜌^𝑛‖².(3.3) Hence it is obvious that <𝑔(𝑣),𝑣>0 for all 𝑣∈𝑍_Δ with ‖𝑣‖²=(‖𝑢^𝑛‖²+‖𝑢^𝑛𝑥‖²+‖𝜌^𝑛‖²)+1. It follows from Lemma 3.1 that there exists 𝑣^∗∈𝑍_Δ such that 𝑔(𝑣^∗)=0. If we take 𝑢^𝑛+1=2𝑣₁−𝑢^𝑛, 𝜌^𝑛+1=2𝑣^∗2−𝜌^𝑛, then (𝑢^𝑛+1,𝜌^𝑛+1) satisfies (2.2)–(2.5). This completes the proof.

Next we show that the difference scheme (2.2)–(2.5) is convergent and stable.

Let 𝑣(𝑥,𝑡) and (𝑥,𝑡) be the solution of problem (1.4)–(1.6), that is, 𝑣^𝑛𝑗=𝑢(𝑥_𝑗,𝑡_𝑛), ^𝑛𝑗=𝜌(𝑥_𝑗,𝑡_𝑛), then the truncation of the difference scheme (2.2)–(2.5) is𝑟^𝑛𝑗=𝑣^𝑛𝑗_𝑡−𝑣^𝑛𝑗_𝑥𝑥𝑡+^𝑗𝑛+1/2𝑥̂−𝜐𝑣^𝑗𝑛+1/2_𝑥𝑥+13𝑣^{𝑛+1/2𝑗+1}𝑠+𝑣^𝑗𝑛+1/2+𝑣^{𝑛+1/2𝑗−1}𝑣^𝑗𝑛+1/2𝑥̂,(3.4)_𝑛𝑗=^𝑛𝑗_𝑡+𝑣^𝑗𝑛+1/2𝑥̂+𝛾^𝑗𝑛+1/2.(3.5) Making use of Taylor expansion, we know that |𝑟^𝑛𝑗|+|𝑠^𝑛𝑗|=𝑂(𝜏²+ℎ²) hold if ℎ, 𝜏→0.

Lemma 3.3. Suppose that 𝑢₀∈𝐻¹, 𝜌₀∈𝐿₂, the solution of (1.4)–(1.6) satisfies ‖𝑢‖_𝐿2≤𝐶, ‖𝑢_𝑥‖_𝐿2≤𝐶, ‖𝜌‖_𝐿2≤𝐶 and ‖𝑢‖_𝐿∞≤𝐶.

Proof. See Lemma 1.1 in [21].

Theorem 3.4. Suppose 𝑢₀∈𝐻¹, 𝜌₀∈𝐿₂, then the solution 𝑢^𝑛 and 𝜌^𝑛 to the difference scheme (2.2)–(2.5) converges to the solution of problem (1.4)–(1.6), and the rate of convergence is 𝑂(𝜏²+ℎ²).

Proof. Subtracting (2.2) from (3.4) and subtracting (2.3) from (3.5), letting 𝑒^𝑛𝑗=𝑣^𝑛𝑗−𝑢^𝑛𝑗, 𝜂^𝑛𝑗=𝜙^𝑛𝑗−𝜌^𝑛𝑗, we have 𝑟^𝑛𝑗=𝑒^𝑛𝑗_𝑡−𝑠𝑒^𝑛𝑗_𝑥𝑥𝑡+𝜂^𝑗𝑛+1/2𝑥̂−𝜐𝑒^𝑗𝑛+1/2_𝑥𝑥+𝑄^𝑗𝑛+1/2,(3.6)_𝑛𝑗=𝜂^𝑛𝑗_𝑡+𝑒^𝑗𝑛+1/2𝑥̂+𝛾𝜂^𝑗𝑛+1/2,(3.7) where 𝑄^𝑗𝑛+1/2=13𝑣^{𝑛+1/2𝑗+1}+𝑣^𝑗𝑛+1/2+𝑣^{𝑛+1/2𝑗−1}⋅𝑣^𝑗𝑛+1/2𝑥̂−13𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1}⋅𝑢^𝑗𝑛+1/2𝑥̂.(3.8) Computing the inner product of (3.6) with 2𝑒^𝑛+1/2 we get 𝑟^𝑛,2𝑒^𝑛+1/2=𝜏1‖‖𝑒^𝑛+1‖‖²−‖𝑒^𝑛‖²+𝜏1‖‖𝑒^𝑥𝑛+1‖‖²−‖‖𝑒^𝑛𝑥‖‖²+2𝜐‖‖𝑒^𝑥𝑛+1/2‖‖²+𝜂^𝑛+1/2𝑥̂,2𝑒^𝑛+1/2+𝑄^𝑛+1/2,2𝑒^𝑛+1/2.(3.9) Similarly to (2.11), we have 𝜂𝑥̂^𝑛+1/2,2𝑒^𝑛+1/2=−2𝑒𝑥̂^𝑛+1/2,𝜂^𝑛+1/2.(3.10) Then (3.9) can be changed to ‖‖𝑒^𝑛+1‖‖²−‖𝑒^𝑛‖²+‖‖𝑒^𝑥𝑛+1‖‖²−‖‖𝑒^𝑛𝑥‖‖²=−2𝜐𝜏‖‖𝑒_𝑥𝑛+1/2−2𝜏𝑒𝑥̂^𝑛+1/2,𝜂^𝑛+1/2‖‖²+𝜏𝑟^𝑛,2𝑒^𝑛+1/2+𝜏−𝑄^𝑛+1/2,2𝑒^𝑛+1/2.(3.11) It follow from Lemma 3.3, Theorems 2.4 and 2.5 that |||𝑣_{𝑛+1/2𝑗+1}+𝑣^𝑗𝑛+1/2+𝑣^{𝑛+1/2𝑗−1}|||≤𝐶,|||𝑢_𝑗𝑛+1/2|||≤𝐶,(𝑗=0,1,2,…,𝐽).(3.12)
Since−𝑄^𝑛+1/2,2𝑒^𝑛+1/2=−23ℎ^𝐽−1_𝑗=1𝑣^{𝑛+1/2𝑗+1}+𝑣^𝑗𝑛+1/2+𝑣^{𝑛+1/2𝑗−1}𝑣^𝑗𝑛+1/2𝑥̂𝑒^𝑗𝑛+1/2+23ℎ^𝐽−1_𝑗=1𝑢^{𝑛+1/2𝑗+1}+𝑢^𝑗𝑛+1/2+𝑢^{𝑛+1/2𝑗−1}𝑢^𝑗𝑛+1/2𝑥̂𝑒^𝑗𝑛+1/2=−23ℎ^𝐽−1_𝑗=1𝑣^{𝑛+1/2𝑗+1}+𝑣^𝑗𝑛+1/2+𝑣^{𝑛+1/2𝑗−1}𝑒^𝑗𝑛+1/2𝑥̂𝑒^𝑗𝑛+1/2−23ℎ^𝐽−1_𝑗=1𝑒^{𝑛+1/2𝑗+1}+𝑒^𝑗𝑛+1/2+𝑒^{𝑛+1/2𝑗−1}𝑢^𝑗𝑛+1/2𝑥̂𝑒^𝑗𝑛+1/2,(3.13) and the Schwarz inequality, we obtain −𝑄^𝑛+1/2,2𝑒^𝑛+1/2≤23𝐶ℎ^𝐽−1_𝑗=1|||𝑒_𝑗𝑛+1/2𝑥̂|||⋅||𝑒^𝑗𝑛+1/2+2||3𝐶ℎ^𝐽−1_𝑗=1|||𝑒^{𝑛+1/2𝑗+1}≤𝐶‖‖𝑒_𝑛+1/2|||+||𝑒_𝑗𝑛+1/2||+|||𝑒^{𝑛+1/2𝑗−1}|||||𝑒_𝑗𝑛+1/2||‖‖²+‖‖𝑒^𝑥𝑛+1/2‖‖²≤𝐶‖‖𝑒^𝑛+1‖‖²+‖𝑒^𝑛‖²+‖‖𝑒^𝑥𝑛+1‖‖²+‖‖𝑒^𝑛𝑥‖‖².(3.14) According to 𝑟^𝑛,2𝑒^𝑛+1/2=𝑟^𝑛,𝑒^𝑛+1+𝑒^𝑛≤‖𝑟^𝑛‖²+12‖‖𝑒^𝑛+1‖‖²+‖𝑒^𝑛‖².(3.15) It follows from (3.14), (3.15), and (3.11) that ‖‖𝑒^𝑛+1‖‖²−‖𝑒^𝑛‖²+‖‖𝑒^𝑥𝑛+1‖‖²−‖‖𝑒^𝑛𝑥‖‖²≤𝐶𝜏‖‖𝑒_𝑛+1−2𝜏𝑒𝑥̂^𝑛+1/2,𝜂^𝑛+1/2‖‖²+‖𝑒^𝑛‖²+‖‖𝑒^𝑥𝑛+1‖‖²+‖‖𝑒^𝑛𝑥‖‖²+𝜏‖𝑟^𝑛‖².(3.16) Computing the inner product of (3.7) with 2𝜂^𝑛+1/2, we obtain ‖𝜂^𝑛+1‖²−‖𝜂^𝑛‖²+2𝜏𝑒𝑥̂^𝑛+1/2,𝜂^𝑛+1/2=𝜏⟨𝑠^𝑛,2𝜂^𝑛+1/2⟩−2𝛾𝜏‖‖𝜂^𝑛+1/2‖‖²≤𝐶𝜏‖‖𝜂^𝑛+1‖‖²+‖𝜂^𝑛‖²+𝜏‖𝑠^𝑛‖².(3.17) Adding (3.17) to (3.16) we have ‖‖𝑒^𝑛+1‖‖²−‖𝑒^𝑛‖²+‖‖𝑒^𝑥𝑛+1‖‖²−‖‖𝑒^𝑛𝑥‖‖²+‖‖𝜂^𝑛+1‖‖²−‖𝜂^𝑛‖²≤𝜏‖𝑟^𝑛‖²+𝜏‖𝑠^𝑛‖²+𝐶𝜏‖‖𝑒^𝑛+1‖‖²+‖𝑒^𝑛‖²+‖‖𝑒^𝑥𝑛+1‖‖²+‖‖𝑒^𝑛𝑥‖‖²+‖‖𝜂^𝑛+1‖‖²+‖𝜂^𝑛‖².(3.18) Let 𝐵^𝑛=‖𝑒^𝑛‖²+‖𝑒^𝑛𝑥‖²+‖𝜂^𝑛‖², we get 𝐵^𝑛+1−𝐵^𝑛≤𝜏‖𝑟^𝑛‖²+𝜏‖𝑠^𝑛‖²+𝐶𝜏𝐵^𝑛+1+𝐵^𝑛.(3.19) If 𝜏 is sufficiently small which satisfies 1−𝐶𝜏>0, then 𝐵^𝑛+1−𝐵^𝑛≤𝐶𝜏𝐵^𝑛+𝐶𝜏‖𝑟^𝑛‖²+𝐶𝜏‖𝑠^𝑛‖².(3.20) Summing up (3.20) from 0 to 𝑛−1, we have 𝐵^𝑛≤𝐵⁰+𝐶𝜏^𝑛−1_𝑙=0‖‖𝑟^𝑙‖‖²+𝐶𝜏^𝑛−1_𝑙=0‖‖𝑠^𝑙‖‖²+𝐶𝜏^𝑛−1_𝑙=0𝐵^𝑙.(3.21) Since 𝜏^𝑛−1_𝑙=0𝜏_𝑛−1‖‖𝑟^𝑙‖‖²≤𝑛𝜏max_{0≤𝑙≤𝑛−1}‖‖𝑟^𝑙‖‖²≤𝑇⋅𝑂𝜏²+ℎ²²,_𝑙=0‖‖𝑠^𝑙‖‖²≤𝑛𝜏max_{0≤𝑙≤𝑛−1}‖‖𝑠^𝑙‖‖²≤𝑇⋅𝑂𝜏²+ℎ²²,(3.22) and 𝐵⁰=𝑂(𝜏²+ℎ²)², we obtain 𝐵^𝑛≤𝑂𝜏²+ℎ²²+𝐶𝜏^𝑛−1_𝑙=0𝐵^𝑙.(3.23) From Lemma 2.3, we get 𝐵^𝑛≤𝑂𝜏²+ℎ²²,(3.24) which implies ‖𝑒^𝑛‖≤𝑂𝜏²+ℎ²,‖‖𝑒^𝑛𝑥‖‖≤𝑂𝜏²+ℎ²,‖𝜂^𝑛‖≤𝑂𝜏²+ℎ².(3.25) Using Lemma 2.2, we get ‖𝑒^𝑛‖_∞≤𝑂𝜏²+ℎ².(3.26)

Similarly to Theorem 3.4, we can prove the results as follows.

Theorem 3.5. Under the conditions of Theorem 3.4, the solution 𝑢^𝑛 and 𝜌^𝑛 of (2.2)–(2.5) is stable in the senses of norm ‖⋅‖_∞ and ‖⋅‖_𝐿², respectively.

Theorem 3.6. The solution 𝑢^𝑛 of (2.2)–(2.5) is unique.

4. Numerical Simulations

The difference scheme (2.2)–(2.5) is a nonlinear system about 𝑢^𝑗𝑛+1 that can be easily solved by Newton iterative algorithm. When 𝑡=0, the damping does not effect and the dissipative term will not appear. So the initial conditions (1.4)–(1.6) are same as those of (1.1): 𝑢₀(𝑥)=52sech²√56𝑥,𝜌⁰(𝑥)=53sech²√56𝑥,(𝑣=1.5).(4.1) Let 𝑥_𝐿=−20, 𝑥_𝑅=20, 𝑇=5.0. Since we do not know the exact solution of (1.4), an error estimates method in [23] is used: A comparison between the numerical solutions on a coarse mesh and those on a refine mesh is made. We consider the solution on mesh 𝜏=ℎ=1/160 as the reference solution. We denote the C-N scheme in this paper as Scheme I and the difference scheme in [21] as Scheme II. In Tables 1 and 2 we give the ratios in the sense of 𝑙_∞ at various time step when 𝜐=𝛾=0.2 and 𝜐=𝛾=0.5, respectively.

Table 1 

The error comparison in the sense of 𝑙_∞ at various time step when 𝜐=𝛾=0.2.

Table 2 

The error comparison in the sense of 𝑙_∞ at various time step when 𝜐=𝛾=0.5.

In Tables 3 and 4 we verify the second convergence of the scheme I using the method in [25] when 𝜐=𝛾=0.2 and 𝜐=𝛾=0.5, respectively.

Table 3 

The verification of the second convergence when 𝜐=𝛾=0.2.

Table 4 

The verification of the second convergence when 𝜐=𝛾=0.5.

When 𝜐=𝛾=0.2 and 𝜐=𝛾=0.5, a wave figure comparison of 𝑢 and 𝜌 at various time step with 𝜏=ℎ=0.05 is as follow: (see Figures 1, 2, 3, and 4).

Figure 1 

When 𝜐=𝛾=0.2, the wave graph of 𝑢 at various time.

Figure 2 

When 𝜐=𝛾=0.2, the wave graph of 𝜌 at various time.

Figure 3 

When 𝜐=𝛾=0.5, the wave graph of 𝑢 at various time.

Figure 4 

When 𝜐=𝛾=0.5, the wave graph of 𝜌 at various time.

5. Conclusion

In this paper, we propose Crank-Nicolson nonlinear-implicit finite difference scheme of the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term. The two-levels finite difference scheme is of second-order convergence and unconditionally stable, which can start by itself. From the Tables 1 and 2 we conclude that the C-N scheme is more efficient than the Scheme 2 in [21]. From the Tables 3 and 4 we conclude that the C-N scheme is of second-order convergence obviously. Figures 1–4 show that the height of wave crest is more and more low with time elapsing due to the effect of damping term and dissipative term and when 𝜐, 𝛾 become bigger the droop of the height of wave crest is faster.

Acknowledgments

This work was supported by the Sichuan province application of technology research and development project (no. 2010JY0058), the youth research foundation of Sichuan University (no. 2009SCU11113) and the research fund of key disciplinary of application mathematics of Xihua University (Grant no. XZD0910-09-1).