Numerical Analysis of a Linear-Implicit Average Scheme for Generalized Benjamin-Bona-Mahony-Burgers Equation

Abstract

A linear-implicit finite difference scheme is given for the initial-boundary problem of GBBM-Burgers equation, which is convergent and unconditionally stable. The unique solvability of numerical solutions is shown. A priori estimate and second-order convergence of the finite difference approximate solution are discussed using energy method. Numerical results demonstrate that the scheme is efficient and accurate.

1. Introduction

The generalized Benjamin-Bona-Mahony-Burgers (GBBM-Burgers) equation is in the form [1] 𝑢_𝑡−𝑢_𝑥𝑥𝑡−𝛼𝑢_𝑥𝑥+𝛽𝑢_𝑥+𝑢^𝑝𝑢_𝑥=0, (1.1) where 𝛼>0,𝛽 are constants, 𝑝≥1 is an integer, and 𝑢(𝑥,𝑡) represents the velocity of fluid in the horizontal direction 𝑥. When 𝑝=1, (1.1) is called as the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation. In the special case, when 𝛼=0, (1.1) is described as the generalized Benjamin-Bona-Mahony equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢_𝑥+𝑢^𝑝𝑢_𝑥=0. (1.2) The Equation (1.2) which is usually called as the generalized regularized long-wave equation proposed by Peregrine [2] and Benjamin et al. [3], so-called generalized Benjamin-Bona-Mahony equation, has been studied by many authors [4–7]. This equation features a balance between the nonlinear dispersive effect but takes no account of dissipation.

In recent years, a vast amount of work and computation has been devoted to the initial value problem for the GBBM-Burgers equation. In [1], Al-Khaled et al. studied the GBBM-Burgers by Decomposition method. In [8], Hayashi et al. investigated large time asymptotics of solutions to the BBM-Burgers equation. In [9], Jiang and Xu investigated the asymptotic behavior of solutions of the initial-boundary value problem for the GBBM-Burgers equations. In [10], Yin et al. studied the large time behavior of traveling wave solutions to the Cauchy problem of the GBBM-Burgers equations. In [11], Mei studied the large time behavior of global solutions to the Cauchy problem of GBBM-Burgers equations. In [12], Kondo and Webler studied the global existence of solutions for multidimensional GBBM-Burgers equations. Kinami et al. discussed the Cauchy problem of the GBBM-Burgers equations by Fourier transform method and energy method [13]. However, there are few studies on finite difference approximations for (1.1) which we consider in this paper.

In a recent work [14], we have made some preliminary computation by proposing a linearized difference scheme for GRLW equation which is unconditionally stable and reduces the computational work, and the numerical results are encouraging. In this paper, we continue our work and propose a linear-implicit difference scheme for generalized BBM-Burgers equation which is unconditionally stable and second-order convergent.

In this paper, we consider the following initial-boundary value problem of the GBBM-Burgers equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡−𝛼𝑢_𝑥𝑥+𝛽𝑢_𝑥+𝑢^𝑝𝑢_𝑥=0,𝑥∈𝑥_𝐿,𝑥_𝑅[],𝑢(),𝑡∈0,𝑇𝑥,0=𝑢₀(𝑥),𝑥∈𝑥_𝐿,𝑥_𝑅,𝑢𝑥_𝐿,𝑡=𝑢𝑥_𝑅,𝑡=0,𝑡∈[0,𝑇]. (1.3) An outline of the paper is as follows. In Section 2, we describe a linear-implicit finite difference scheme for the GBBM-Burgers equation and prove the error estimates of 2 order. In Section 3, we show that the scheme is uniquely solvable. In Section 4, convergence and stability of the scheme are proved. In Section 5, numerical results are provided to test the theoretical results.

2. Finite Difference Scheme and Estimate for the Difference Solution

As usual, the following notations will be used: 𝑥_𝑗=𝑥_𝐿+𝑗ℎ,𝑡_𝑛𝑇=𝑛𝜏,0≤𝑗≤𝐽,0≤𝑛≤𝑁=𝜏, (2.1) where ℎ=(𝑥_𝑅−𝑥_𝐿)/𝐽 and 𝜏 are the uniform spatial and temporal step sizes, respectively, 𝑢^𝑛_𝑗𝑥=𝑢^𝑛_𝑗+1−𝑢^𝑛_𝑗ℎ,𝑢^𝑛_𝑗𝑥=𝑢^𝑛_𝑗−𝑢^𝑛_𝑗−1ℎ,𝑢^𝑛_𝑗=𝑢𝑥^𝑛𝑗+1−𝑢^𝑛_𝑗−1,𝑢2ℎ^𝑛𝑗𝑡̂=𝑢_𝑗^𝑛+1−𝑢_𝑗^𝑛−1,2𝜏𝑢^𝑛_𝑗=𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−12,𝑢^𝑛,𝑣^𝑛=ℎ_𝑗𝑢^𝑛_𝑗𝑣^𝑛_𝑗,𝑢^𝑛2=𝑢^𝑛,𝑢^𝑛,𝑢^𝑛_∞=sup_𝑗𝑢^𝑛_𝑗. (2.2) Let 𝑢^𝑛_𝑗 denote the approximation of 𝑢(𝑥_𝑗,𝑡_𝑛), 𝑍⁰_ℎ={𝑢=(𝑢_𝑗)∣𝑢₀=𝑢_𝐽=0,1≤𝑗≤𝐽}. In this paper, we will denote 𝐶 as a generic constant independent of step sizes ℎ and 𝜏.

We propose a three-level linear-implicit difference scheme for the solution of the problem (1.3) 𝑢^𝑛_𝑗𝑡̂−𝑢^𝑛_𝑗𝑥𝑥𝑡̂−𝛼𝑢^𝑛_𝑗𝑥𝑥+𝛽𝑢^𝑛_𝑗+1𝑥𝑝+2𝑢^𝑛_𝑗^𝑝𝑢^𝑛_𝑗𝑥+𝑢^𝑛_𝑗^𝑝𝑢^𝑛_𝑗𝑥=0,1≤𝑗≤𝐽−1,1≤𝑛≤𝑁−1, (2.3)𝑢⁰_𝑗=𝑢₀𝑥_𝑗,1≤𝑗≤𝐽−1, (2.4)𝑢¹_𝑗−𝑢¹_𝑗𝑥𝑥=𝑢₀𝑥_𝑗+𝑑²𝑢₀𝑑𝑥²𝑥_𝑗−𝜏𝛽𝑑𝑢₀𝑥𝑑𝑥^𝑗𝑑−𝛼²𝑢₀𝑑𝑥²𝑥_𝑗+𝑢^𝑝₀𝑥_𝑗𝑑𝑢₀𝑥𝑑𝑥^𝑗, (2.5)𝑢^𝑛₀=𝑢^𝑛_𝐽=0,1≤𝑛≤𝑁−1. (2.6)

For convenience, the last term of (2.3) is defined by 𝜓𝑢^𝑛,𝑢^𝑛=1𝑢𝑝+2^𝑛𝑗𝑝𝑢^𝑛_𝑗𝑥+𝑢^𝑛_𝑗^𝑝𝑢^𝑛_𝑗𝑥. (2.7)

Lemma 2.1 (see [15]). For any two mesh functions 𝐮,𝐯∈𝐙^𝟎_𝐡, one has 𝐮_𝐱,𝐯=−𝐮,𝐯_𝐱,(𝐮)_𝐱𝐱,𝐯=−𝐮_𝐱,𝐯_𝐱,(𝐮)_𝐱𝐱,𝐮=−𝐮_𝐱,𝐮_𝐱=−𝐮_𝐱^𝟐. (2.8)

Lemma 2.2. For any mesh function 𝑢∈𝑍⁰_ℎ, one has 𝜓𝑢^𝑛,𝑢^𝑛,𝑢^𝑛=0. (2.9)

Proof. For 𝑢^𝑛∈𝑍⁰_ℎ, one has 𝜓𝑢^𝑛,𝑢^𝑛,𝑢^𝑛=18𝑝+2_𝑗𝑢^𝑛_𝑗^𝑝𝑢^𝑛+1_𝑗+1−𝑢^𝑛+1_𝑗−1+𝑢^𝑛−1_𝑗+1−𝑢^𝑛−1_𝑗−1+𝑢^𝑛_𝑗+1^𝑝𝑢^𝑛+1_𝑗+1+𝑢^𝑛−1_𝑗+1−𝑢^𝑛_𝑗−1^𝑝𝑢^𝑛+1_𝑗−1+𝑢^𝑛−1_𝑗−1𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1=18𝑝+2_𝑗𝑢^𝑛_𝑗^𝑝𝑢^𝑛+1_𝑗+1+𝑢^𝑛−1_𝑗+1+𝑢^𝑛_𝑗+1^𝑝𝑢^𝑛+1_𝑗+1+𝑢^𝑛−1_𝑗+1𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1−18𝑝+2_𝑗𝑢^𝑛_𝑗+1^𝑝𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1+𝑢^𝑛_𝑗^𝑝𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1𝑢^𝑛+1_𝑗+1+𝑢^𝑛−1_𝑗+1=0. (2.10)

Lemma 2.3 (Discrete Sobolev Inequality [16]). For any discrete function 𝑢_ℎ and for any given 𝜀>0, there exists a constant 𝐾(𝜀,𝑛), depending only 𝜀 and 𝑛, such that 𝑢^𝑛_∞≤𝜀𝑢^𝑛_𝑥+𝐾(𝜀,𝑛)𝑢^𝑛. (2.11)

Theorem 2.4. Assume 𝑢₀∈𝐻¹₀, then there is the estimation for the solution of difference scheme (2.3)–(2.6), 𝑢^𝑛≤𝐶,𝑢^𝑛_𝑥≤𝐶,𝑢^𝑛_∞≤𝐶. (2.12)

Proof. Computing the inner product of (2.3) with 2𝑢^𝑛 (i.e., 𝑢^𝑛+1+𝑢^𝑛−1), we obtain 1𝑢2𝜏^𝑛+12−𝑢^𝑛−12+1𝑢2𝜏^𝑥𝑛+12−𝑢_𝑥^𝑛−12−𝛼𝑢^𝑛_𝑥𝑥,2𝑢^𝑛+𝛽ℎ_𝑗𝑢^𝑛_𝑗𝑥𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1+𝜓𝑢^𝑛,𝑢^𝑛,2𝑢^𝑛=0. (2.13) Now, computing the fourth term of the left-hand side in (2.13), we have ℎ_𝑗𝑢^𝑛_𝑗𝑥𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1=ℎ_𝑗𝑢^𝑛_𝑗𝑥𝑢_𝑗^𝑛+1−_𝑗𝑢_𝑗^𝑛−1𝑥𝑢^𝑛_𝑗. (2.14) According to Lemmas 2.1 and 2.2, and using (2.14), we get 1𝑢2𝜏^𝑛+12−𝑢^𝑛−12+1𝑢2𝜏^𝑥𝑛+12−𝑢_𝑥^𝑛−12+𝛽ℎ_𝑗𝑢^𝑛_𝑗𝑥𝑢_𝑗^𝑛+1−_𝑗𝑢_𝑗^𝑛−1𝑥𝑢^𝑛_𝑗=−2𝛼𝑢^𝑛_𝑥²≤0. (2.15) We let 𝐸^𝑛=12𝑢^𝑛+12+𝑢^𝑛2+12𝑢_𝑥^𝑛+12+𝑢^𝑛_𝑥²+𝛽ℎ𝜏_𝑗𝑢^𝑛_𝑗𝑥𝑢_𝑗^𝑛+1. (2.16) It follows from (2.15) that 𝐸^𝑛=12𝑢^𝑛+12+𝑢^𝑛2+12𝑢_𝑥^𝑛+12+𝑢^𝑛_𝑥²+𝛽ℎ𝜏_𝑗𝑢^𝑛_𝑗𝑥𝑢_𝑗^𝑛+1≤𝐸^𝑛−1≤⋯≤𝐸⁰. (2.17) Then we have 12𝑢^𝑛+12+𝑢^𝑛2+12𝑢_𝑥^𝑛+12+𝑢^𝑛_𝑥²1≤𝐶+2𝛽𝜏𝑢^𝑛_𝑥²+𝑢^𝑛+12. (2.18) Using (2.18), we obtain 121−𝛽𝜏𝑢^𝑛+12+𝑢^𝑛2+12𝑢_𝑥^𝑛+12+1−𝛽𝜏𝑢^𝑛_𝑥²≤𝐶. (2.19) Equation (2.19) yields 𝑢^𝑛≤𝐶,𝑢^𝑛_𝑥≤𝐶. (2.20) Using Lemma 2.3, the proof of Theorem 2.4 is completed.

Remark 2.5. Theorem 2.4 implies that scheme (2.3)–(2.6) is unconditionally stable.

3. Solvability

Next, we will discuss the solvability of the scheme (2.3) based on the technique of Omrani et al. [17].

Theorem 3.1. The finite difference scheme (2.3) is uniquely solvable.

Proof. It is obvious that 𝑢⁰ and 𝑢¹ are uniquely determined by (2.4)-(2.5). Now suppose 𝑢⁰,𝑢¹,…,𝑢^𝑛(1≤𝑛≤𝑁−1) be solved uniquely. Considering the equation of (2.3) for 𝑢^𝑛+1, we have 1𝑢2𝜏^𝑗𝑛+1−1𝑢2𝜏^{𝑗𝑛+1𝑥𝑥}−𝛼2𝑢_𝑗^𝑛+1_𝑥𝑥+12𝑢𝑝+2^𝑛𝑗𝑝𝑢_𝑗^𝑛+1𝑥+𝑢^𝑛_𝑗^𝑝𝑢_𝑗^𝑛+1𝑥=0. (3.1) Computing the inner product of (3.1) with 𝑢^𝑛+1, we have 1𝑢2𝜏^𝑛+12+1𝑢2𝜏^𝑥𝑛+12+𝛼2𝑢_𝑥^𝑛+12+𝜙𝑢^𝑛,𝑢^𝑛+1,𝑢^𝑛+1=0, (3.2) where 𝜙(𝑢^𝑛,𝑢^𝑛+1)=(1/2(𝑝+2))[(𝑢^𝑛_𝑗)^𝑝(𝑢_𝑗^𝑛+1)𝑥+((𝑢^𝑛_𝑗)^𝑝𝑢_𝑗^𝑛+1)𝑥].
In view of difference properties and the boundary conditions (2.6), we obtain 𝜙𝑢^𝑛,𝑢^𝑛+1,𝑢^𝑛+1=12ℎ𝑝+2^𝐽−1_𝑗=1𝑢^𝑛_𝑗^𝑝𝑢_𝑗^𝑛+1𝑥+𝑢^𝑛_𝑗^𝑝𝑢_𝑗^𝑛+1𝑥𝑢_𝑗^𝑛+1=14ℎ𝑝+2^𝐽−1_𝑗=1𝑢^𝑛_𝑗^𝑝𝑢^𝑛+1_𝑗+1𝑢_𝑗^𝑛+1+𝑢^𝑛_𝑗+1^𝑝𝑢^𝑛+1_𝑗+1𝑢_𝑗^𝑛+1−14ℎ𝑝+2^𝐽−1_𝑗=1𝑢^𝑛_𝑗^𝑝𝑢^𝑛+1_𝑗−1𝑢_𝑗^𝑛+1+𝑢^𝑛_𝑗−1^𝑝𝑢^𝑛+1_𝑗−1𝑢_𝑗^𝑛+1=0. (3.3) It follows from (3.2) and (3.3) that 𝑢^𝑛+12+𝑢_𝑥^𝑛+12+𝛼𝜏𝑢_𝑥^𝑛+12=0. (3.4) Noting that 𝛼>0 and following from (3.4), we have 𝑢^𝑛+12+𝑢_𝑥^𝑛+12=0. (3.5) That is (3.1) has only a trivial solution. Therefore, the scheme (2.3) determines 𝑢_𝑗^𝑛+1 uniquely. This completes the proof.

Remark 3.2. All results above in this paper are correct for IBV problem of the BBM-Burgers equation with finite or infinite boundary.

4. Convergence and Stability of the Difference Scheme

First, we consider the truncation error of the difference scheme (2.3)–(2.6).

Suppose 𝑣^𝑛_𝑗=𝑢(𝑥_𝑗,𝑡_𝑛). Making use of Taylor expansion, we find 𝐸𝑟^𝑛_𝑗=𝑣^𝑛_𝑗𝑡̂−𝑣^𝑛_𝑗𝑥𝑥𝑡̂−𝛼𝑣^𝑛_𝑗𝑥𝑥+𝛽𝑣^𝑛_𝑗+1𝑥𝑝+2𝑣^𝑛_𝑗^𝑝𝑣^𝑛_𝑗𝑥+𝑣^𝑛_𝑗^𝑝𝑣^𝑛_𝑗,𝑢_0𝑗𝑥=𝑢₀𝑥_𝑗,𝑢¹_𝑗−𝑢¹_𝑗𝑥𝑥=𝑢₀𝑥_𝑗+𝑑²𝑢₀𝑑𝑥²𝑥_𝑗−𝜏𝛽𝑑𝑢₀𝑥𝑑𝑥^𝑗𝑑−𝛼²𝑢₀𝑑𝑥²𝑥_𝑗+𝑢^𝑝₀𝑥_𝑗𝑑𝑢₀𝑥𝑑𝑥^𝑗+𝑟_𝑖, (4.1) where 𝐸𝑟^𝑛_𝑗 and 𝑟_𝑖 are the truncation errors of the difference scheme (2.3)–(2.6). It can be easily obtained that (see [18, 19]) 𝐸𝑟^𝑛_𝑗=𝑂ℎ²+𝜏²𝑟_𝑛𝑗,=𝑂ℎ²+𝜏². (4.2)(4.3)

Lemma 4.1. Assume 𝑢(𝑥,𝑡) is smooth enough, then the local truncation error of the finite difference scheme (2.3)–(2.6) is 𝐸𝑟^𝑛_𝑗=𝑂ℎ²+𝜏². (4.4)

Lemma 4.2 (see [16]). Suppose that the discrete function 𝑤_ℎ satisfies recurrence formula 𝑤_𝑛−𝑤_𝑛−1≤𝐴𝜏𝑤_𝑛+𝐵𝜏𝑤_𝑛−1+𝐶_𝑛𝜏, (4.5) where 𝐴,𝐵,𝐶_𝑛(𝑛=1,⋯𝑁) are nonnegative constants. Then 𝑤_𝑛∞≤𝑤₀+𝜏^𝑁_𝑘=1𝐶_𝑘𝑒^{2(𝐴+𝐵)𝜏}, (4.6) where 𝜏 is small, such that (𝐴+𝐵)𝜏≤((𝑁−1)/2𝑁)(𝑁>1).

Theorem 4.3. Assume 𝑢₀∈𝐻¹₀[𝑥_𝐿,𝑥_𝑅] and 𝑢∈𝐶^(4,3), then the solution of the difference scheme (2.3)–(2.6) converges to the solution of the problem (1.3) with order 𝑂(ℎ²+𝜏²) by the ||⋅||_∞ norm.

Proof. Let 𝑒^𝑛_𝑗=𝑣^𝑛_𝑗−𝑢^𝑛_𝑗. Subtracting (2.3)-(2.5) from (4.1)–(4.3), respectively, we have 𝐸𝑟^𝑛_𝑗=𝑒^𝑛_𝑗𝑡̂−𝑒^𝑛_𝑗𝑥𝑥𝑡̂−𝛼𝑒^𝑛_𝑗𝑥𝑥+𝛽𝑒^𝑛_𝑗+1𝑥𝑝+2𝑣^𝑛_𝑗^𝑝𝑣^𝑛_𝑗𝑥+𝑣^𝑛_𝑗^𝑝𝑣^𝑛_𝑗−1𝑥𝑢𝑝+2^𝑛𝑗𝑝𝑢^𝑛_𝑗𝑥+𝑢^𝑛_𝑗^𝑝𝑢^𝑛_𝑗,𝑒_0𝑗𝑥𝑒=0,_1𝑗=𝑟_𝑗. (4.7) For a simple notation, the last two terms of (4.7) are defined by 1𝐼=𝑣𝑝+2^𝑛𝑗𝑝𝑣^𝑛_𝑗−1𝑥𝑝+2𝑢^𝑛_𝑗^𝑝𝑢^𝑛_𝑗,1𝑥𝐼𝐼=𝑣𝑝+2^𝑛𝑗𝑝𝑣^𝑛_𝑗−1𝑥𝑝+2𝑢^𝑛_𝑗^𝑝𝑢^𝑛_𝑗𝑥. (4.8) Computing the inner product of (4.7) with 𝑒^𝑛+1+𝑒^𝑛−1 (i.e., 2𝑒^𝑛), we get 𝐸𝑟^𝑛_𝑗,2𝑒^𝑛=1𝑒2𝜏^𝑛+12−𝑒^𝑛−12+1𝑒2𝜏^𝑥𝑛+12−𝑒_𝑥^𝑛−12−𝛼𝑒^𝑛_𝑥𝑥,2𝑒^𝑛+𝛽ℎ_𝑗𝑒^𝑛_𝑗𝑥𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1+𝐼+𝐼𝐼,2𝑒^𝑛. (4.9) Similarly to the proof of Theorem 2.4, we obtain 𝛼𝑒^𝑛_𝑥𝑥,2𝑒^𝑛=−2𝛼𝑒^𝑛_𝑥²,𝛽ℎ_𝑗𝑒^𝑛_𝑗𝑥𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1=𝛽ℎ_𝑗𝑒^𝑛_𝑗𝑥𝑒_𝑗^𝑛+1−ℎ_𝑗𝑒^𝑛_𝑗−1𝑥𝑒^𝑛_𝑗. (4.10) According to Theorem 2.4, we obtain 𝐼,2𝑒^𝑛=1𝑝+2ℎ_𝑗𝑣^𝑛_𝑗^𝑝𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1𝑥+𝑣^𝑛_𝑗^𝑝−𝑢^𝑛_𝑗^𝑝𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1𝑥𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1≤𝐶ℎ_𝑗𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1𝑥+𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1𝑥𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1≤𝐶𝑒_𝑥^𝑛+12+𝑒_𝑥^𝑛−12+𝑒^𝑛+12+𝑒^𝑛−12,𝐼𝐼,2𝑒^𝑛=1𝑝+2_𝑗𝑣^𝑛_𝑗^𝑝𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1𝑥+((𝑣^𝑛_𝑗)^𝑝−(𝑢^𝑛_𝑗)^𝑝)(𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1)𝑥𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−11=−3ℎ_𝑗𝑣^𝑛_𝑗^𝑝𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1+𝑣^𝑛_𝑗^𝑝−𝑢^𝑛_𝑗^𝑝𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−1𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1𝑥≤𝐶ℎ_𝑗𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1+𝑒^𝑛_𝑗𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1𝑒𝑥≤𝐶_𝑥^𝑛+12+𝑒_𝑥^𝑛−12+𝑒^𝑛+12+𝑒^𝑛2+𝑒^𝑛−12. (4.11) In addition, there exists obviously that 𝐸𝑟^𝑛_𝑗,𝑒^𝑛+1+𝑒^𝑛−1≤𝐸𝑟^𝑛2+12𝑒^𝑛+12+𝑒^𝑛−12. (4.12) Substituting (4.10)–(4.12) into (4.9), we have 1𝑒2𝜏^𝑛+12−𝑒^𝑛−12+1𝑒2𝜏^𝑥𝑛+12−𝑒_𝑥^𝑛−12≤𝐸𝑟^𝑛2+12𝑒^𝑛+12+𝑒^𝑛−12+𝛽𝑒^𝑛_𝑥²+12𝛽𝑒^𝑛+12+𝑒^𝑛2+𝐶𝑒_𝑥^𝑛+12+𝑒^𝑛_𝑥²+𝑒_𝑥^𝑛−12+𝑒^𝑛+12+𝑒^𝑛2+𝑒^𝑛−12. (4.13) Let 𝐵^𝑛=12𝑒^𝑛+12+𝑒^𝑛2+12𝑒_𝑥^𝑛+12+𝑒^𝑛_𝑥². (4.14) Then (4.13) can be rewritten as 𝐵^𝑛−𝐵^𝑛−1≤𝜏𝐸𝑟^𝑛2+𝐶𝜏𝐵^𝑛+𝐵^𝑛−1. (4.15) By Lemma 4.2, it can immediately be obtained that 𝐵^𝑁≤𝐵⁰+𝑇sup_{1≤𝑛≤𝑁}𝐸𝑟^𝑛2𝑒^𝐶𝑇. (4.16) To complete the proof, it is enough to find 𝐵⁰ estimate. From (4.7), we obtain 𝑒⁰=0. (4.17) Using (4.3) and (4.8), we get 𝑒¹≤𝑂ℎ²+𝜏². (4.18) It follows from (4.17) and (4.18) that 𝐵⁰≤𝑂𝜏²+ℎ²². (4.19) Thus 𝑒^𝑛≤𝑂𝜏²+ℎ²,𝑒^𝑛_𝑥≤𝑂𝜏²+ℎ². (4.20) According to Lemma 2.3, there exists that 𝑒^𝑛_∞≤𝑂𝜏²+ℎ². (4.21)

Similarly, the following theorem can be proved.

Theorem 4.4. Under the conditions of Theorem 4.3, the solution of finite difference scheme (2.3)–(2.6) is stable by the ||⋅||_∞ norm.

5. Numerical Experiments

In this section, we will compute several numerical experiments to verify the correction of our theoretical analysis in the above sections.

Example 5.1 (see [20]). Consider the following initial-boundary problem of BBM-Burgers equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡−𝛼𝑢_𝑥𝑥+𝑢_𝑥+𝑢𝑢_𝑥[][],𝑢()=0,𝑥∈0,1,𝑡∈0,10𝑥,0=𝑢₀(𝑥)[]𝑢()()[].,𝑥∈0,1,0,𝑡=𝑢1,𝑡=0,𝑡∈0,10 (5.1)(5.2)(5.3) We denote the scheme proposed in [20] as Scheme I and the scheme (2.3) in present paper as Scheme II. In computations, we choose the initial condition 𝑢₀(𝑥)=exp(−𝑥²) [20]. The maximal errors of both schemes are listed in Table 1. We get that a second-order linear scheme is as accurate as Scheme I which is a nonlinear one.

Table 1 

The maximal errors of numerical solutions at 𝑡=10 with 𝜏=0.1 for 𝛼=0.5 when 𝑝=1.

Example 5.2 (see [13]). Consider the GBBM-Burgers equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡−𝛼𝑢_𝑥𝑥+𝛽𝑢_𝑥+𝑢^𝑝𝑢_𝑥=0,𝑥∈[0,1],𝑡∈[0,𝑇], (5.4) with an initial condition 𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈[0,1], (5.5) and boundary conditions 𝑢(0,𝑡)=𝑢(1,𝑡)=0,𝑡∈[0,𝑇]. (5.6) In computations, we choose the initial condition 𝑢₀(𝑥)=1/(1+𝑥⁴) [13]. Without loss of generality, We take 𝑝=2,4,8 and 𝛼=0.5, 𝛽=1. Since we do not know the exact solution of (5.4)–(5.6), an error estimate method in [21] is used. A comparison between the numerical solutions on a coarse mesh and those on a refine mesh is made. In order to obtain the error estimates, we consider the solution on mesh ℎ=1/160 as reference solution and obtain error estimates on mesh ℎ=1/4,1/8,1/16, and 1/32, respectively. We denote the scheme proposed in [13] as Scheme III and make a comparison with the scheme (2.3) in present paper as Scheme II when 𝑝=4 in Table 2. The corresponding errors in the sense of 𝐿_∞-norm and 𝐿₂-norm are listed in Tables 3, 4, and 5, respectively. These three tables verify the second-order convergence and good stability of the numerical solutions.
Figures 1 and 2 plot the numerical solutions computed by the linearly implicit scheme (2.3) with 𝜏=0.1, ℎ=0.03125, and 𝛼=0.5 when 𝑝=2,8 at 𝑡=2,4,6,8, and 10, respectively. From Figures 1 and 2, it is easy to observe that the height of the numerical approximation to 𝑢 is more and more low with time elapsing due to the effect of dissipative term 𝛼𝑢_𝑥𝑥. Both of them simulates that the continuous energy 𝐸(𝑡) of the problem (1.3) in Theorem 2.4 decreases in time. Numerical experiments show our scheme is accurate and efficient.

Table 2 

The maximal errors of numerical solutions at 𝑡=10 with 𝜏=0.1 for 𝛼=0.5 when 𝑝=4.

Table 3 

The errors of numerical solutions at 𝑡=10 with 𝜏=0.1 when 𝑝=2.

Table 4 

The errors of numerical solutions at 𝑡=10 with 𝜏=0.1 when 𝑝=4.

Table 5 

The errors of numerical solutions at 𝑡=10 with 𝜏=0.1 when 𝑝=8.

Figure 1 

Numerical solution of 𝑢(𝑥,𝑡) with ℎ=0.03125,𝜏=0.1 when 𝑝=2.

Figure 2 

Numerical solution of 𝑢(𝑥,𝑡) with ℎ=0.03125,𝜏=0.1 when 𝑝=8.

6. Conclusions

In this paper, we have presented a three-level linear-implicit finite difference scheme for the GBBM-Burgers equation, which has a wide range of applications in physics. The convergence and stability as well as second-order error estimate of the finite difference approximate solutions were discussed in detail. Numerical experiments show our scheme is accurate and efficient.

Acknowledgments

This work is supported by the fund of National Natural Science (11171193, 11171180, and 10901096) and the fund of Natural Science of Shandong Province (ZR2009AL019, ZR2011AM016), and the Youth Research Foundation of WFU (no. 2011Z17). The authors thank the referees for their valuable comments.