Optimal Error Estimate of Chebyshev-Legendre Spectral Method for the Generalised Benjamin-Bona-Mahony-Burgers Equations

Abstract

Combining with the Crank-Nicolson/leapfrog scheme in time discretization, Chebyshev-Legendre spectral method is applied to space discretization for numerically solving the Benjamin-Bona-Mahony-Burgers (gBBM-B) equations. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. Optimal error estimate of the Chebyshev-Legendre method is proved for the problem with Dirichlet boundary condition. The convergence rate shows “spectral accuracy.” Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.

1. Introduction

We are interested in numerically solving initial boundary value problem of the generalised Benjamin-Bona-Mahony-Burgers (gBBM-B) equations in the following form:𝑢_𝑡−𝜇𝑢_𝑥𝑥𝑡−𝛼𝑢_𝑥𝑥+𝛽𝑢_𝑥+𝑔(𝑢)_𝑥=0, (1.1) where 𝑢=𝑢(𝑥,𝑡) represents the fluid velocity in horizontal direction 𝑥, parameters 𝜇∈[0,1], 𝛼>0 and 𝛽 are any given constants, and 𝑔(𝑢) is a nonlinear function with certain smoothness. Notation 𝑢_𝑡 denotes the first derivative of 𝑢 with respect to temporal 𝑡 and 𝑢_𝑥, 𝑢_𝑥𝑥 are the first and second derivatives with respect to space variable 𝑥.

When 𝑔(𝑢)=(1/2)𝑢² with 𝛼=0 and 𝛽=1, (1.1) is the alternative regularized long-wave equation proposed by Peregrine [1] and Benjamin et al. [2]. Equation (1.1) features a balance between nonlinear and dispersive effects but takes no account of dissipation for the case 𝛼=0. In the physical sense, (1.1) with the dissipative term 𝛼𝑢_𝑥𝑥 is proposed if the good predictive power is desired, such problem arises in the phenomena for both the bore propagation and the water waves.

The well posedness and some asymptotic properties of solutions to this problem were discussed by Karch [3], Zhang [4], and so on. For more knowledge about BBM-B equations, please consult with [5, 6] and so forth.

In [7], Al-Khaled et al. implemented the Adomian decomposition method for obtaining explicit and numerical solutions of the BBM-B equation (1.1). By applying the classical Lie method of infinitesimals, Bruzón et al. [8–10] obtained, for a generalization of a family of BBM equations, many exact solutions expressed by various single and combined nondegenerative Jacobi elliptic functions. Tari and Ganji [11] have applied two methods known as “variational iteration” and “homotopy perturbation” methods to derive approximate explicit solution for (1.1) with 𝑔(𝑢)=(1/2)𝑢². El-Wakil et al. [12] and Kazeminia et al. [13] used the “exp-function” method with the aid of symbolic computational system to obtain the solitary solutions and periodic solutions for (1.1) with 𝑔(𝑢)=(1/2)𝑢². Variational iteration combining with the exp-function method to solve the generalized BBM equation with variable coefficients was conducted by Gómez and Salas [14]. In [15], Fakhari et al. solved the BBM-B equation by the homotopy analysis method to evaluate the nonlinear equation (1.1) with 𝑔(𝑢)=(1/2)𝑢², 𝛼=0, and 𝛽=1. A tanh method and sinc-Galerkin procedure were used by Alqruan and Al-Khaled [16] to solve gBBM-B equations. Omrani and Ayadi [17] used Crank-Nicolson-type finite difference method for numerical solutions of the BBM-B equation in one space dimension. In [18], a local discontinuous Galerkin finite element method was used and an optimal error estimate was derived by the authors for numerical solution to BBM-B equation. To our knowledge, there is little work on spectral method for numerical solution of the gBBM-B equations.

For completeness, (1.1) may be complemented with certain initial and boundary conditions. Here in this paper we will deal with the initial boundary value problem𝑢_𝑡−𝜇𝑢_𝑥𝑥𝑡−𝛼𝑢_𝑥𝑥+𝛽𝑢_𝑥+𝑔(𝑢)_𝑥=𝑓,𝑥∈Λ∶=[−1,1],𝑡>0, (1.2) with Dirichlet-type boundary conditions𝑢(1,𝑡)=𝑢(−1,𝑡)=0,𝑡≥0, (1.3) and initial condition𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈Λ, (1.4) The appearance of the right-hand side term in (1.2) is just for convenience in order to test the numerical efficiency of the proposed method. For other kinds of boundary conditions, a penalty method may be used as in [19–21].

Let 𝐿²(Λ) be a square integrable function space with inner product and norm as follows:(𝑢,𝑣)=_Λ𝑢(𝑥)𝑣(𝑥)𝑑𝑥,‖𝑢‖=(𝑢,𝑢). (1.5) For any positive integer 𝑟, 𝐻^𝑟(Λ) denotes the Sobolev-type space defined as𝐻^𝑟(Λ)=𝑢∈𝐿²(Λ)∶𝜕^𝑘𝑢𝜕𝑥^𝑘∈𝐿²(Λ),0≤𝑘≤𝑟. (1.6) We denote the norm and seminorm of 𝐻^𝑟(Λ) by ‖⋅‖_𝑟 and |⋅|_𝑟, respectively. For any real 𝑟, 𝐻^𝑟(Λ) is defined by space interpolation [22]. 𝐻¹₀(Λ) is a subspace of 𝐻¹(Λ) in which the function vanishes at ±1.

We reformulate the problem (1.2)–(1.4) in weak form: find 𝑢(𝑡)∈𝐻¹₀(Λ), such that, forall𝜙∈𝐻¹₀(Λ),𝑢_𝑡,𝜙+𝜇𝑢_𝑥𝑡,𝜙_𝑥+𝛼𝑢_𝑥,𝜙_𝑥=𝛽𝑢,𝜙_𝑥+𝑔(𝑢),𝜙_𝑥+(],=𝑢𝑓,𝜙,𝑡∈0,𝑇𝑢,𝜙₀,𝜙,𝑡=0. (1.7)

Due to the high accuracy, spectral/pseudospectral methods are increasingly popular during the last two decades [23–27]. In the context of spectral methods, Legendre and Chebyshev orthogonal polynomials are extensively used for the nonperiodic cases. The Legendre method is natural in the theoretical analysis due to the unity weight function. However, it is well known that the applications of Legendre method are limited by the lack of fast transform between the physical space and the spectral space and by the lack of explicit evaluation formulation of the Legendre-Gauss-Lobatto (LGL) nodes. A Chebyshev-Legendre method that implements the Legendre method at Chebyshev nodes was proposed by Don and Gottlieb [19] for the parabolic and hyperbolic equations. The approach enjoys advantages of both the Legendre and Chebyshev methods. The Chebyshev-Legendre methods were also applied to elliptic problems [28], nonlinear conservative laws in [29, 30], Burgers-like equations in [21], KdV equations in [31], generalised Burgers-Fisher equation [32], nonclassical parabolic equation [33], and optimal control problems [34]. In this paper, Chebyshev-Legendre spectral method will be applied to solving the initial boundary value problem (1.2)–(1.4). The proposed approach is based on Legendre-Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. The computational complexity is reduced and both accuracy and efficiency are achieved by using the proposed method. Compared with the generalised Burgers equation, the gBBM-B equation has an additional term −𝜇𝑢_𝑥𝑥𝑡, which serves as a stabilizer. Due to this reason, numerical analysis of the Chebyshev-Legendre schemes for gBBM-B equations is actually much easier than that for generalised Burgers equations in [20, 21].

This paper is organized as follows. In Section 2, we set up Chebyshev-Legendre spectral method for gBBM-B equations and show how to implement the scheme. Section 3 is preliminary that involves some lemmas which will be used later. In Section 4, error analysis is performed for both semidiscretization and fully discretization schemes. We obtain an optimal convergence rate in sense of 𝐻¹-norm. In Section 5, numerical experiments are presented to support the theoretical results. Conclusion is given in Section 6.

2. Chebyshev-Legendre Spectral Method

In this section, we first set up semidiscretization and fully discretization Chebyshev-Legendre spectral method for problem (1.2) and then give how to implement the proposed scheme.

2.1. Chebyshev-Legendre Spectral Method

Let ℙ_𝑁 be the set of all algebraic polynomials of degree at most 𝑁. We introduce the operator of interpolation at the CGL nodes {𝜂_𝑖=cos(𝑖𝜋/𝑁)}_{0≤𝑖≤𝑁}, denoted by Π^𝐶_𝑁, which satisfies Π^𝐶_𝑁𝑓∈ℙ_𝑁 andΠ^𝐶_𝑁𝑓𝜂_𝑖=𝑓𝜂_𝑖,0≤𝑖≤𝑁. (2.1)

Denote𝑉_𝑁=ℙ_𝑁(Λ)∩𝐻¹₀(Λ). (2.2)

The weak form (1.7) leads to the following semidiscretization Legendre-Galerkin scheme: find 𝑢_𝑁(𝑡)∈𝑉_𝑁, such that, forall𝜙_𝑁∈𝑉_𝑁,𝑢_𝑁𝑡,𝜙_𝑁+𝜇𝑢_𝑁𝑥𝑡,𝜙_𝑁𝑥+𝛼𝑢_𝑁𝑥,𝜙_𝑁𝑥=𝛽𝑢_𝑁,𝜙_𝑁𝑥+𝑔𝑢_𝑁,𝜙_𝑁𝑥+𝑓,𝜙_𝑁(],𝑢,𝑡∈0,𝑇_𝑁,𝜙_𝑁=𝑢₀,𝜙_𝑁,𝑡=0. (2.3)

Now we give a Chebyshev-Legendre spectral scheme for the problem (1.2).

The semidiscretization Chebyshev-Legendre spectral method for (1.2) is to find 𝑢_𝑁(𝑡)∈𝑉_𝑁, such that, forall𝜙_𝑁∈𝑉_𝑁,𝑢_𝑁𝑡,𝜙_𝑁+𝜇𝑢_𝑁𝑥𝑡,𝜙_𝑁𝑥+𝛼𝑢_𝑁𝑥,𝜙_𝑁𝑥=𝛽𝑢_𝑁,𝜙_𝑁𝑥+Π^𝐶_𝑁𝑔𝑢_𝑁,𝜙_𝑁𝑥+Π^𝐶_𝑁𝑓,𝜙_𝑁(],𝑢,𝑡∈0,𝑇_𝑁,𝜙_𝑁=Π^𝐶_𝑁𝑢₀,𝜙_𝑁,𝑡=0. (2.4)

Remark 2.1. The difference between Legendre method (2.3) and Chebyshev-Legendre method (2.4) lies in the treatment of three terms: nonlinear term, source term and initial data. But such treatment leads to two advantages: one is improvement of convergence rate in theoretical analysis, the other is free from computing the LGL nodes. We only need Chebyshev transform and Chebyshev-Legendre transform in computing, and the former can perform through FFT.

For the time advance, we adopt the second-order Crank-Nicolson/leapfrog. For a given time step 𝜏, let 𝑆_𝑡={𝑘𝜏∶𝑘=1,2,…,𝑀_𝑡,𝑡=𝑀_𝑡𝜏}; the notations 𝑣𝑡(̂𝑡) and 𝑣(𝑡) are used as𝑣𝑡̂(𝑡)=𝑣(𝑡+𝜏)−𝑣(𝑡−𝜏)(𝑡)=𝑣()()2𝜏,𝑣𝑡+𝜏+𝑣𝑡−𝜏2. (2.5) Let 𝑡_𝑘=𝑘𝜏 and 𝑢^𝑘=𝑢(𝑡_𝑘). The fully discretization Chebyshev-Legendre spectral method for (1.2) is to find 𝑢^𝑘_𝑁(𝑘=0,1,…,𝑀_𝑇=⌈𝑇/𝜏⌉)∈𝑉_𝑁, such that, for all 𝜙_𝑁∈𝑉_𝑁,𝑢^𝑘_𝑁̂𝑡,𝜙_𝑁+𝜇𝑢^𝑘_𝑁𝑥̂𝑡,𝜙_𝑁𝑥+𝛼𝑢^𝑘_𝑁𝑥,𝜙_𝑁𝑥=𝛽𝑢^𝑘_𝑁,𝜙_𝑁𝑥+Π^𝐶_𝑁𝑔𝑢^𝑘_𝑁,𝜙_𝑁𝑥+Π^𝐶_𝑁𝑓^𝑘,𝜙_𝑁,𝑘=1,…,𝑀_𝑇𝑢−1,_1𝑁−𝑢⁰_𝑁𝜏,𝜙_𝑁𝑢+𝜇^1𝑁𝑥−𝑢⁰_𝑁𝑥𝜏,𝜙_𝑁𝑥𝑢+𝛼^1𝑁𝑥+𝑢⁰_𝑁𝑥2,𝜙_𝑁𝑥𝑢=𝛽^1𝑁+𝑢⁰_𝑁2,𝜙_𝑁𝑥+Π^𝐶_𝑁𝑔𝑢⁰_𝑁,𝜙_𝑁𝑥+Π^𝐶_𝑁𝑓¹+𝑓⁰2,𝜙_𝑁,𝑢⁰_𝑁=Π^𝐶_𝑁𝑢₀. (2.6)

2.2. Implementation of the Chebyshev-Legendre Spectral Method

Let 𝐿_𝑘 be the 𝑘th degree Legendre polynomial that is mutually orthogonal in 𝐿²(Λ), that is,𝐿_𝑘,𝐿_𝑗=_Λ𝐿_𝑘(𝑥)𝐿_𝑗(𝑥)2𝑑𝑥=𝛿2𝑘+1^𝑘𝑗. (2.7) We define (in [35])Φ_𝑘(𝑥)=𝐿_𝑘(𝑥)−𝐿_𝑘+2(𝑥). (2.8) One useful property of the Legendre polynomials is(2𝑛+1)𝐿_𝑛(𝑥)=𝐿^′_𝑛+1(𝑥)−𝐿^′_𝑛−1(𝑥), (2.9) which gives the following relation:Φ^′_𝑘(𝑥)=−(2𝑘+3)𝐿_𝑘+1(𝑥). (2.10)

It is easy to verify that, for 𝑁≥2,𝑉_𝑁=spanΦ₀,Φ₁,…,Φ_𝑁−2. (2.11)

Let us denote𝑢^𝑖_𝑁=^𝑁−2_𝑘=0𝑢^𝑖_𝑘Φ_𝑘,𝐮^𝑖=𝑢^𝑖₀,𝑢^𝑖₁,…,𝑢^𝑖_𝑁−2^𝑇,𝑔^𝑖_𝑘=Π^𝐶_𝑁𝑔𝑢^𝑖_𝑁,Φ^′_𝑘,𝐠^𝑖=𝑔^𝑖₀,𝑔^𝑖₁,…,𝑔^𝑖_𝑁−2^𝑇,𝑓^𝑖_𝑘=Π^𝐶_𝑁𝑓^𝑖,Φ_𝑘,𝐟^𝑖̂=𝑓^𝑖₀,𝑓^𝑖₁,…,𝑓^𝑖_𝑁−2^𝑇,𝑢_0,𝑘=Π^𝐶_𝑁𝑢₀,Φ_𝑘,𝐮₀=𝑢_0,0,𝑢_0,1,…,𝑢_0,𝑁−2^𝑇,𝑚_𝑖𝑗=Φ_𝑗,Φ_𝑖,𝐌=𝑚_{𝑖𝑗𝑖,𝑗=0,1,…,𝑁−2},𝑠_𝑖𝑗=Φ_𝑗,Φ^′_𝑖,𝐒=𝑠_{𝑖𝑗𝑖,𝑗=0,1,…,𝑁−2},𝑝_𝑖𝑗=Φ^′_𝑗,Φ^′_𝑖,𝐏=𝑝_{𝑖𝑗𝑖,𝑗=0,1,…,𝑁−2}. (2.12) Then, scheme (2.6) leads to the following linear system series: For 𝑘=1,2,…,𝑀_𝑇𝐌+𝜇+𝛼𝜏𝐏−𝛽𝜏𝐒𝐮^𝑘+1=𝐌+𝜇−𝛼𝜏𝐏+𝛽𝜏𝐒𝐮^𝑘−1+2𝜏𝐠^𝑘+𝜏𝐟̂^𝑘+1+𝐟̂^𝑘−1, (2.13) starting with𝐌+𝜇+𝛼𝜏2𝐏−𝛽𝜏2𝐒𝐮¹=𝐌+𝜇−𝛼𝜏2𝐏+𝛽𝜏2𝐒𝐮⁰+𝜏𝐠⁰+𝜏2𝐟¹̂+𝐟⁰̂,𝐌𝐮⁰=𝐮₀. (2.14)

By using integration by parts and orthogonality of Legendre polynomials, one can easily determine that the matrix 𝐌 is pentadiagonal and 𝐏 diagonal. Moreover, the nonzero entries of the matrices 𝐌, 𝐏 can be determined by using the orthogonal relation (2.7) and (2.10) as follows:𝑚_𝑗𝑘=𝑚_𝑘𝑗=2+22𝑘+1−22𝑘+5,𝑗=𝑘,𝑠2𝑘+5,𝑗=𝑘±2,0,otherwise,_𝑗𝑘=𝑝−2,𝑗=𝑘−1,2,𝑗=𝑘+1,0,otherwise,_𝑗𝑘=𝑝_𝑘𝑗=4𝑘+6,𝑗=𝑘;0,otherwise. (2.15)

Hence, the matrices in the left-hand side of (2.13) are banded and the linear system (2.13) can be easily solved.

For evaluation of the last three terms on the right-side hand in (2.13), it can be split into two steps: one step is the transform from its values at CGL nodes to the coefficients of its Chebyshev expansion, which can be done by using fast Chebyshev transform or fast Fourier transform. The other step involves the fast Legendre transform between the coefficients of the Chebyshev expansion and of the Legendre expansion, which has been addressed by Alpert and Rokhlin [36]. It is worthy to note that Shen suggested an algorithm [28] that seems more attractive for moderate 𝑁.

3. Preliminaries

In this section, we introduce a suitable comparison function and give some lemmas needed in error analysis. We denote by 𝐶 a generic positive constant independent of 𝑁 or any function. Now we introduce two projection operators and their approximation properties. One is 𝑃^𝐿_𝑁∶𝐿²(Λ)→ℙ_𝑁 such that𝑃^𝐿_𝑁𝑣,𝜙=𝑣,𝜙,∀𝜙∈ℙ_𝑁. (3.1) The approximation property (Theorem 6.1 and (6.8) in [24], page 258, 261) is the following.

Lemma 3.1. For any 𝑢∈𝐻^𝑟(Λ)(𝑟>1), there exists a positive constant 𝐶 independent of 𝑁 such that 𝑢−𝑃^𝐿_𝑁𝑢≤𝐶𝑁^−𝑟‖𝑢‖_𝑟,𝑢−𝑃^𝐿_𝑁𝑢_𝑠≤𝐶𝑁^{3𝑠/2−𝑟}‖𝑢‖_𝑟,𝑠≤1,𝐶𝑁^{2𝑠−1/2−𝑟}‖𝑢‖_𝑟,𝑠≥1. (3.2)(3.3)

The other projection operator is defined as follows:𝑃_𝑁𝑢(𝑥)=^𝑥₋₁𝑃^𝐿_𝑁−1𝑢_𝑦𝑦𝑑𝑦. (3.4) It is easy to know that 𝑃_𝑁∶𝐻¹₀(Λ)→𝑉_𝑁 such that𝑃_𝑁𝑣_𝑥,𝜙_𝑥=𝑣_𝑥,𝜙_𝑥,∀𝜙∈𝑉_𝑁. (3.5) Its approximation property (Theorem  6.2 in [24], page 262) is the following.

Lemma 3.2. For any 𝑢∈𝐻^𝑟(Λ)∩𝐻¹₀(Λ), there exists a positive constant 𝐶 independent of 𝑁 such that 𝑢−𝑃_𝑁𝑢_𝑠≤𝐶𝑁^𝑠−𝑟‖𝑢‖_𝑟,0≤𝑠≤1≤𝑟. (3.6)

For the interpolation operator Π^𝐶_𝑁 at the Chebyshev-Gauss-Lobatto points, we cite the following approximation result [29].

Lemma 3.3. If 𝑢∈𝐻¹(Λ), then 𝑁Π^𝐶_𝑁𝑢−𝑢+Π^𝐶_𝑁𝑢₁≤𝐶‖𝑢‖₁, (3.7) moreover, if 𝑢∈𝐻^𝑠(Λ)(𝑠≥1), then Π^𝐶_𝑁𝑢−𝑢_𝑙≤𝐶𝑁^𝑙−𝑠‖𝑢‖_𝑠,0≤𝑙≤1. (3.8)

In general, the discretization inner product and norm are defined as follows:(𝑢,𝑣)_𝑁=^𝑁_𝑗=0𝑢𝑦_𝑗𝑣𝑦_𝑗𝜔_𝑗,‖𝑢‖_𝑁=(𝑢,𝑢)_𝑁, (3.9) where 𝑦_𝑗 and 𝜔_𝑗(𝑗=0,1,…,𝑁) are the Legendre-Gauss-Lobatto points and the corresponding quadrature weights. Associating with this quadrature rule, we denote by Π^𝐿_𝑁 the Legendre interpolation operator. The following result is cited from [26].

Lemma 3.4. If 𝑢∈𝐻^𝜎(Λ)(𝜎≥1), then Π^𝐿_𝑁𝑢−𝑢_𝑙≤𝐶𝑁^𝑙−𝜎‖𝑢‖_𝜎)−(),0≤𝑙≤1,(𝑢,𝑣𝑢,𝑣_𝑁≤𝐶𝑁^−𝜎‖𝑢‖_𝜎‖𝑣‖,∀𝑣∈P_𝑁. (3.10) Further, if 𝑢∈ℙ_𝑁, then ‖𝑢‖≤‖𝑢‖_𝑁≤12+𝑁‖𝑢‖,‖𝑢‖_𝐿∞(Λ)≤𝑁+1√2‖𝑢‖. (3.11)(3.12)

4. Error Analysis of the Schemes

In this section, we set up the stability and convergence results first for the semidiscretization Chebyshev-Legendre spectral method (2.4) and then for the fully discretization scheme (2.6).

4.1. Stability and Convergence of Scheme (2.4)

Since the initial value and the right-hand side term cannot be exactly evaluated, we consider here how stable the numerical solution of (2.4) depending on the initial value and the right-hand side term. As in [21], we suppose that 𝑢_𝑁 and 𝑓 are computed with errors 𝑢 and 𝑓 respectively. That is,𝑢_𝑡,𝜙_𝑁+𝜇𝑢_𝑥𝑡,𝜙_𝑁𝑥+𝛼𝑢_𝑥,𝜙_𝑁𝑥=𝛽𝑢,𝜙_𝑁𝑥+Π^𝐶_𝑁𝐺,𝜙_𝑁𝑥+Π^𝐶_𝑁𝑓,𝜙_𝑁,∀𝜙_𝑁∈𝑉_𝑁, (4.1) where 𝐺∶=[𝑔(𝑢_𝑁+𝑢)−𝑔(𝑢_𝑁)]. Since 𝑢∈𝑉_𝑁 leads to (𝑢,𝑢_𝑥)=0, we come out to, by taking 𝜙_𝑁=𝑢 in (4.1),12𝑑‖𝑑𝑡‖𝑢²+𝜇|𝑢|²₁+𝛼|𝑢|²₁=Π^𝐶_𝑁𝐺,𝑢_𝑥+Π^𝐶_𝑁𝑓,𝑢. (4.2)

Now we need to estimate the nonlinear term Π^𝐶_𝑁𝐺 in the right-hand side (4.2). Let 𝐶₀ be a positive constant and𝑢_𝑀=max_{0≤𝑠≤𝑇}‖𝑢(𝑠)‖_𝐿∞(Λ)+𝑢_𝑥(𝑠)_𝐿∞(Λ),𝐶_𝑔𝑧₁,𝑧₂=max_{|𝑧|≤𝑧1+𝑧2}𝑔^′(𝑧)+𝑧₁+𝑧₂max_{|𝑧|≤𝑧1+𝑧2}𝑔^′′(𝑧). (4.3) For any given 𝑡∈(0,𝑇], if‖𝑢(𝑠)‖_𝐿∞(Λ)≤𝐶₀,∀𝑠∈(0,𝑡), (4.4) then|𝐺|=𝑔′𝑢_𝑁+𝜃𝑢𝑢≤𝐶_𝑔𝑢_𝑀,𝐶₀||,𝐺_′𝑢≤𝑔^′′𝑢_𝑁+𝜃𝑢𝑢_𝑁𝑥𝑢+𝑔^′𝑢_𝑁+𝑢𝑢_𝑥≤𝐶_𝑔𝑢_𝑀,𝐶₀𝑢_𝑥(𝑠)+|𝑢(𝑠)|. (4.5) As an immediate consequence of the above estimates and inequality (3.8), it is true that, for large 𝑁,Π^𝐶_𝑁𝐺≤Π^𝐶_𝑁𝐺−𝐺+‖𝐺‖≤𝐶𝑁⁻¹|𝐺|₁+‖𝐺‖≤𝐶𝑁⁻¹𝐶_𝑔𝑢_𝑀,𝐶₀|𝑢|₁+‖𝑢‖+𝐶_𝑔𝑢_𝑀,𝐶₀‖𝑢‖≤𝐶𝐶_𝑔𝑢_𝑀,𝐶₀𝑁⁻¹+1‖𝑢‖+𝐶𝐶_𝑔𝑢_𝑀,𝐶₀𝑁|𝑢|₁. (4.6) Hence, it is true that, for large 𝑁,Π^𝐶_𝑁𝐺,𝑢_𝑥≤𝐶^∗‖𝑢‖²+𝛼2|𝑢|²₁. (4.7) Therefore, integrating (4.2) in time leads to‖𝑢(𝑠)‖²+𝜇|𝑢(𝑠)|²₁+𝛼^𝑠₀|𝑢(𝑡)|²₁𝑑𝑡≤‖𝑢(0)‖²+𝜇|𝑢(0)|²₁+𝐶^𝑠₀‖𝑢(𝑡)‖²𝑑𝑡+2^𝑠₀Π^𝐶_𝑁𝑓(𝑡)²𝑑𝑡. (4.8) Denoting𝐸(𝑢,𝑠)=‖𝑢(𝑠)‖²+𝜇|𝑢(𝑠)|²₁+𝛼^𝑠₀|𝑢(𝑡)|²₁𝜌()=‖‖𝑑𝑡,𝑢,𝑠𝑢(0)²+𝜇|𝑢(0)|²₁+2^𝑠₀Π^𝐶_𝑁𝑓(𝑡)²𝑑𝑡, (4.9) then, we have𝐸(𝑢,𝑡)≤𝜌(𝑢,𝑡)+𝐶^𝑡₀𝐸(𝑢(𝑠),𝑠)𝑑𝑠. (4.10) We have the following stability result.

Theorem 4.1. If 𝑔∈𝐶²(ℝ) and 𝜌(𝑢,𝑇)≤2𝐶²₀𝑒^−𝐶𝑇(𝑁+1)², (4.11) then there holds the following inequality: 𝐸(𝑢,𝑡)≤𝜌(𝑢,𝑡)𝑒^𝐶𝑡,∀𝑡∈(0,𝑇]. (4.12)

Proof. We need to verify that ‖𝑢(𝑠)‖_𝐿∞(Λ)≤𝐶₀,∀𝑠∈(0,𝑡). (4.13) Otherwise, there must exist 𝑡₁<𝑇 such that max_{0≤𝑠≤𝑡1}‖𝑢(𝑠)‖_𝐿∞(Λ)≤𝐶₀,𝑢(𝑡₁)_𝐿∞(Λ)=𝐶₀, (4.14) while by (4.10), (4.11), and the Gronwall inequality, we have 𝐸𝑢,𝑡₁≤𝜌𝑢,𝑓,𝑡₁𝑒^𝐶𝑡1<𝜌𝑢,𝑓,𝑇𝑒^𝐶𝑇≤2𝐶²₀(𝑁+1)⁻². (4.15) Thus, from Lemma 3.4, 𝑢(𝑡₁)_𝐿∞(Λ)≤𝑁+1√2𝑢𝑡₁≤𝑁+1√2𝐸𝑢,𝑡₁<𝐶₀, (4.16) which is contradictory with (4.14). Thus, (4.35) holds, and we derive (4.12) from the Gronwall inequality.

Remark 4.2. Condition (4.11) in Theorem 4.1 means that errors occurrs during evaluation of the initial value and the right-hand side term should not be larger than 2𝐶²₀(𝑒^−𝐶𝑇/(𝑁+1)²), which becomes small when 𝑁 goes large. This condition may be caused by the nonlinearity of the problem. Also, the condition could be satisfied because the orthogonal polynomial approximation goes faster than 1/(𝑁+1)² in general. As for the result (4.12) in Theorem 4.1, it is analogous to those in [29–33] just for different energy norm 𝐸(𝑢,𝑡).

Next we turn to convergence of the semidiscretization scheme (2.4). Let 𝑢(𝑡) and 𝑢_𝑁(𝑡) be the solutions to (1.7) and (2.4), respectively. We denote 𝑢^∗=𝑃_𝑁𝑢,𝑒(𝑡)=𝑢_𝑁(𝑡)−𝑢^∗(𝑡). Then we have the following result.

Theorem 4.3. If 𝑢∈𝐶¹(0,𝑇;𝐻^𝑟(Λ)∩𝐻¹₀(Λ)),𝑓∈𝐶(0,𝑇;𝐻^𝑟(Λ)∩𝐻¹₀(Λ))(𝑟>1), and 𝑔(𝑧)∈𝐶¹(ℝ) which satisfies the assumption of Theorem 4.1, then the following error estimate ‖𝑒(𝑡)‖²+𝜇|𝑒(𝑡)|²₁+𝛼^𝑡₀|𝑒(𝑠)|²₁𝑑𝑠≤𝐶1+𝜇𝑁²𝑁^−2𝑟,0≤𝑡≤𝑇, (4.17) holds.

Proof. From (1.7), (2.4), and (3.5), we know that 𝑒_𝑡,𝜙+𝜇𝑒_𝑥𝑡,𝜙_𝑥+𝛼𝑒_𝑥,𝜙_𝑥−𝛽𝑒,𝜙_𝑥=Π^𝐶_𝑁𝑔𝑢_𝑁−𝑔(𝑢),𝜙_𝑥+𝑢−𝑢^∗_𝑡,𝜙+𝛽𝑢−𝑢^∗,𝜙_𝑥+𝑓−Π^𝐶_𝑁𝑓,𝜙. (4.18) Taking 𝜙=𝑒 in (4.18), we have 12𝑑𝑒‖𝑑𝑡‖²+𝜇|𝑒|²₁+𝛼|𝑒|²₁≤𝐼₁+𝐼₂+𝐼₃+𝐼₄,𝐼₁∶=Π^𝐶_𝑁𝑔𝑢_𝑁−𝑔(𝑢),𝑒_𝑥≤Π^𝐶_𝑁𝑔𝑢_𝑁−𝑔(𝑢)|𝑒|₁≤1𝛼Π^𝐶_𝑁𝑔𝑢_𝑁−𝑔(𝑢)²+𝛼4|𝑒|²₁,𝐼₂∶=𝑢−𝑢^∗_𝑡,𝑒≤𝑢_𝑡−𝑃_𝑁𝑢_𝑡‖𝑒‖≤𝐶𝑁^−2𝑟𝑢_𝑡²_𝑟+‖𝑒‖²,𝐼₃∶=𝛽𝑢−𝑢^∗,𝑒_𝑥≤𝑢−𝑃_𝑁𝑢|𝑒|₁≤1𝛼𝐶𝑁^−2𝑟‖𝑢‖²_𝑟+𝛼4|𝑒|²₁,𝐼₄∶=𝑓−Π^𝐶_𝑁𝑓,𝑒≤𝑓−Π^𝐶_𝑁𝑓²+‖𝑒‖²≤𝐶𝑁^−2𝑟𝑓²_𝑟+‖𝑒‖². (4.19)(4.20) Next, we estimate ‖Π^𝐶_𝑁𝑔(𝑢_𝑁)−𝑔(𝑢)‖, Π^𝐶_𝑁𝑔𝑢_𝑁−𝑔(𝑢)≤Π^𝐶_𝑁𝑔𝑢_𝑁−𝑔𝑢_𝑁+𝑔𝑢_𝑁−𝑔𝑢^∗+𝑔𝑢^∗−𝑔(𝑢)≤𝐶𝑁^−𝑟𝑔𝑢_𝑁𝑟+𝑔^′𝑢_𝑁+𝜃𝑢^∗‖𝑒‖+𝑔′𝑢^∗+𝜃𝑢𝑢^∗−𝑢≤𝐶𝑁^−𝑟‖𝑢‖_𝑟+𝐶′‖𝑒‖. (4.21) Combining with the estimates above, inequality (4.19) becomes 𝑑𝑒‖𝑑𝑡‖²+𝜇|𝑒|²₁+𝛼|𝑒|²₁≤𝐶𝑁^−2𝑟‖𝑢‖²_𝑟+𝑢_𝑡²_𝑟+𝑓²_𝑟+𝐶′‖𝑒‖². (4.22) Denoting 𝐸(𝑡)=‖𝑒(𝑡)‖²+𝜇|𝑒(𝑡)|²₁+𝛼^𝑡₀|𝑒(𝑠)|²₁𝜌(𝑡)=‖‖𝑑𝑠,𝑒(0)²+𝜇|𝑒(0)|²₁+𝐶𝑁^−2𝑟𝑡₀‖𝑢(𝑠)‖²_𝑟+𝑢_𝑡(𝑠)²_𝑟+𝑓(𝑡)²_𝑟𝑑𝑠, (4.23) we have 𝐸(𝑡)≤𝜌(𝑡)+𝐶^𝑡₀𝐸(𝑠)𝑑𝑠. (4.24) Thanks to the Gronwall inequality, we get 𝐸(𝑡)≤𝜌(𝑡)𝑒^𝐶𝑡,0<𝑡≤𝑇. (4.25) Because of 𝑒(0)=𝑃_𝑁𝑢₀−Π^𝐶_𝑁𝑢₀=(𝑢₀−𝑃_𝑁𝑢₀)−(𝑢₀−Π^𝐶_𝑁𝑢₀), we combine with the approximation properties (3.6)–(3.3) to obtain ‖𝑒(0)‖≤𝑢₀−𝑃_𝑁𝑢₀+𝑢₀−Π^𝐶_𝑁𝑢₀≤𝐶𝑁^−𝑟‖𝑢‖_𝑟,|𝑒(0)|₁≤𝑢₀−𝑃_𝑁𝑢₀₁+𝑢₀−Π^𝐶_𝑁𝑢₀₁≤𝐶𝑁^1−𝑟‖𝑢‖_𝑟. (4.26) Then it is easy to show the desired result.

Remark 4.4. The result in Theorem 4.3 is optimal in sense of 𝐻¹ estimation.

4.2. Stability and Convergence of the Fully Discretization Scheme

In this section we consider the stability and convergence of the fully discretization scheme of (2.6). We assume that all functions below are valued at time 𝑠 unless otherwise specified. Suppose that 𝑢_𝑁 and 𝑓 in (2.6) have the errors 𝑢 and 𝑓, respectively. Then, we have𝑢𝑡,̂𝜙+𝜇𝑢_𝑥𝑡,̂𝜙_𝑥+𝛼𝑢_𝑥,𝜙_𝑥=𝛽𝑢,𝜙_𝑥+Π^𝐶_𝑁𝑔,𝜙_𝑥+Π^𝐶_𝑁𝑓,𝜙, (4.27) where 𝑔=𝑔(𝑢^𝑘_𝑁+𝑢)−𝑔(𝑢^𝑘_𝑁). Taking 𝜙=𝑢 in (4.27), we get12‖𝑢‖²̂𝑡+𝜇|𝑢|²₁𝑡̂+𝛼𝑢²₁=Π^𝐶_𝑁𝑔,𝑢_𝑥+Π^𝐶_𝑁𝑓,𝑢. (4.28) Now we need to estimate ‖Π^𝐶_𝑁𝑔‖, which would be easy to do, provided that Π^𝐶_𝑁 is replaced by Legendre interpolation operator or the norm is in the weighted Chebyshev one. But here we cannot deal with this directly. Just like in [21], by taking 𝜙=𝑢𝑡̂ in (4.27), we again get𝑢𝑡̂²+𝜇𝑢𝑡̂²₁+12𝛼|𝑢|²₁𝑡̂=𝛽𝑢,𝑢_𝑥𝑡̂+Π^𝐶_𝑁𝑔,𝑢_𝑥𝑡̂+Π^𝐶_𝑁𝑓,𝑢𝑡̂. (4.29) Combining (4.28) and (4.29) through the factor 𝑁⁻², we arrive at‖𝑢‖²+𝜇|𝑢|²₁+𝑁⁻²𝛼|𝑢|²₁𝑡̂+2𝑁⁻²𝑢𝑡̂²+𝑁⁻²𝜇𝑢𝑡̂²₁+𝛼𝑢²₁=2Π^𝐶_𝑁𝑔,𝑢_𝑥+2Π^𝐶_𝑁𝑓,𝑢+𝑁⁻²2𝛽𝑢,𝑢_𝑥𝑡̂+𝑁⁻²2Π^𝐶_𝑁𝑔,𝑢_𝑥𝑡̂+𝑁⁻²2Π^𝐶_𝑁𝑓,𝑢𝑡̂. (4.30) By using the Hölder inequality, we have2Π^𝐶_𝑁𝑔,𝑢_𝑥≤2Π^𝐶_𝑁𝑔𝑢₁≤3𝛼Π^𝐶_𝑁𝑔²+𝛼3𝑢²₁,2Π^𝐶_𝑁𝑓,𝑢≤2Π^𝐶_𝑁𝑓₋₁𝑢₁≤3𝛼Π^𝐶_𝑁𝑓²₋₁+𝛼3𝑢²₁,𝑁⁻²2𝛽𝑢,𝑢_𝑥𝑡̂≤𝑁⁻²2𝛽𝑢₁𝑢𝑡̂≤𝑁⁻²2𝛽²𝑢²₁+12𝑁⁻²𝑢𝑡̂²,𝑁⁻²2Π^𝐶_𝑁𝑔,𝑢_𝑥𝑡̂≤𝑁⁻²2Π^𝐶_𝑁𝑔₁𝑢𝑡̂≤𝑁⁻²2Π^𝐶_𝑁𝑔²₁+𝑁⁻²12𝑡̂𝑢²,𝑁⁻²2Π^𝐶_𝑁𝑓,𝑢𝑡̂≤𝑁⁻²2Π^𝐶_𝑁𝑓₋₁𝑢𝑡̂₁≤𝑁⁻²𝜇⁻¹Π^𝐶_𝑁𝑓²₋₁+𝑁⁻²𝜇𝑢𝑡̂²₁. (4.31) Thus, if 𝑁≥𝛽6/𝛼, we have‖𝑢‖²+𝜇|𝑢|²₁+𝑁⁻²𝛼|𝑢|²₁𝑡̂+𝑁⁻²𝑢𝑡̂²+𝑁⁻²𝜇𝑢𝑡̂²₁+𝛼𝑢²₁≤3𝛼Π^𝐶_𝑁𝑔²+𝑁⁻²2Π^𝐶_𝑁𝑔²₁+3𝛼Π^𝐶_𝑁𝑓²₋₁+𝑁⁻²𝜇⁻¹Π^𝐶_𝑁𝑓²₋₁≤𝐶Π^𝐶_𝑁𝑔²+𝑁⁻²Π^𝐶_𝑁𝑔²₁+Π^𝐶_𝑁𝑓²₋₁+𝑁⁻²Π^𝐶_𝑁𝑓²₋₁, (4.32) where 𝐶 is a positive constant dependent on 𝛼⁻¹, 𝜇⁻¹. Summing (4.32) for 𝑠∈𝑆_𝑡−𝜏 gives𝐸(𝑢,𝑡)≤𝜌𝑢,𝑓,𝑡+𝐶𝜏_{𝑠∈𝑆𝑡−𝜏}Π^𝐶_𝑁𝑔(𝑠)²+𝑁⁻²Π^𝐶_𝑁𝑔²₁, (4.33) where𝐸(𝑢,𝑡)=‖𝑢(𝑡)‖²+𝜇|𝑢(𝑡)|²₁+𝑁⁻²𝛼|𝑢(𝑡)|²₁+2𝜏_{𝑠∈𝑆𝑡−𝜏}𝑁⁻²𝑢𝑡(̂𝑠)²+𝑁⁻²𝜇𝑢𝑡̂(𝑠)²₁+𝛼𝑢(𝑠)²₁,𝜌𝑢,𝑓,𝑡=‖𝑢(0)‖²+𝜇|𝑢(0)|²₁+𝑁⁻²𝛼|𝑢(0)|²₁+‖𝑢(𝜏)‖²+𝜇|𝑢(𝜏)|²₁+𝑁⁻²𝛼|𝑢(𝜏)|²₁+𝐶𝜏_{𝑠∈𝑆𝑡−𝜏}Π^𝐶_𝑁𝑓(𝑠)²₋₁+𝑁⁻²Π^𝐶_𝑁𝑓(𝑠)²₋₁. (4.34) For any given 𝑡∈𝑆_𝑇, if‖𝑢(𝑠)‖_𝐿∞(Λ)≤𝐶₀,∀𝑠∈𝑆_𝑡−𝜏, (4.35) then, by (3.7) and (3.8),Π^𝐶_𝑁𝑔+𝑁⁻¹Π^𝐶_𝑁𝑔₁≤𝑔+Π^𝐶_𝑁𝑔−𝑔+𝑁⁻¹Π^𝐶_𝑁𝑔₁≤𝑔+𝐶𝑁⁻¹𝑔₁≤𝐶_𝑔𝑢_𝑀,𝐶₀‖𝑢‖+𝑁⁻¹|𝑢|₁,∀𝑠∈𝑆_𝑡−𝜏. (4.36) Thus, we have shown that for any 𝑡∈𝑆_𝑇, if (4.35) holds, then𝐸(𝑢,𝑡)≤𝜌𝑢,𝑓,𝑡+𝐶^∗𝜏_{𝑠∈𝑆𝑡−𝜏}𝐸(𝑢,𝑠), (4.37) where 𝐶^∗ is a positive constant dependent on 𝐶_𝑔(𝑢_𝑀,𝐶₀), 𝛼⁻¹ and 𝜇⁻¹.

Theorem 4.5. Let 𝜏 be suitably small and 𝑁≥𝛽6/𝛼. If 𝑔∈𝐶²(ℝ) and 𝜌(𝑢,𝑓,𝑇)≤2𝐶²₀(𝑒^{−𝐶∗𝑇}/(𝑁+1)²), then𝐸(𝑢,𝑡)≤𝜌𝑢,𝑓,𝑡𝑒^𝐶∗𝑡,∀𝑡∈𝑆_𝑇. (4.38)

Proof. We verify the result by induction over 𝑡∈𝑆_𝑇. It is easy to see that the result is true for 𝑡=𝜏. Assume that it is true for all 𝑠∈𝑆_𝑡−𝜏 that 𝐸(𝑢,𝑠)≤𝜌𝑢,𝑓,𝑠𝑒^𝐶∗𝑠. (4.39) Then, from the inverse inequality (3.12) we have ‖𝑢(𝑠)‖²_𝐿∞(Λ)≤(𝑁+1)²2‖𝑢(𝑠)‖²≤(𝑁+1)²2𝜌𝑢,𝑓,𝑠𝑒^𝐶∗𝑠≤𝐶²₀, (4.40) which means that (4.35) holds. Therefore, we have, from (4.37) and (4.39), that 𝐸(𝑢,𝑡)≤𝜌𝑢,𝑓,𝑡+𝐶^∗𝜏_{𝑠∈𝑆𝑇−𝜏}𝐸(𝑢,𝑠)≤𝜌𝑢,𝑓,𝑡+𝐶^∗𝜏_{𝑠∈𝑆𝑇−𝜏}𝜌𝑢,𝑓,𝑠𝑒^𝐶∗𝑠≤𝜌𝑢,𝑓,𝑡1+𝐶^∗𝜏_{𝑠∈𝑆𝑇−𝜏}𝑒^𝐶∗𝑠≤𝜌𝑢,𝑓,𝑡𝑒^𝐶∗𝑡. (4.41) Thus, the proof is completed.

Next we consider the convergence of scheme (2.6). Let 𝑢(𝑡) and 𝑢^𝑘_𝑁 be the solutions to (1.7) and (2.6), respectively. Setting 𝑣(𝑡)=𝑃_𝑁𝑢(𝑡),𝑒^𝑘=𝑣^𝑘−𝑢^𝑘_𝑁, we have𝑒^𝑘̂𝑡,𝜙+𝜇𝑒^𝑘_𝑥̂𝑡,𝜙_𝑥+𝛼𝑒^𝑘_𝑥̂,𝜙_𝑥=𝛽𝑒^𝑘̂,𝜙_𝑥+Π^𝐶_𝑁𝐺,𝜙_𝑥+Π^𝐶_𝑁𝑓^𝑘−𝑓^𝑘+𝑃,𝜙_𝑁𝑢^𝑘̂𝑡−𝑢^𝑘_𝑡,𝜙+𝜇𝑃_𝑁𝑢^𝑘_𝑥̂𝑡−𝑢_𝑥𝑡,𝜙_𝑥+𝛽𝑢^𝑘−𝑃_𝑁𝑢^𝑘,𝜙_𝑥+𝑔𝑢^𝑘−Π^𝐶_𝑁𝑔𝑢^𝑘_𝑁+𝑒^𝑘,𝜙_𝑥, (4.42) where 𝐺=𝑔(𝑢^𝑘_𝑁+𝑒^𝑘)−𝑔(𝑢^𝑘_𝑁) and 𝑔(𝑢^𝑘)=(𝑔(𝑢(𝑡_𝑘+𝜏))+𝑔(𝑢(𝑡_𝑘−𝜏)))/2. It is easy to obtain the following estimate:Π^𝐶_𝑁𝑓^𝑘−𝑓^𝑘≤𝐶𝑁^−𝑟𝑓_𝑟,𝑃_𝑁𝑢^𝑘̂𝑡−𝑢^𝑘_𝑡≤𝑃_𝑁𝑢^𝑘̂𝑡−𝑢^𝑘̂𝑡+𝑢^𝑘̂𝑡−𝑢^𝑘_𝑡≤𝐶𝑁^−𝑟‖𝑢‖_𝑟+𝑢^𝑘̂𝑡−𝑢_𝑡𝑡_𝑘+𝑢_𝑡𝑡_𝑘−𝑢^𝑘_𝑡≤𝐶𝑁^−𝑟‖𝑢‖_𝑟+𝐶^′𝜏²𝑢_𝑡𝑡𝑡𝑡_𝑘+𝐶^′′𝜏²𝑢_𝑡𝑡𝑡_𝑘,𝑃_𝑁𝑢^𝑘_𝑥̂𝑡−𝑢^𝑘_𝑥𝑡≤𝑃_𝑁𝑢^𝑘_𝑥̂𝑡−𝑢^𝑘_𝑥̂𝑡+𝑢^𝑘_𝑥̂𝑡−𝑢^𝑘_𝑥𝑡≤𝐶𝑁^1−𝑟‖𝑢‖_𝑟+𝑢^𝑘_𝑥̂𝑡−𝑢_𝑥𝑡𝑡_𝑘+𝑢_𝑥𝑡𝑡_𝑘−𝑢^𝑘_𝑥𝑡≤𝐶𝑁^1−𝑟‖𝑢‖_𝑟+𝐶^′𝜏²𝑢_{𝑥𝑡𝑡𝑡}𝑡_𝑘+𝐶^′′𝜏²𝑢_𝑥𝑡𝑡𝑡_𝑘,𝑢^𝑘−𝑃_𝑁𝑢^𝑘≤𝐶𝑁^−𝑟‖𝑢‖_𝑟,𝑔𝑢^𝑘−Π^𝐶_𝑁𝑔𝑢^𝑘_𝑁+𝑒^𝑘≤𝑔𝑢^𝑘−Π^𝐶_𝑁𝑔𝑢^𝑘+Π^𝐶_𝑁𝑔𝑢^𝑘−𝑔𝑢^𝑘_𝑁+𝑒^𝑘≤𝐶𝑁^−𝑟‖𝑢‖_𝑟+𝐶^′𝑔𝑢^𝑘−𝑔𝑢𝑡_𝑘+𝐶^′′𝑔𝑢^𝑘−𝑔𝑢^𝑘_𝑁+𝑒^𝑘≤𝐶𝑁^−𝑟‖𝑢‖_𝑟+𝐶^′𝜏²‖𝑢‖+𝐶^′′𝑁^−𝑟‖𝑢‖_𝑟. (4.43) Combining with the stability results of Theorem 4.5, we can obtain the following convergence result.

Theorem 4.6. Let 𝑢 and 𝑢^𝑘_𝑁 be the solutions of (1.7) and (2.6), respectively. If 𝑢(𝑥,𝑡)∈𝐶³(0,𝑇;𝐻^𝑟(Λ)∩𝐻¹₀(Λ))(𝑟≥2) and 𝑔∈𝐶²(ℝ) and 𝜏 is suitably small, then one has𝑢𝑡_𝑘−𝑢^𝑘_𝑁+𝜇𝑢𝑡_𝑘−𝑢^𝑘_𝑁1≤𝐶𝜏²+𝑁^1−𝑟. (4.44)

5. Numerical Experiments

In this section, we will present some numerical experiments to confirm the effectiveness and robustness of scheme (2.6) or (2.13).

Example 5.1. We consider the initial boundary problem (1.2)–(1.4) with the exact solution as follows: 𝑢(𝑥,𝑡)=𝑡^𝛾sin(𝜋𝑥). (5.1) The data 𝑢₀(𝑥) is zero function, and the source term 𝑓 is determined by (1.2) and 𝑔(𝑢)=𝑢^𝑛/𝑛, that is, 𝑓(𝑥,𝑡)=𝑡^𝛾−1sin(𝜋𝑥)𝛾+𝜇𝜋²𝛾+𝛼𝜋²𝑡+𝛽𝜋𝑡^𝛾cos(𝜋𝑥)+𝜋𝑡^𝑛sin^𝑛−1(𝜋𝑥)cos(𝜋𝑥). (5.2)
Because the exact solution is known, we can compute the error between the numerical solution and the exact solution. In Table 1, the convergent rate in 𝐿_∞ and 𝐿₂ errors for spatial discretization is computed by fixing time step 𝜏=2×10⁻⁵ and changing 𝑁 from 2² to 2⁴ with the parameters 𝜇=0, 𝛼=1, 𝛽=1, 𝛾=3, and 𝑛=2 at 𝑡=1. In Table 2, the convergent rate in 𝐿_∞ and 𝐿₂ errors for spatial discretization is computed by fixing time step 𝜏=2×10⁻⁵ and changing 𝑁 from 2² to 2⁴ with the parameters 𝜇=1, 𝛼=1, 𝛽=1, 𝛾=3, and 𝑛=2 at 𝑡=1. The results show clearly “spectral accuracy.”
For temporal discretization, the convergent rate in 𝐿_∞ and 𝐿₂ errors is computed by fixing 𝑁=32 and changing time step 𝜏 from 2×10⁻² to 1×10⁻⁴ with the parameters 𝜇=0, 𝛼=𝛽=1, 𝛾=3, and 𝑛=2 at 𝑡=1. In Table 3, the results show clearly second-order convergence for time discretization.
In Table 4, the results show clearly second-order convergence for time discretization and spectral accuracy for space discretization again.

Table 1 

Convergence rates at 𝑡=1 with 𝜇=0, 𝛼=1, 𝛽=1, 𝛾=3, 𝑛=2.

Table 2 

Convergence rates at 𝑡=1 with 𝜇=1, 𝛼=1, 𝛽=1, 𝛾=3, 𝑛=2.

Table 3 

Convergence rates at 𝑡=1 with 𝜇=1, 𝛼=𝛽=1,𝛾=3, and 𝑛=2.

Table 4 

Convergence rates at 𝑡=1 with 𝜇=0.1, 𝛼=1, 𝛽=0.01, 𝛾=3, and 𝑛=2.

Example 5.2. We consider the initial boundary problem (1.2)–(1.4) with the exact solution as follows: 𝑢(𝑥,𝑡)=1−𝑥²𝑒^𝑥−𝑡. (5.3) The data 𝑢₀(𝑥)=(1−𝑥²)𝑒^𝑥 and the source term 𝑓 is determined by (1.2) and 𝑔(𝑢)=𝑢^𝑛/𝑛.
In Table 5, the results show clearly second-order convergence for time discretization and spectral accuracy for space discretization again.

Table 5 

Convergence rates at 𝑡=1 with 𝜇=0.1, 𝛼=1, 𝛽=0.01, 𝛾=3, and 𝑛=2.

Example 5.3. We consider the gBBM-B equation 𝑢_𝑡−𝜇𝑢_𝑥𝑥𝑡−𝛼𝑢_𝑥𝑥+𝛽𝑢_𝑥+𝛾𝑢𝑢_𝑥=0, (5.4) with initial profile as 𝑢₀(𝑥)=𝑒^−40𝑥2. (5.5) In Figures 1 and 2, the results show how the numerical diffusion does with the increasing diffusion 𝛼 and reaction coefficient 𝛽.

(a) 𝛼=0.02

(b) 𝛼=0.1

(c) 𝛼=0.2

Figure 1 

The evolution of 𝑢(𝑥,𝑡) with different 𝛼.

(a) 𝛽=0.05

(b) 𝛽=0.1

(c) 𝛽=0.2

Figure 2 

The evolution of 𝑢(𝑥,𝑡) with different 𝛽.

6. Conclusion

In this paper, we have developed the Chebyshev-Legendre spectral method to the generalised Benjamin-Bona-Mahony-Burgers (gBBM-B) equations. As is well known, Chebyshev method is popular for its explicit formulae and fast Chebyshev transform. But the Chebyshev weight makes much trouble in error analysis. Legendre method is natural in the theoretical analysis due to the unity weight function. However, it is well known that the applications of Legendre method are limited by the lack of fast transform between the physical space and the spectral space and by the lack of explicit evaluation formulation of the LGL nodes. Chebyshev-Legendre method adopts advantages of both Chebyshev method and Legendre method. Hence, it gives a better scheme. Also, error analysis shows that an optimal convergence rate can be obtained by using the Chebyshev-Legendre method. The numerical results show that this scheme is an efficient one. For the problems of high dimension and with complex geometry boundary, Chebyshev-Legendre method could perform well. This will be left for our further research.

Acknowledgments

The authors thank the anonymous referees for their valuable suggestions and helpful comments. The work of the first author was partly supported by the National Natural Science Foundation of China under Grant no. 11161026. The work of Y. Wu was partially supported by the National Basic Research Program of China, 973 Program no. 2011CB706903.