Explicit Multistep Mixed Finite Element Method for RLW Equation

Abstract

An explicit multistep mixed finite element method is proposed and discussed for regularized long wave (RLW) equation. The spatial direction is approximated by the mixed Galerkin method using mixed linear space finite elements, and the time direction is discretized by the explicit multistep method. The optimal error estimates in 𝐿² and 𝐻¹ norms for the scalar unknown 𝑢 and its flux 𝑞=𝑢_𝑥 based on time explicit multistep method are derived. Some numerical results are given to verify our theoretical analysis and illustrate the efficiency of our method.

1. Introduction

In this paper, we consider the following initial boundary problem of RLW equation: 𝑢_𝑡+𝛾𝑢_𝑥+𝛿𝑢𝑢_𝑥−𝛽𝑢_𝑥𝑥𝑡()𝑢()()=0,𝑥,𝑡∈𝐼×𝐽,𝑎,𝑡=𝑢𝑏,𝑡=0,𝑡∈𝑢()𝐽,𝑥,0=𝑢₀(𝑥),𝑥∈𝐼, (1) where 𝐼=(𝑎,𝑏) is a bounded open interval, 𝐽=(0,𝑇] with 0<𝑇<∞. The initial value 𝑢₀(𝑥) is given function, and the coefficients 𝛽, 𝛾, 𝛿 are all positive constants.

In recent years, large nonlinear phenomena are found in many research fields, for example, physics, biology, fluid dynamics, and so forth. These phenomena can be described by the mathematical model of some nonlinear evolution equations. In particular, some attention has also been paid to nonlinear RLW equations [1, 2] which play a very important role in the study of nonlinear dispersive waves. Solitary waves are wave packet or pulses, which propagate in nonlinear dispersive media. Due to dynamical balance between the nonlinear and dispersive effects, these waves retain an unchanged waveform. A soliton is a very special type of solitary wave, which also keeps its waveform after collision with other solitons. The regularized long wave (RLW) equation is an alternative description of nonlinear dispersive waves to the more usual Korteweg-de Vries (KdV) equation. Mathematical theories and numerical methods for (1) were considered in [1–24]. The existence and uniqueness of the solution of RLW equation are discussed in [5]. Their analytical solutions were found under restricted initial and boundary conditions, and therefore they got interest from a numerical point of view. Several numerical methods for the solution of the RLW equation have been introduced in the literature. These include a variety of difference methods [5–8, 17], finite element methods based on Galerkin and collocation principles [9–13], mixed finite element methods [14–16], meshfree method [18], adomian decomposition method [19], and so on.

In [25], Chatzipantelidis studied the explicit multistep methods for some nonlinear partial differential equations and discussed some mathematical theories. Akrivis et al. [26] studied the multistep method for some nonlinear evolution equations. Mei and Chen [20] presented the explicit multistep method based on Galerkin method for regularized long wave (RLW) equation. In this paper, our purpose is to propose and study an explicit multistep mixed method, which combines a mixed Galerkin method in the spatial direction and the explicit multistep method in the time direction, for RLW equation. We derive optimal error estimates in 𝐿² and 𝐻¹ norms for the scalar unknown 𝑢 and its flux 𝑞=𝑢_𝑥 for the fully discrete explicit multistep mixed scheme and compare our method’s accuracy with some other numerical schemes. Compared to the numerical methods in [20, 25, 26], we not only obtain the approximation solution for 𝑢, but also get the approximation solution for 𝑞=𝑢_𝑥.

The layout of the paper is as follows. In Section 2, an explicit multistep mixed scheme and numerical process are given. The optimal error estimates in 𝐿² and 𝐻¹ norms for the scalar unknown 𝑢 and its flux 𝑞=𝑢_𝑥 for the fully discrete explicit multistep mixed scheme are proved in Section 3. In Section 4, some numerical results are shown to confirm our theoretical analysis. Finally, some concluding remarks are given in Section 5. Throughout this paper, 𝐶 will denote a generic positive constant which does not depend on the spatial mesh parameter ℎ or time discretization parameter Δ𝑡.

2. The Mixed Numerical Scheme

With the auxiliary variable 𝑞=𝑢_𝑥, we reformulate (1) as the following first-order coupled system: 𝑢_𝑥𝑢=𝑞,_𝑡+𝛾𝑞+𝛿𝑢𝑞−𝛽𝑞_𝑥𝑡=0. (2) We consider the following mixed weak formulation of (2). Find {𝑢,𝑞}∶[0,𝑇]↦𝐻¹₀×𝐻¹ satisfying: 𝑢_𝑥,𝑣_𝑥=𝑞,𝑣_𝑥,∀𝑣∈𝐻¹₀,𝑞_𝑡,𝑤+𝛽𝑞_𝑥𝑡,𝑤_𝑥=𝛾𝑞,𝑤_𝑥+𝛿𝑢𝑞,𝑤_𝑥,∀𝑤∈𝐻¹. (3)(4) Noting the Dirichlet boundary conditions 𝑢_𝑡(𝑎,𝑡)=𝑢_𝑡(𝑏,𝑡)=0 and 𝑞_𝑡=𝑢_𝑥𝑡, we can get (𝑢_𝑡,−𝑤_𝑥)=(𝑢_𝑥𝑡,𝑤)=(𝑞_𝑡,𝑤) easily and then get the scheme (4).

Let 𝑉_ℎ and 𝑊_ℎ be finite dimensional subspaces of 𝐻¹₀ and 𝐻¹, respectively, defined by 𝑉_ℎ=𝑣_ℎ𝑣_ℎ∈𝐶⁰𝐼,𝑣_ℎ𝐼𝑗∈𝑃_𝑘𝐼_𝑗,∀𝐼_𝑗∈𝑇_ℎ,𝑣_ℎ(𝑎)=𝑣_ℎ(𝑏)=0⊂𝐻¹₀,𝑊_ℎ=𝑤_ℎ𝑤_ℎ∈𝐶⁰𝐼,𝑤_ℎ𝐼𝑗∈𝑃_𝑟𝐼_𝑗,∀𝐼_𝑗∈𝑇_ℎ⊂𝐻¹, (5) where 𝑇_ℎ is a partition of 𝐼=[𝑎,𝑏] into 𝑁 subintervals 𝐼_𝑗=[𝑥_𝑗,𝑥_𝑗+1], 𝑗=0,1,2,…,(𝑁−1), ℎ_𝑗=ℎ_𝑗+1−ℎ_𝑗, ℎ=max_{0≤𝑗≤𝑁−1}ℎ_𝑗, and 𝑃_𝑚(𝐼_𝑗) denotes the polynomials of degree less than or equal to 𝑚 in 𝐼_𝑗.

The semidiscrete mixed finite element method for (3) and (4) consists in determining {𝑢^ℎ,𝑞^ℎ}∶[0,𝑇]↦𝑉_ℎ×𝑊_ℎ such that 𝑢^ℎ_𝑥,𝑣^ℎ_𝑥=𝑞^ℎ,𝑣^ℎ_𝑥,∀𝑣^ℎ∈𝑉_ℎ,𝑞^ℎ_𝑡,𝑤^ℎ+𝛽𝑞^ℎ_𝑥𝑡,𝑤^ℎ_𝑥=𝛾𝑞^ℎ,𝑤^ℎ_𝑥+𝛿𝑢^ℎ𝑞^ℎ,𝑤^ℎ_𝑥,∀𝑤^ℎ∈𝑊_ℎ. (6)(7)

In the following discussion, we will give an explicit multistep mixed scheme. We take linear finite element spaces 𝑉_ℎ=span{𝜑₀,𝜑₁,…,𝜑_𝑁} and 𝑊_ℎ=span{𝜑₀,𝜑₁,…,𝜑_𝑁}, and then 𝑢_ℎ and 𝑞_ℎ can be expressed as the following formulation: 𝑢_ℎ(𝑥,𝑡)=^𝑁_𝑖=0𝑢_𝑖(𝑡)𝜑_𝑖(𝑥),()𝑞𝑥,𝑡∈Ω×𝐽,_ℎ(𝑥,𝑡)=^𝑁_𝑖=0𝑞_𝑖(𝑡)𝜑_𝑖(𝑥),(𝑥,𝑡)∈Ω×𝐽. (8) Substitute (8) into (6) and (7), and take 𝑣^ℎ=𝜑_𝑗 and 𝑤^ℎ=𝜑_𝑗 in (6) and (7), respectively, to obtain ^𝑁_𝑖=0^𝑏_𝑎𝜑_𝑖⋅𝜑_𝑗𝑑𝑥+𝛽^𝑏_𝑎𝜑_𝑖𝑥⋅𝜑_𝑗𝑥𝑑𝑥𝜕𝑞_𝑖−𝛾𝜕𝑡_𝑏𝑎𝜑_𝑖⋅𝜑_𝑗𝑥𝑑𝑥+𝛿^𝑏_𝑎^𝑁_𝑖=0𝑢_𝑖𝜑_𝑖𝜑_𝑖⋅𝜑_𝑗𝑥𝑑𝑥𝑞_𝑖=0,_{𝑁𝑖=0𝑏𝑎}𝜑_𝑖𝑥⋅𝜑_𝑗𝑥𝑑𝑥𝑢_𝑖−^𝑏_𝑎𝜑_𝑖⋅𝜑_𝑗𝑥𝑑𝑥𝑞_𝑖=0, (9) where 𝑗=0,1,2,…,𝑁.

We subdivide the space variable domain [𝑎,𝑏] into uniform subintervals with 𝑁+1 grid points 𝑥_𝑘,𝑘=0,…,𝑁, such that 𝑎=𝑥₀<𝑥₁<⋯<𝑥_𝑁=𝑏, ℎ=𝑥_𝑘+1−𝑥_𝑘=(𝑏−𝑎)/𝑁. Using the local coordinate transformation 𝑥=𝑥_𝑘+𝜇ℎ, 0≤𝜇≤1, we transform a subinterval [𝑥_𝑘,𝑥_𝑘+1] into a standard interval [0,1]. Furthemore, we have ^𝑘+1_𝑖=𝑘¹₀𝜑_𝑖⋅𝜑_𝑗𝛽𝑑𝑥+ℎ²¹⁰𝜑_𝑖𝑥⋅𝜑_𝑗𝑥𝑑𝑥𝜕𝑞_𝑖−1𝜕𝑡ℎ𝛾¹₀𝜑_𝑖⋅𝜑_𝑗𝑥𝑑𝑥+𝛿¹₀^𝑁_𝑖=0𝑢_𝑖𝜑_𝑖𝜑_𝑖⋅𝜑_𝑗𝑥𝑑𝑥𝑞_𝑖=0,_{𝑘+1𝑖=𝑘}1ℎ²¹⁰𝜑_𝑖𝑥⋅𝜑_𝑗𝑥𝑑𝑥𝑢_𝑖−1ℎ^𝑏_𝑎𝜑_𝑖⋅𝜑_𝑗𝑥𝑑𝑥𝑞_𝑖=0. (10) We take linear basis functions defined as follows: 𝐿₁=1−𝜇,𝐿₂=𝜇, (11) and then the variables 𝑢 and 𝑞 over the element [𝑥_𝑘,𝑥_𝑘+1] are written as 𝑢^𝑒=²_𝑗=1𝐿_𝑗𝑢_𝑗,𝑞^𝑒=²_𝑗=1𝐿_𝑗𝑞_𝑗. (12) Then, we get the following equations: ²_𝑖=1¹₀𝐿_𝑖⋅𝐿_𝑗𝛽𝑑𝜇+ℎ²¹⁰𝐿_𝑖𝜇⋅𝐿_𝑗𝜇𝑑𝜇𝜕𝑞_𝑖−1𝜕𝑡ℎ𝛾¹₀𝐿_𝑖⋅𝐿_𝑗𝜇𝑑𝜇+𝛿¹₀²_𝑖=1𝑢_𝑖𝐿_𝑖𝐿_𝑖⋅𝐿_𝑗𝜇𝑑𝜇𝑞_𝑖=0,_2𝑖=11ℎ²¹⁰𝐿_𝑖𝜇⋅𝐿_𝑗𝜇𝑑𝜇𝑢_𝑖−1ℎ^𝑏_𝑎𝐿_𝑖⋅𝐿_𝑗𝜇𝑑𝜇𝑞_𝑖=0. (13) Then, the system (13) has the following matrix form: 𝐴^𝑒_𝑖𝑗+𝛽𝐵^𝑒_𝑖𝑗𝜕𝐪^𝑒−𝜕𝑡𝛾𝐶^𝑒_𝑖𝑗+𝛿𝐷^𝑒_𝑖𝑗𝑢^𝑒𝐪^𝑒𝐵=0,_𝑒𝑖𝑗𝐮^𝑒−𝐶^𝑒_𝑖𝑗𝐪^𝑒=0, (14) with the following element matrices: 𝐮^𝑒=𝑢₁,𝑢₂^𝑇,𝐪^𝑒=𝑞₁,𝑞₂^𝑇,𝐴^𝑒_𝑖𝑗=¹₀𝐿_𝑖⋅𝐿_𝑗𝑑𝜇,𝐵^𝑒_𝑖𝑗=1ℎ²¹⁰𝐿_𝑖𝜇⋅𝐿_𝑗𝜇𝐶𝑑𝜇,_𝑒𝑖𝑗=1ℎ¹₀𝐿_𝑖⋅𝐿_𝑗𝜇𝐷𝑑𝜇,_𝑒𝑗𝑘=𝛿ℎ¹₀²_𝑖=1𝑢_𝑖𝐿_𝑖𝐿_𝑖𝜇⋅𝐿_𝑗𝑑𝜇. (15) Assembling contributions from all elements, we obtain the following coupled system of nonlinear matrix equations: 𝐴+𝛽𝐵𝜕𝐪_ℎ−𝐮𝜕𝑡𝛾𝐶+𝛿𝐷_ℎ𝐪_ℎ=0,𝐵𝐮_ℎ−𝐶𝐪_ℎ=0. (16) To formulate a fully discrete scheme, we consider a uniform partition of 𝐽=[0,𝑇] with time step length Δ𝑡=𝑇/𝑁, 𝑁∈𝐙⁺, and time levels 𝑡^𝑛=𝑛Δ𝑡, 𝑛=0,…,𝑁. We now discuss a fully discrete scheme based on a linear explicit multistep method. We now define 𝑈^𝑛∈𝑉_ℎ and 𝑍^𝑛∈𝑊_ℎ as approximations to 𝑢(𝑡^𝑛) and 𝑞(𝑡_𝑛), respectively, and formulate the following fully discrete linear explicit multistep mixed scheme: 𝐴+𝛽𝐵^𝑝_𝑖=0𝛼_𝑖𝑍^𝑛+𝑖=Δ𝑡^𝑝−1_𝑖=0𝜎_𝑖𝛾𝐶+𝛿𝐷𝑈^𝑛+𝑖𝑍^𝑛+𝑖,𝐵𝑈^𝑛+𝑝=𝐶𝑍^𝑛+𝑝, (17) with given the initial approximations 𝑈⁰,…,𝑈^𝑝−1 and 𝑍⁰,…,𝑍^𝑝−1. In the explicit multistep mixed system (17), the parameter variable 𝛼_𝑖 and 𝜎_𝑖 is described by the coefficients of the term 𝜒^𝑖, for the following polynomials 𝛼(𝜒) and 𝜎(𝜒), respectively: 𝛼𝜒∶=^𝑝_𝑗=11𝑗𝜒^𝑝−𝑗𝜒−1^𝑗,𝜎𝜒∶=𝜒^𝑝−𝜒−1^𝑝. (18)

In this paper, we consider the explicit 2-step mixed method for the RLW equation. For 𝑝=2, we obtained easily 𝛼₀=32,𝛼₁=−2,𝛼₂=12,𝜎₀=−1,𝜎₁=2. (19) Substituting (19) into (17), we obtain the following 2-step mixed scheme: 3𝐴+𝛽𝐵2𝑍^𝑛+2−2𝑍^𝑛+1+12𝑍^𝑛=Δ𝑡𝛾𝐶2𝑍^𝑛+1−𝑍^𝑛+𝛿2𝐷𝑈^𝑛+1𝑍^𝑛+1−𝐷𝑈^𝑛𝑍^𝑛,𝐵𝑈^𝑛+2=𝐶𝑍^𝑛+2. (20)

Remark 1. There have been many numerical schemes for the RLW equation, but we have not seen the related research on explicit multistep mixed element method for RLW equation in the literature. From the viewpoint of numerical theory, we propose a mixed element scheme (6) and (7), which is different from some other mixed finite element methods in [14–16], for the RLW equation and derive some a priori error estimates based on the explicit multistep mixed element method. From the perspective of numerical calculation, our method is efficient for RLW equation.

3. Two-Step Mixed Scheme and Optimal Error Estimates

3.1. Two-Step Mixed Scheme and Some Lemmas

In this section, we will discuss some a priori error estimates based on explicit 2-step mixed finite element method for the RLW equation. For the fully discrete procedure, let 0=𝑡₀<𝑡₁<⋯<𝑡_𝑁=𝑇  be a given partition of the time interval [0,𝑇] with step length Δ𝑡=𝑇/𝑁, for some positive integer 𝑁. For a smooth function 𝜙 on [0,𝑇], define 𝜙^𝑛=𝜙(𝑡_𝑛).

The system (3) and (4) has the following formulation at 𝑡=𝑡_𝑛+1: 𝑢_𝑥^𝑛+1,𝑣_𝑥=𝑞^𝑛+1,𝑣_𝑥,∀𝑣∈𝐻¹₀,𝑞_𝑡^𝑛+1,𝑤+𝛽𝑞^𝑛+1_𝑥𝑡,𝑤_𝑥=𝛾𝑞^𝑛+1,𝑤_𝑥+𝛿𝑢^𝑛+1𝑞^𝑛+1,𝑤_𝑥,∀𝑤∈𝐻¹. (21) Based on system (17), we get an equivalent formulation for system (21) as 𝑢_𝑥^𝑛+1,𝑣_𝑥=𝑞^𝑛+1,𝑣_𝑥,∀𝑣∈𝐻¹₀,3𝑞^𝑛+1−4𝑞^𝑛+𝑞^𝑛−12Δ𝑡,𝑤+𝛽3𝑞_𝑥^𝑛+1−4𝑞^𝑛_𝑥+𝑞_𝑥^𝑛−12Δ𝑡,𝑤_𝑥=−𝜏^𝑛+1,𝑤−𝛽𝜅₁^𝑛+1,𝑤_𝑥+𝛾2𝑞^𝑛−𝑞^𝑛−1,𝑤_𝑥+𝛿2𝑢^𝑛𝑞^𝑛−𝑢^𝑛−1𝑞^𝑛−1,𝑤_𝑥+𝛾𝑅₁^𝑛+1,𝑤_𝑥+𝛿𝑅₂^𝑛+1,𝑤_𝑥,∀𝑤∈𝐻¹, (22)(23) where 𝜏^𝑛+1=𝑞_𝑡^𝑛+1−3𝑞^𝑛+1−4𝑞^𝑛+𝑞^𝑛−1,𝜅2Δ𝑡_1𝑛+1=𝑞_𝑥𝑡𝑡_𝑛+1−3𝑞_𝑥^𝑛+1−4𝑞^𝑛_𝑥+𝑞_𝑥^𝑛−1,𝑅2Δ𝑡_1𝑛+1=𝑞^𝑛+1−2𝑞^𝑛−𝑞^𝑛−1,𝑅₂^𝑛+1=𝑢^𝑛+1𝑞^𝑛+1−2𝑢^𝑛𝑞^𝑛−𝑢^𝑛−1𝑞^𝑛−1. (24) We now find a pair {𝑈^𝑛+1,𝑍^𝑛+1} in 𝑉_ℎ×𝑊_ℎ satisfying 𝑈_𝑥^𝑛+1,𝑣^ℎ_𝑥=𝑍^𝑛+1,𝑣^ℎ_𝑥,∀𝑣^ℎ∈𝑉_ℎ,3𝑍^𝑛+1−4𝑍^𝑛+𝑍^𝑛−12Δ𝑡,𝑤^ℎ+𝛽3𝑍_𝑥^𝑛+1−4𝑍^𝑛_𝑥+𝑍_𝑥^𝑛−12Δ𝑡,𝑤^ℎ_𝑥=𝛾2𝑍^𝑛−𝑍^𝑛−1,𝑤^ℎ_𝑥+𝛿2𝑈^𝑛𝑍^𝑛−𝑈^𝑛−1𝑍^𝑛−1,𝑤^ℎ_𝑥,∀𝑤^ℎ∈𝑊_ℎ. (25)(26)

For the theoretical analysis of a priori error estimates, we define the following projections.

Lemma 2 (see [15, 27, 28]). One defines the elliptic projection 𝑢^ℎ∈𝑉_ℎ by 𝑢_𝑥−𝑢^ℎ_𝑥,𝑣^ℎ_𝑥=0,𝑣^ℎ∈𝑉_ℎ. (27) With 𝜂=𝑢−𝑢^ℎ, the following estimates are well known for 𝑗=0,1: 𝜂_𝑗≤𝐶ℎ^{𝑘+1−𝑗}‖𝑢‖_𝑘+1. (28)

Lemma 3 (see [15, 27, 28]). Furthermore, one also defines a Ritz projection 𝑞^ℎ∈𝑊_ℎ of 𝑞 as the solution of 𝐴𝑞−𝑞^ℎ,𝑤^ℎ=0,𝑤^ℎ∈𝑊_ℎ, (29) where 𝐴(𝑞,𝑤)=(𝑞_𝑥,𝑤_𝑥)+𝜆(𝑞,𝑤), and 𝜆 is taken appropriately so that 𝐴(𝑤,𝑤)≥𝜇₀‖𝑤‖²₁,𝑤∈𝐻¹, (30) where 𝜇₀ is a positive constant. Moreover, it is easy to verify that 𝐴(⋅,⋅) is bounded.
With 𝜌=𝑞−𝑞^ℎ, the following estimates hold: 𝜕^𝑖𝜌𝜕𝑡^𝑖_𝑗≤𝐶ℎ^{𝑟+1−𝑗}𝜕^𝑖𝑞𝜕𝑡^𝑖_𝑟+1,𝑖=0,1,2,3;𝑗=0,1. (31)

For fully discrete error estimates, we now write the errors as 𝑢𝑡_𝑛−𝑈^𝑛=𝑢𝑡_𝑛−𝑢^ℎ𝑡_𝑛+𝑢^ℎ𝑡_𝑛−𝑈^𝑛=𝜂^𝑛+𝜍^𝑛,𝑞𝑡_𝑛−𝑍^𝑛=𝑞𝑡_𝑛−𝑞^ℎ𝑡_𝑛+𝑞^ℎ𝑡_𝑛−𝑍^𝑛=𝜌^𝑛+𝜉^𝑛. (32) Combine (27), (29), (22), (23), (25), and (26) at 𝑡=𝑡_𝑛+1 to get the following error equations: 𝜍_𝑥^𝑛+1,𝑣^ℎ_𝑥=𝜌^𝑛+1+𝜉^𝑛+1,𝑣^ℎ_𝑥,∀𝑣^ℎ∈𝑉_ℎ,3𝜉^𝑛+1−4𝜉^𝑛+𝜉^𝑛−12Δ𝑡,𝑤^ℎ+𝛽3𝜉_𝑥^𝑛+1−4𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12Δ𝑡,𝑤^ℎ_𝑥=−1−𝛽𝜆3𝜌^𝑛+1−4𝜌^𝑛+𝜌^𝑛−12Δ𝑡+𝜏^𝑛+1,𝑤^ℎ−𝛽𝜅₂^𝑛+1,𝑤^ℎ_𝑥+𝛾2𝜉^𝑛−𝜉^𝑛−1,𝑤^ℎ_𝑥+𝛿2𝑢𝑡_𝑛𝑞𝑡_𝑛−𝑈^𝑛𝑍^𝑛−𝑢𝑡_𝑛−1𝑞𝑡_𝑛−1−𝑈^𝑛−1𝑍^𝑛−1,𝑤^ℎ_𝑥+𝛾𝑅₁^𝑛+1,𝑤^ℎ_𝑥+𝛿𝑅₂^𝑛+1,𝑤^ℎ_𝑥,∀𝑤^ℎ∈𝑊_ℎ, (33)(34) where 𝜅₂^𝑛+1=𝑞^ℎ_𝑥𝑡𝑡_𝑛+1−3𝑞_𝑥^ℎ,𝑛+1−4𝑞_𝑥^ℎ,𝑛+𝑞_𝑥^ℎ,𝑛−1.2Δ𝑡 (35)

Lemma 4. For 𝜏^𝑛+1, 𝜅₂^𝑛+1, 𝑅₁^𝑛+1, and 𝑅₂^𝑛+1, the following estimates hold: 𝜏^𝑛+1≤𝐶Δ𝑡²𝑞_{𝑡𝑡𝑡𝐿∞(𝐿2)},𝜅₂^𝑛+1≤𝐶Δ𝑡²ℎ^𝑟𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐻𝑟+1)}+𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐿2)},𝑅₁^𝑛+1≤𝐶Δ𝑡²𝑞_{𝑡𝑡𝐿∞(𝐿2)},𝑅₂^𝑛+1≤𝐶Δ𝑡²𝑢𝑞_{𝑡𝑡𝐿∞(𝐿2)}+𝑢_𝑡𝑞_{𝑡𝐿∞(𝐿2)}+𝑢_𝑡𝑡𝑞_{𝐿∞(𝐿2)}. (36)

Proof. Using the Taylor expansion, we have 𝑞𝑡_𝑛=𝑞𝑡_𝑛+1+𝑞_𝑡𝑡_𝑛+1𝑡_𝑛−𝑡_𝑛+1+𝑞_𝑡𝑡𝑡_Δ12𝑡_𝑛−𝑡_𝑛+1²,𝑡_𝑛<𝑡_Δ1<𝑡_𝑛+1,𝑞𝑡_𝑛−1=𝑞𝑡_𝑛+1+𝑞_𝑡𝑡_𝑛+1𝑡_𝑛−1−𝑡_𝑛+1+𝑞_𝑡𝑡𝑡_Δ22𝑡_𝑛−1−𝑡_𝑛+1²,𝑡_𝑛−1<𝑡_Δ2<𝑡_𝑛+1. (37)(38) Combining (37) and (38) and noting that −2Δ𝑡=2(𝑡_𝑛−𝑡_𝑛+1)=𝑡_𝑛−1−𝑡_𝑛+1, we obtain 𝑞𝑡_𝑛+1=2𝑞𝑡_𝑛−𝑞𝑡_𝑛−1+𝑞_𝑡𝑡𝑡_Δ1−2𝑞_𝑡𝑡𝑡_Δ2Δ𝑡². (39) From (39), we have 𝑞𝑡_𝑛+1−2𝑞𝑡_𝑛−𝑞𝑡_𝑛−1≤𝐶Δ𝑡²𝑞_{𝑡𝑡𝐿∞(𝐿2)}. (40) Using a similar estimate as the one for ‖𝑅₁^𝑛+1‖, we have 𝑢𝑡_𝑛+1𝑞𝑡_𝑛+1=2𝑢𝑡_𝑛𝑞𝑡_𝑛−𝑢𝑡_𝑛−1𝑞𝑡_𝑛−1+𝑢𝑞_𝑡𝑡𝑡_Δ3−2𝑢𝑞_𝑡𝑡𝑡_Δ4Δ𝑡², (41) where 𝑡_𝑛<𝑡_Δ3<𝑡_𝑛+1, 𝑡_𝑛−1<𝑡_Δ4<𝑡_𝑛+1.
From (41), we have 𝑢𝑡_𝑛+1𝑞𝑡_𝑛+1−2𝑢𝑡_𝑛𝑞𝑡_𝑛−𝑢𝑡_𝑛−1𝑞𝑡_𝑛−1≤𝐶Δ𝑡²𝑢𝑞_{𝑡𝑡𝐿∞(𝐿2)}+𝑢_𝑡𝑞_{𝑡𝐿∞(𝐿2)}+𝑢_𝑡𝑡𝑞_{𝐿∞(𝐿2)}. (42) Using the Taylor expansion and noting that −2Δ𝑡=2(𝑡_𝑛−𝑡_𝑛+1)=𝑡_𝑛−1−𝑡_𝑛+1, we have 4𝑞𝑡_𝑛=4𝑞𝑡_𝑛+1−4𝑞_𝑡𝑡_𝑛+1Δ𝑡+2𝑞_𝑡𝑡𝑡_𝑛+1Δ𝑡²−2𝑞_𝑡𝑡𝑡𝑡_Δ53Δ𝑡³,𝑡_𝑛<𝑡_Δ5<𝑡_𝑛+1,𝑞𝑡_𝑛−1=𝑞𝑡_𝑛+1−2𝑞_𝑡𝑡_𝑛+1𝑞Δ𝑡+2^𝑡𝑡𝑡_𝑛+12Δ𝑡²−4𝑞_𝑡𝑡𝑡𝑡_Δ63Δ𝑡³,𝑡_𝑛−1<𝑡_Δ6<𝑡_𝑛+1. (43) Using (43), we obtain 𝑞_𝑡^𝑛+1=3𝑞^𝑛+1−4𝑞^𝑛+𝑞^𝑛−1−2Δ𝑡2𝑞_𝑡𝑡𝑡𝑡_Δ53−4𝑞_𝑡𝑡𝑡𝑡_Δ63Δ𝑡². (44) By (44), we have 𝜏^𝑛+1≤𝐶Δ𝑡²𝑞_𝑡𝑡𝑡(𝑡)_{𝐿∞(𝐿2)}. (45) Using the similar method to the estimate for ‖𝜏^𝑛+1‖ and (31), we obtain 𝜅₂^𝑛+1≤𝐶Δ𝑡²𝑞^ℎ_{𝑥𝑡𝑡𝑡𝐿∞(𝐿2)}≤𝐶Δ𝑡²𝜌_{𝑥𝑡𝑡𝑡𝐿∞(𝐿2)}+𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐿2)}≤𝐶Δ𝑡²ℎ^𝑟𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐻𝑟+1)}+𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐿2)}. (46)

3.2. Optimal Error Estimates

In this subsection, we derive the fully discrete optimal error estimates and obtain the following theorem.

Theorem 5. Assuming that 𝑈⁰, 𝑈¹∈𝑉_ℎ and 𝑍⁰, 𝑍¹∈𝑊_ℎ are given, then for 1≤𝐽≤𝑀, 𝑗=0,1, one has 𝑢^𝐽+1−𝑈^𝐽+1_𝑗≤𝐶ℎ^{min(𝑘+1−𝑗,𝑟+1)}+Δ𝑡²,𝑞^𝐽+1−𝑍^𝐽+1_𝑗+2𝑞^𝐽+1−𝑍^𝐽+1−𝑞^𝐽−𝑍^𝐽_𝑗≤𝐶ℎ^{min(𝑘+1,𝑟+1−𝑗)}+Δ𝑡². (47)

Proof. Taking 𝑣^ℎ=𝜍^𝑛+1 in (33) and using Cauchy-Schwarz’s inequality and Poincaré’s inequality, we get 𝜍^𝑛+1≤𝐶𝜍_𝑥^𝑛+1≤𝐶𝜌^𝑛+1+𝜉^𝑛+1. (48) Set 𝑤^ℎ=𝜉^𝑛+1 in (34) to obtain 3𝜉^𝑛+1−4𝜉^𝑛+𝜉^𝑛−12Δ𝑡,𝜉^𝑛+1+𝛽3𝜉_𝑥^𝑛+1−4𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12Δ𝑡,𝜉_𝑥^𝑛+1=−1−𝛽𝜆3𝜌^𝑛+1−4𝜌^𝑛+𝜌^𝑛−12Δ𝑡+𝜏^𝑛+1,𝜉^𝑛+1−𝛽𝜅^𝑛+1,𝜉_𝑥^𝑛+1+𝛾2𝜌^𝑛−𝜌^𝑛−1,𝜉_𝑥^𝑛+1+𝛾2𝜉^𝑛−𝜉^𝑛−1,𝜉_𝑥^𝑛+1+𝛾𝑅₁^𝑛+1,𝜉_𝑥^𝑛+1+𝛿𝑅₂^𝑛+1,𝜉_𝑥^𝑛+1+𝛿2𝑢𝑡_𝑛𝑞𝑡_𝑛−𝑈^𝑛𝑍^𝑛−𝑢𝑡_𝑛−1𝑞𝑡_𝑛−1−𝑈^𝑛−1𝑍^𝑛−1,𝜉_𝑥^𝑛+1. (49) Use (48) as well as the Cauchy-Schwarz and Young’s inequalities to obtain 3𝜉^𝑛+1−4𝜉^𝑛+𝜉^𝑛−12Δ𝑡,𝜉^𝑛+1+𝛽3𝜉_𝑥^𝑛+1−4𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12Δ𝑡,𝜉_𝑥^𝑛+1≤𝐶𝜉^𝑛+12+𝜉_𝑥^𝑛+12+𝜉^𝑛2+2𝜌^𝑛−𝜌^𝑛−12+2𝜉^𝑛−𝜉^𝑛−12+3𝜌^𝑛+1−4𝜌^𝑛+𝜌^𝑛−12Δ𝑡²+𝜏^𝑛+12+𝜅^𝑛+12+𝑅₁^𝑛+12+𝑅₂^𝑛+12+𝛿2𝑢𝑡_𝑛𝑞𝑡_𝑛−𝑈^𝑛𝑍^𝑛−𝑢𝑡_𝑛−1𝑞𝑡_𝑛−1−𝑈^𝑛−1𝑍^𝑛−1,𝜉_𝑥^𝑛+1. (50) Note that (a)3𝜉^𝑛+1−4𝜉^𝑛+𝜉^𝑛−12Δ𝑡,𝜉^𝑛+1=1𝜉4Δ𝑡^𝑛+12+2𝜉^𝑛+1−𝜉^𝑛2−𝜉^𝑛2+2𝜉^𝑛−𝜉^𝑛−12+𝜉^𝑛+1−2𝜉^𝑛+𝜉^𝑛−12,(b)𝛽3𝜉_𝑥^𝑛+1−4𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12Δ𝑡,𝜉_𝑥^𝑛+1=1𝜉4Δ𝑡^𝑥𝑛+12+2𝜉_𝑥^𝑛+1−𝜉^𝑛_𝑥²−𝜉^𝑛_𝑥²+2𝜉^𝑛_𝑥−𝜉_𝑥^𝑛−12+𝜉_𝑥^𝑛+1−2𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12,(c)3𝜌^𝑛+1−4𝜌^𝑛+𝜌^𝑛−12Δ𝑡²≤32Δ𝑡^𝑡𝑛+1_𝑡𝑛𝜌_𝑡²+1𝑑𝑠2Δ𝑡^𝑡𝑛_𝑡𝑛−1𝜌_𝑡²2𝑢𝑡𝑑𝑠,_𝑛𝑞𝑡_𝑛−𝑈^𝑛𝑍^𝑛−𝑢𝑡_𝑛−1𝑞𝑡_𝑛−1−𝑈^𝑛−1𝑍^𝑛−1,𝜉_𝑥^𝑛+1≤2𝑢𝑡_𝑛𝑞𝑡_𝑛−𝑍^𝑛−𝑢𝑡_𝑛𝑞𝑡_𝑛−1−𝑍^𝑛−1,𝜉_𝑥^𝑛+1+𝑢𝑡_𝑛−𝑢𝑡_𝑛−1𝑞𝑡_𝑛−1−𝑍^𝑛−1,𝜉_𝑥^𝑛+1+2𝑢𝑡_𝑛−𝑈^𝑛𝑍^𝑛−𝑢𝑡_𝑛−1−𝑈^𝑛−1𝑍^𝑛,𝜉_𝑥^𝑛+1+𝑢𝑡_𝑛−1−𝑈^𝑛−1𝑍^𝑛−𝑍^𝑛−1,𝜉_𝑥^𝑛+1=̇𝑇₁+𝑇₂+𝑇₃+𝑇₄. (51)(52) We now estimate 𝑇₁, 𝑇₂, 𝑇₃, and 𝑇₄ as 𝑇₁≤𝑢𝑡_𝑛2𝜉^𝑛−𝜉^𝑛−1,𝜉_𝑥^𝑛+1+𝑢𝑡_𝑛2𝜌^𝑛−𝜌^𝑛−1,𝜉_𝑥^𝑛+1≤𝑢𝑡_𝑛∞2𝜉^𝑛−𝜉^𝑛−1𝜉_𝑥^𝑛+1+𝑢𝑡_𝑛∞2𝜌^𝑛−𝜌^𝑛−1𝜉_𝑥^𝑛+1,𝑇₂=^𝑡𝑛_𝑡𝑛−1𝑢_𝑡𝑑𝑠𝜌^𝑛−1+𝜉^𝑛−1,𝜉_𝑥^𝑛+1≤Δ𝑡𝑢_𝑡∞𝜌^𝑛−1+𝜉^𝑛−1𝜉_𝑥^𝑛+1,𝑇₃≤𝑍^𝑛2𝜍^𝑛−𝜍^𝑛−1,𝜉_𝑥^𝑛+1+𝑍^𝑛2𝜂^𝑛−𝜂^𝑛−1,𝜉_𝑥^𝑛+1≤𝑍^𝑛_∞2𝜍^𝑛−𝜍^𝑛−1𝜉_𝑥^𝑛+1+𝑍^𝑛_∞2𝜂^𝑛−𝜂^𝑛−1𝜉_𝑥^𝑛+1,𝑇₄≤^𝑡𝑛_𝑡𝑛−1𝑍_𝑡𝑑𝑠𝜂^𝑛−1+𝜍^𝑛−1,𝜉_𝑥^𝑛+1≤Δ𝑡𝑍_𝑡∞𝜂^𝑛−1+𝜍^𝑛−1𝜉_𝑥^𝑛+1. (53) Substituting (53) into (52), we obtain 2𝑢𝑡_𝑛𝑞𝑡_𝑛−𝑈^𝑛𝑍^𝑛−𝑢𝑡_𝑛−1𝑞𝑡_𝑛−1−𝑈^𝑛−1𝑍^𝑛−1,𝜉_𝑥^𝑛+1≤𝐶2𝜉^𝑛−𝜉^𝑛−12+𝜉^𝑛−12+𝜌^𝑛2+𝜌^𝑛−12+𝜍^𝑛2+𝜍^𝑛−12+𝜂^𝑛2+𝜂^𝑛−12+𝜉_𝑥^𝑛+12. (54) Substituting (36), (51), and (54) into (50), using (48), and summing from 𝑛=1,2,…,𝐽, the resulting inequality becomes (1−𝐶Δ𝑡)𝜉^𝐽+12+2𝜉^𝐽+1−𝜉^𝐽2+𝜉_𝑥^𝐽+12+2𝜉_𝑥^𝐽+1−𝜉^𝐽_𝑥²≤𝐶Δ𝑡^𝐽_𝑛=1𝜂^𝑛−12+𝜌^𝑛−12+𝐶^𝑡𝐽+1₀𝜌_𝑡²𝑑𝑠+𝐶Δ𝑡⁴𝑞_𝑡𝑡²_𝐿∞𝐿2+𝑞_𝑡𝑡𝑡²_𝐿∞𝐿2+𝑢𝑞_{𝑡𝑡𝐿∞(𝐿2)}+𝑢_𝑡𝑞_{𝑡𝐿∞(𝐿2)}+𝑢_𝑡𝑡𝑞_{𝐿∞(𝐿2)}+ℎ^𝑟𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐻𝑟+1)}+𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐿2)}+𝐶Δ𝑡^𝐽_𝑛=1𝜉^𝑛2+𝜉^𝑛_𝑥²+2𝜉^𝑛−𝜉^𝑛−12+2𝜉^𝑛_𝑥−𝜉_𝑥^𝑛−12. (55) Choose Δ𝑡₀ in such a way that for 0<Δ𝑡≤Δ𝑡₀, (1−𝐶Δ𝑡)>0. Then, as an application of Gronwall’s lemma, we obtain 𝜉^𝐽+12+2𝜉^𝐽+1−𝜉^𝐽2+𝜉_𝑥^𝐽+12+2𝜉_𝑥^𝐽+1−𝜉^𝐽_𝑥²≤𝐶Δ𝑡^𝐽_𝑛=1𝜂^𝑛−12+𝜌^𝑛−12+𝐶^𝑡𝐽+1₀𝜌_𝑡²𝑑𝑠+𝐶Δ𝑡⁴𝑞_𝑡𝑡²_𝐿∞𝐿2+𝑞_𝑡𝑡𝑡²_𝐿∞𝐿2+𝑢𝑞_{𝑡𝑡𝐿∞(𝐿2)}+𝑢_𝑡𝑞_{𝑡𝐿∞(𝐿2)}+𝑢_𝑡𝑡𝑞_{𝐿∞(𝐿2)}+ℎ^𝑟𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐻𝑟+1)}+𝑞_{𝑥𝑡𝑡𝑡𝐿∞(𝐿2)}. (56) Combine (28), (31), and (56) with the triangle inequality to complete the 𝐿² and 𝐻¹ error estimates for 𝑞. Furthermore, use (48) and the triangle inequality to complete the optimal error estimates for ‖𝑢(𝑡_𝑛)−𝑈^𝑛‖ and ‖𝑢(𝑡_𝑛)−𝑈^𝑛‖₁.

Remark 6. Compared to a variety of difference methods in [17], our method is studied based on mixed element scheme (6) and (7).

Remark 7. Although some convergence proofs of multistep methods for RLW/BBM are provided in [20, 25, 29], our convergence results of multistep methods are proved based on a mixed finite element scheme. Based on the current discussion, we have to provide the detailed proofs for multistep mixed finite element methods in this paper.

4. Numerical Results

In order to test the viability of the proposed method, we consider a test problem. We write conservation laws as [3, 4] 𝑄₁=^𝑏_𝑎𝑢𝑑𝑥≃ℎ^𝑁_𝑗=1𝑢^𝑛_𝑗,𝑄₂=^𝑏_𝑎𝑢²+𝜇𝑢_𝑥²𝑑𝑥≃ℎ^𝑁_𝑗=1𝑢^𝑛_𝑗²+𝜇𝑢_𝑥^𝑛_𝑗²,𝑄₃=^𝑏_𝑎𝑢³+3𝑢²𝑑𝑥≃ℎ^𝑁_𝑗=1𝑢^𝑛_𝑗³+3(𝑢)^𝑛_𝑗², (57) where 𝑄₁, 𝑄₂, and 𝑄₃ are usually called mass, momentum, and energy, respectively, which are observed to check the conservation of the numerical scheme.

We consider RLW equation (1) and let, in (1), 𝛿=𝛾=𝛽=1. Then, the solitary wave solution of (1) is 𝑢(𝑥,𝑡)=3𝑐sech²𝑘𝑥−𝑥₀−𝑣𝑡, (58) where 1𝑘=2𝑐1+𝑐,𝑣=1+𝑐. (59)

We consider the motion of a single solitary wave and take as initial condition, with 𝑐=0.1 and 𝑥₀=0, 𝑢(𝑥,0)=0.3sech²𝑥2√.11 (60) The corresponding exact solution with initial condition (60) is 𝑢(𝑥,𝑡)=0.3sech²𝑥−1.1𝑡2√.11 (61)

In this procedure, we take space-time domain as −40≤𝑥≤60 and 0≤𝑡≤20.

In Table 1, we take spatial mesh parameter ℎ=0.125 and time discretization parameter Δ𝑡=0.1 and list the three invariants 𝑄₁, 𝑄₂, and 𝑄₃ and the optimal error estimate in 𝐿² and 𝐿^∞ norms for 𝑢 at different times 𝑡=0, 4, 8, 12, 16, and 20. At the same time, we show some numerical results at time 𝑡=20 obtained by other numerical methods in Table 1. From Table 1, we find that our method is more accurate than the numerical methods in [17, 18, 28] but is less than the numerical methods in [23, 24]. From the shown results in Table 1, we can see that 𝑄₁, 𝑄₂ and 𝑄₃ keep almost constants, so the conservation for our method is very well.

Table 1 

Solitary wave Amp. 0.3 and the errors in 𝐿² and 𝐿^∞ norms for 𝑢, 𝑄₁, 𝑄₂, and 𝑄₃ at 𝑡=20, ℎ=0.125, Δ𝑡=0.1, and −40≤𝑥≤60.

In Tables 2 and 3, we take spatial mesh parameter ℎ=0.125 and obtain the optimal error results in 𝐿² and 𝐿^∞ norms for 𝑢 at different times 𝑡=0, 4, 8, 12, 16, and 20 with different time discretization parameters Δ𝑡=0.1,0.2, and 0.4. From Tables 2 and 3, we see easily that the convergence rate for time is close to order 2. Similarly, the results for 𝑞 are shown in Tables 4 and 5.

Table 2 

Convergence order and error in 𝐿² norm for 𝑢 of time with ℎ=0.125 and 𝑐=0.1.

Table 3 

Convergence order and error in 𝐿^∞ norm for 𝑢 of time with ℎ=0.125 and 𝑐=0.1.

Table 4 

Convergence order and error in 𝐿² norm for 𝑞 of time with ℎ=0.125 and 𝑐=0.1.

Table 5 

Convergence order and error in 𝐿^∞ norm for 𝑞 of time with ℎ=0.125 and 𝑐=0.1.

In Tables 6 and 7, the optimal error results in 𝐿² and 𝐿^∞ norms for 𝑢 at different times 𝑡=0, 4, 8, 12, 16, and 20 with different spatial mesh parameters ℎ=0.2, 0.4, and 0.8 and time discretization parameter Δ𝑡=0.01 are shown. It is easy to see that the convergence rate for space is close to order 2. The similar results for 𝑞 are listed in Tables 8 and 9.

Table 6 

Convergence order and error in 𝐿² norm for 𝑢 of space with Δ𝑡=0.01 and 𝑐=0.1.

Table 7 

Convergence order and error in 𝐿^∞ norm for 𝑢 of space with Δ𝑡=0.01 and 𝑐=0.1.

Table 8 

Convergence order and error in 𝐿² norm for 𝑞 of space with Δ𝑡=0.01 and 𝑐=0.1.

Table 9 

Convergence order and error in 𝐿^∞ norm for 𝑞 of space with Δ𝑡=0.01 and 𝑐=0.1.

Figure 1 shows the surface for the exact solution 𝑢 in space-time domain ((𝑥,𝑡)∈[−20,60]×[0,20]), and the corresponding surface for the numerical solution 𝑢_ℎ with ℎ=2 and Δ𝑡=0.4 is described in Figure 2. From Figures 1 and 2, we see easily that the exact solution 𝑢 is approximated very well by the numerical solution 𝑢_ℎ. In Figures 3 and 4, we show the surface for the exact solution 𝑞 and the numerical solution 𝑞_ℎ, respectively, with ℎ=2 and Δ𝑡=0.4 and obtain a good approximation solution 𝑞_ℎ for the exact solution 𝑞.

Figure 1 

Surface for exact solution 𝑢.

Figure 2 

Surface for numerical solution 𝑢ℎ.

Figure 3 

Surface for exact solution 𝑞.

Figure 4 

Surface for numerical solution 𝑞ℎ.

The comparison between the exact solution 𝑢 and the numerical solution 𝑢_ℎ is described at different times 𝑡=4,8,12,16, and 20 with ℎ=0.125 and Δ𝑡=0.2 in Figure 5. The similar comparison for the exact solution 𝑞 and the numerical solution 𝑞_ℎ is shown in Figure 6. Figures 5 and 6 show that the solitary wave for 𝑢 and 𝑞 moves to the right with unchanged form and velocity, respectively. Furthermore, the exact solutions 𝑢 and 𝑞 are approximated well by the numerical solutions 𝑢_ℎ and 𝑞_ℎ, respectively.

Figure 5 

Comparison between 𝑢 and 𝑢ℎ at times 𝑡=4, 8, 12, 16, and 20 with ℎ=0.125 and Δ𝑡=0.2.

Figure 6 

Comparison between 𝑞 and 𝑞ℎ at times 𝑡=4 8, 12, 16, and 20 with ℎ=0.125 and Δ𝑡=0.2.

In Figure 7, we show the comparison between 𝑢 and 𝑢_ℎ at different spaces 𝑥=0,10,20, and 30 with ℎ=0.125 and Δ𝑡=0.2 to verify the efficiency for the proposed scheme in this paper. Figure 8 describes a similar result for 𝑞.

Figure 7 

Comparison between 𝑢 and 𝑢ℎ at spaces 𝑥=0, 10, 20, and 30 with ℎ=0.125 and Δ𝑡=0.2.

Figure 8 

Comparison between 𝑞 and 𝑞ℎ at spaces 𝑥=0, 10, 20, and 30 with ℎ=0.125 and Δ𝑡=0.2.

From the previous analysis in Tables 1–9 and Figures 1–8, we can see that the numerical results confirm the theoretical results of Theorem 5 and our method is efficient for RLW equation.

5. Concluding Remarks

In this paper, we propose and analyze an explicit multistep mixed finite element method, which combines spatial mixed finite element method and time explicit multistep method, for RLW equation. We discuss the numerical process for our method, prove the theoretical results for the fully discrete explicit multistep mixed scheme, obtain the optimal convergence order, and compare our method’s accuracy with some other numerical schemes. Compared with the numerical method in [20, 25, 26], our method can obtain the optimal error estimates in 𝐿² and 𝐻¹ norms for the scalar unknown 𝑢 and its flux 𝑞=𝑢_𝑥 simultaneously. From our numerical results, we can see that our method is efficient for RLW equation.

Acknowledgments

This work was supported by the National Natural Science Fund of China (11061021), the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011 and NJZY13199), the Natural Science Fund of Inner Mongolia Province (2012MS0108 and 2012MS0106), and the Program of Higher-Level Talents of Inner Mongolia University (125119).