A High-Accuracy Linear Conservative Difference Scheme for Rosenau-RLW Equation

Abstract

We study the initial-boundary value problem for Rosenau-RLW equation. We propose a three-level linear finite difference scheme, which has the theoretical accuracy of 𝑂(𝜏²+ℎ⁴). The scheme simulates two conservative properties of original problem well. The existence, uniqueness of difference solution, and a priori estimates in infinite norm are obtained. Furthermore, we analyze the convergence and stability of the scheme by energy method. At last, numerical experiments demonstrate the theoretical results.

1. Introduction

In the study of the dynamics of compact discrete systems, wave-wave and wave-wall interactions cannot be described by the well-known KdV equation. To overcome this shortcoming of KdV equation, Rosenau proposed the following Rosenau equation in [1, 2]: 𝑢_𝑡+𝑢_{𝑥𝑥𝑥𝑥𝑡}+𝑢_𝑥+𝑢𝑢_𝑥=0. (1) Rosenau equation (1) is usually used to describe the dense discrete system and simulate the long-chain transmission model through an L-C flow in radio and computer fields. The existence and uniqueness of solution to (1) were proved by Park in [3]. Rosenau equation is also regarded as the transformation of the Regularized Long Wave (RLW) equation (see [4]): 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢_𝑥+𝑢𝑢_𝑥=0, (2) which is usually used to simulate the long wave in nonlinear emanative medium. RLW equation plays important role in the study of nonlinear diffusion wave because it could model lots of physical phenomena. As RLW equation and KdV equation have the same approximative order when they are used to describe motivations, RLW equation could simulate almost all of the applications of KdV equation [5]. Therefore, there are many works about Rosenau equation (1) and RLW equation (2) (see, e.g., [6–23]).

Rosenau-RLW equation is the generalization of Rosenau equation (1) and RLW equation (2). The authors of [24–27] studied the numerical solution of Rosenau-RLW equation. Motivated by the above works, we consider the initial-boundary value problem of the following Rosenau-RLW equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢_{𝑥𝑥𝑥𝑥𝑡}+𝑢_𝑥+𝑢𝑢_𝑥()∈𝑥=0,𝑥,𝑡_𝐿,𝑥_𝑅×(],𝑢()0,𝑇𝑥,0=𝑢₀(𝑥),𝑥∈𝑥_𝐿,𝑥_𝑅,𝑢𝑥_𝐿,𝑡=𝑢𝑥_𝑅,𝑡=0,𝑢_𝑥𝑥𝑥_𝐿,𝑡=𝑢_𝑥𝑥𝑥_𝑅[],,𝑡=0,𝑡∈0,𝑇 (3)(4)(5) where 𝑢₀(𝑥) is a smooth function.

As the solitary wave solution of (3) is (see, e.g., [24–26]) 𝑢(𝑥,𝑡)=1519sech⁴√1326𝑥−169𝑡,133 (6) the initial-boundary problem (3)–(5) is in accordance with the Cauchy problem of (3) when −𝑥_𝐿≫0, 𝑥_𝑅≫0. It is easy to verify that problem (3)–(5) satisfies the following conservative laws [27]: 𝑄(𝑡)=^𝑥𝑅_𝑥𝐿𝑢(𝑥,𝑡)𝑑𝑥=^𝑥𝑅_𝑥𝐿𝑢₀(𝑥)(0),𝐸(𝑡)=‖𝑢‖𝑑𝑥=𝑄_2𝐿2+𝑢_𝑥²_𝐿2+𝑢_𝑥𝑥²_𝐿2=𝐸(0), (7)(8) where 𝑄(0) and 𝐸(0) are both constants only depending on initial data.

Li and Vu-Quoc pointed in [28] that in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation. It is said in [29] that conservative difference scheme can simulate the conservative law of initial problem well and it could avoid the nonlinear blowup. Therefore, constructing conservative difference scheme is an important and significant job. To our knowledge, the theoretic accuracy of the existing difference scheme for Rosenau-RLW equation (see [24, 25, 27]) is 𝑂(𝜏²+ℎ²). Particularly, in [27], the authors proposed a three-level linear conservative difference scheme for problem (3)–(5), whose theoretic accuracy is 𝑂(𝜏²+ℎ²). One does not need iteration when solving the equation numerically using this scheme. Henceforth, it could save some computing time. Using Richardson extrapolation idea, we will propose a three-level linear difference scheme which has the theoretic accuracy of 𝑂(𝜏²+ℎ⁴) without refined mesh in this paper. Our scheme simulates the two conservative laws (7) and (8) well. And we will study the a priori estimate, existence, and uniqueness of the difference solution. Furthermore, we shall analyze the convergence and stability.

The rest of this paper is organized as follows. We propose the conservative difference scheme in Section 2 and prove the existence and uniqueness of solution to difference scheme by mathematical induction in Section 3. Section 4 is devoted to the prior estimate, convergence, and stability of the difference scheme. In Section 5, we verify our theoretical analysis by numerical experiments.

2. Difference Scheme and Its Conservative Law

Let ℎ and 𝜏 be the uniform stepsize in the spatial and temporal directions, respectively. Denote ℎ=(𝑥_𝑅−𝑥_𝐿)/𝐽, 𝑥_𝑗=𝑥_𝐿+𝑗ℎ(𝑗=−2,−1,0,1,…,𝐽,𝐽+1,𝐽+2), and 𝑡_𝑛=𝑛𝜏(𝑛=0,1,2,…,𝑁, where 𝑁=[𝑇/𝜏]). Let 𝑢^𝑛_𝑗 be the difference approximation of 𝑢(𝑥,𝑡) at (𝑥_𝑗,𝑡_𝑛); that is, 𝑢^𝑛_𝑗≈𝑢(𝑥_𝑗,𝑡_𝑛). Let 𝑍⁰_ℎ=𝑢=𝑢_𝑗∣𝑢₋₂=𝑢₋₁=𝑢₀=𝑢_𝐽=𝑢_𝐽+1=𝑢_𝐽+2=0,𝑗=−2,−1,0,1,…,𝐽,𝐽+1,𝐽+2. (9) We define the difference operators, inner product, and norms that will be used in this paper as follows: 𝑢^𝑛_𝑗𝑥=𝑢^𝑛_𝑗+1−𝑢^𝑛_𝑗ℎ,𝑢^𝑛_𝑗𝑥=𝑢^𝑛_𝑗−𝑢^𝑛_𝑗−1ℎ,𝑢^𝑛_𝑗𝑥=𝑢^𝑛_𝑗+1−𝑢^𝑛_𝑗−1,𝑢2ℎ^{𝑛𝑗̈𝑥}=𝑢^𝑛_𝑗+2−𝑢^𝑛_𝑗−2,𝑢4ℎ_𝑛𝑗𝑡̂=𝑢_𝑗^𝑛+1−𝑢^𝑛_𝑗,2𝜏𝑢^𝑛_𝑗=𝑢_𝑗^𝑛+1+𝑢_𝑗^𝑛−12,𝑢^𝑛,𝑣^𝑛=ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑣^𝑛_𝑗,𝑢^𝑛2=𝑢^𝑛,𝑢^𝑛,𝑢^𝑛_∞=max_{1≤𝑗≤𝐽−1}𝑢^𝑛_𝑗. (10)

Lemma 1. It follows from the Cauchy-Schwarz inequality and summation by parts (see [30]) that, for any 𝑢∈𝑍⁰_ℎ, 𝑢_̈𝑥²≤𝑢𝑥²≤𝑢_𝑥²,𝑢_𝑥²=𝑢_𝑥². (11)

In the paper, 𝐶 denotes a general positive constant which may have a different value in a different occurrence.

Consider the following finite difference scheme for problem (3)–(5): 𝑢^𝑛_𝑗𝑡̂−43𝑢^𝑛_𝑗𝑥𝑥𝑡̂+13𝑢^𝑛_𝑗𝑥𝑥𝑡̂+53𝑢^𝑛_{𝑗𝑥𝑥𝑥𝑥}𝑡̂−23𝑢^𝑛_𝑗𝑥𝑥𝑥𝑥𝑡̂+43𝑢^𝑛_𝑗𝑥−13𝑢^𝑛_𝑗̈𝑥+49𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥+𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥−19𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥+𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥𝑢=0,𝑗=1,2,…,𝐽−1,𝑛=1,2,…,𝑁−1_0𝑗=𝑢₀𝑥_𝑗𝑢𝑗=1,2,…,𝐽−1,_1𝑗−43𝑢¹_𝑗𝑥𝑥+13𝑢¹_𝑗𝑥𝑥+53𝑢¹_{𝑗𝑥𝑥𝑥𝑥}−23𝑢¹_𝑗𝑥𝑥𝑥𝑥=𝑢₀𝑥_𝑗−𝜕²𝑢₀𝜕𝑥²𝑥_𝑗+𝜕⁴𝑢₀𝜕𝑥⁴𝑥_𝑗−𝜏𝜕𝑢₀𝑥𝜕𝑥^𝑗−𝜏𝑢₀𝑥_𝑗𝜕𝑢₀𝑥𝜕𝑥^𝑗,𝑢𝑗=1,2,…,𝐽−1_𝑛∈𝑍⁰_ℎ,𝑢^𝑛_0𝑥𝑥=𝑢^𝑛_𝐽𝑥𝑥𝑢=0,_𝑛0𝑥𝑥=𝑢^𝑛_𝐽𝑥𝑥=0(𝑛=0,1,2,…,𝑁). (12)(13)(14)(15) The discrete boundary condition (15) is reasonable from the boundary condition of (3)–(5). The following theorem shows how the difference scheme (12)–(15) simulates the conservative law numerically.

Theorem 2. Suppose that 𝑢₀∈𝐻²₀[𝑥_𝐿,𝑥_𝑅] and 𝑢(𝑥,𝑡)∈𝐶^5,3. Then the scheme (12)–(15) is conservative for discrete energy; that is, 𝑄^𝑛=𝑄^𝑛−1=⋯=𝑄⁰,𝐸^𝑛=𝐸^𝑛−1=⋯=𝐸⁰ (16)(17)(𝑛=1,2,…,𝑁), where 𝑄^𝑛=ℎ2^𝐽−1_𝑗=1𝑢_𝑗^𝑛+1+𝑢^𝑛_𝑗+2ℎ9𝜏^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥−ℎ𝜏18^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥,𝐸^𝑛=12𝑢^𝑛+12+43𝑢_𝑥^𝑛+12−13𝑢𝑥^𝑛+12+53𝑢^𝑛+1_𝑥𝑥²−23𝑢𝑥_𝑥^𝑛+12+𝑢^𝑛2+43𝑢^𝑛_𝑥²−13𝑢𝑥^𝑛2+53𝑢^𝑛_𝑥𝑥²−23𝑢^𝑛_𝑥𝑥². (18)

Proof. Multiplying ℎ in the two sides of (12) and taking summation for 𝑗, we could obtain from (15) and summation by parts [30] that ℎ^𝐽−1_𝑗=1𝑢_𝑗^𝑛+1−𝑢_𝑗^𝑛−1+42𝜏9ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥−19ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥=0. (19) On the other hand, ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥=ℎ2^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥+ℎ2^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛−1𝑥=ℎ2^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥−ℎ2^𝐽−1_𝑗=1𝑢_𝑗^𝑛−1𝑢^𝑛_𝑗𝑥,ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥=ℎ2^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥+ℎ2^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛−1_̈𝑥=ℎ2^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥−ℎ2^𝐽−1_𝑗=1𝑢_𝑗^𝑛−1𝑢^𝑛_𝑗̈𝑥. (20) From the definition of 𝑄^𝑛, we know that (16) could be deduced from (19)-(20).
Taking an inner product of (12) and 2𝑢^𝑛 (i.e., 𝑢^𝑛+1+𝑢^𝑛−1), we could obtain from boundary condition (15) and summation by parts [30] that 𝑢𝑡^𝑛2̂+43𝑢𝑡^𝑛𝑥2̂−13𝑢𝑥^𝑛2_𝑡+53𝑢𝑡^{𝑛𝑥𝑥2}̂−23𝑢^𝑛_𝑥𝑥𝑡²̂+83𝑢𝑥^𝑛,𝑢^𝑛−23𝑢^𝑛_̈𝑥,𝑢^𝑛+2𝜙𝑢^𝑛,𝑢^𝑛,𝑢^𝑛−2𝜅𝑢^𝑛,𝑢^𝑛,𝑢^𝑛=0, (21) where 𝜙𝑢^𝑛_𝑗,𝑢^𝑛_𝑗=49𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥+𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥,𝜅𝑢^𝑛_𝑗,𝑢^𝑛_𝑗=19𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥+𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥. (22) On the other hand, 𝑢𝑥^𝑛,𝑢^𝑛=0,𝑢^𝑛_̈𝑥,𝑢^𝑛𝜙𝑢=0,_𝑛,𝑢^𝑛,𝑢^𝑛=49ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥+𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥𝑢^𝑛_𝑗=49ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥+49ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥𝑢^𝑛_𝑗=49ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥−49ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥𝜅𝑢=0,_𝑛,𝑢^𝑛,𝑢^𝑛=19ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥+𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥𝑢^𝑛_𝑗=19ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥+49ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥𝑢^𝑛_𝑗=19ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥−19ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥=0. (23)(24)(25) Taking (23)–(25) into (21), we have 1𝑢2𝜏^𝑛+12−𝑢^𝑛−12+2𝑢3𝜏^𝑥𝑛+12−𝑢_𝑥^𝑛−12−1𝑢6𝜏𝑥^𝑛+12−𝑢𝑥^𝑛−12+5𝑢6𝜏^{𝑛+1𝑥𝑥2}−𝑢^𝑛−1_𝑥𝑥²−1𝑢3𝜏𝑥_𝑥^𝑛+12−𝑢𝑥_𝑥^𝑛−12=0. (26) From the definition of 𝐸^𝑛, we know that (17) could be obtained by deducing the above equality from 0 to 𝑁−1.

3. Solvability

Theorem 3. The solution of difference scheme (12)–(15) is unique.

Proof. We will use the mathematical induction to prove our theorem. We first note that 𝑢⁰ and 𝑢¹ are determined uniquely by (13) and (14).
Suppose that 𝑢⁰,𝑢¹,…,𝑢^𝑛−1,𝑢^𝑛(𝑛≤𝑁−1) are the unique solution to scheme (12)–(15). Next we prove that there exists unique 𝑢^𝑛+1 which satisfies (12)–(15).
Consider 1𝑢2𝜏^𝑗𝑛+1−2𝑢3𝜏^{𝑗𝑛+1𝑥𝑥}+1𝑢6𝜏^𝑗𝑛+1𝑥𝑥+5𝑢6𝜏^{𝑗𝑛+1𝑥𝑥𝑥𝑥}−1𝑢3𝜏^{𝑗𝑛+1𝑥𝑥}𝑥𝑥+23𝑢_𝑗^𝑛+1𝑥−16𝑢_𝑗^𝑛+1_̈𝑥+29𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥+𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥−1𝑢18^𝑛𝑗𝑢_𝑗^𝑛+1_̈𝑥+𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥.=0𝑗=1,2,…,𝐽−1,𝑛=1,2,…,𝑁−1 (27)
Taking an inner product of (27) and 𝑢^𝑛+1, we could obtain from boundary condition (15) and summation by parts [30] that 1𝑢2𝜏^𝑛+12+2𝑢3𝜏^𝑥𝑛+12−1𝑢6𝜏𝑥^𝑛+12+5𝑢6𝜏^{𝑛+1𝑥𝑥2}−1𝑢3𝜏𝑥_𝑥^𝑛+12+23𝑢𝑥^𝑛+1,𝑢^𝑛+1−16𝑢_̈𝑥^𝑛+1,𝑢^𝑛+1+2ℎ9^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥+𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥𝑢_𝑗^𝑛+1−ℎ18^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥+𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥𝑢_𝑗^𝑛+1=0. (28) Note that 𝑢𝑥^𝑛+1,𝑢^𝑛+1=0,𝑢_̈𝑥^𝑛+1,𝑢^𝑛+1ℎ=0,_{𝐽−1𝑗=1}𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥+𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥𝑢_𝑗^𝑛+1=ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑢_𝑗^𝑛+1𝑥+ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑥𝑢_𝑗^𝑛+1=ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑢_𝑗^𝑛+1𝑥−ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑢_𝑗^𝑛+1𝑥ℎ=0,_{𝐽−1𝑗=1}𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥+𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥𝑢_𝑗^𝑛+1=ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑢_𝑗^𝑛+1_̈𝑥+ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1_̈𝑥𝑢_𝑗^𝑛+1=ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑢_𝑗^𝑛+1_̈𝑥−ℎ^𝐽−1_𝑗=1𝑢^𝑛_𝑗𝑢_𝑗^𝑛+1𝑢_𝑗^𝑛+1_̈𝑥=0. (29) Furthermore, from Lemma 1, one can easily obtain that 𝑢𝑥²≤𝑢_𝑥²,𝑢_𝑥𝑥²≤𝑢_𝑥𝑥². (30) Henceforth we could have 𝑢^𝑛+12+𝑢_𝑥^𝑛+12+𝑢^𝑛+1_𝑥𝑥²=0, (31) from (28)–(30).
That is, (27) only admits zero solution. So, there exits unique 𝑢_𝑗^𝑛+1 that satisfies (12).

4. Convergence and Stability of the Difference Scheme

Suppose that 𝑣(𝑥,𝑡) is the solution to problem (3)–(5). Let 𝑣^𝑛_𝑗=𝑣(𝑥_𝑗,𝑡_𝑛). Then the truncation error of the difference scheme (12)–(15) is 𝑟^𝑛_𝑗=𝑣^𝑛_𝑗𝑡̂−43𝑣^𝑛_𝑗𝑥𝑥𝑡̂+13𝑣^𝑛_𝑗𝑥𝑥𝑡̂+53𝑣^𝑛_{𝑗𝑥𝑥𝑥𝑥}𝑡̂−23𝑣^𝑛_𝑗𝑥𝑥𝑥𝑥𝑡̂+43𝑣^𝑛_𝑗𝑥−13𝑣^𝑛_𝑗̈𝑥+𝜑𝑣^𝑛_𝑗,𝑣^𝑛_𝑗−𝜅𝑣^𝑛_𝑗,𝑣^𝑛_𝑗,𝑣𝑗=1,2,…,𝐽−1,𝑛=1,2,…,𝑁−1_0𝑗=𝑢₀𝑥_𝑗,𝑣𝑗=1,2,…,𝐽−1_1𝑗−43𝑣¹_𝑗𝑥𝑥+13𝑣¹_𝑗𝑥𝑥+53𝑣¹_{𝑗𝑥𝑥𝑥𝑥}−23𝑣¹_𝑗𝑥𝑥𝑥𝑥=𝑢₀𝑥_𝑗−𝜕²𝑢₀𝜕𝑥²𝑥_𝑗+𝜕⁴𝑢₀𝜕𝑥⁴𝑥_𝑗−𝜏𝜕𝑢₀𝑥𝜕𝑥^𝑗−𝜏𝑢₀𝑥_𝑗𝜕𝑢₀𝑥𝜕𝑥^𝑗+𝑟⁰_𝑗,𝑣𝑗=1,2,…,𝐽−1_𝑛∈𝑍⁰_ℎ,𝑣^𝑛_0𝑥𝑥=𝑣^𝑛_𝐽𝑥𝑥=0,𝑣^𝑛₀𝑥𝑥=𝑣^𝑛_𝐽𝑥𝑥().=0𝑛=0,1,2,…,𝑁 (32) Making use of the Taylor expansion theorem, we know that |𝑟^𝑛_𝑗|=𝑂(𝜏²+ℎ⁴) as ℎ→0, 𝜏→0.

Lemma 4. Assume that 𝑢₀∈𝐻²₀[𝑥_𝐿,𝑥_𝑅], 𝑢(𝑥,𝑡)∈𝐶^5,3. Then the solution to problem (3)–(5) satisfies ‖𝑢‖_𝐿2≤𝐶,𝑢_𝑥𝐿2≤𝐶,𝑢_{𝑥𝑥𝐿2}‖𝑢‖≤𝐶,_𝐿∞≤𝐶,𝑢_𝑥𝐿∞≤𝐶. (33)

Proof. From (8), we know that ‖𝑢‖_𝐿2≤𝐶,𝑢_𝑥𝐿2≤𝐶,𝑢_{𝑥𝑥𝐿2}≤𝐶. (34) Then ‖𝑢‖_𝐿∞≤𝐶,𝑢_𝑥𝐿∞≤𝐶, (35) from the Sobolev inequality.

Theorem 5. Suppose that 𝑢₀∈𝐻²[𝑥_𝐿,𝑥_𝑅], 𝑢(𝑥,𝑡)∈𝐶^5,3. Then the solution to difference scheme (12)–(15) satisfies 𝑢^𝑛≤𝐶,𝑢^𝑛_𝑥≤𝐶,𝑢^𝑛_𝑥𝑥≤𝐶; (36) thus, 𝑢^𝑛_∞≤𝐶,𝑢^𝑛_𝑥∞≤𝐶(𝑛=1,2,…,𝑁). (37)

Proof. From Theorem 2 and (30) we can get 𝑢^𝑛2+𝑢^𝑛_𝑥²+𝑢^𝑛_𝑥𝑥²≤𝐸^𝑛=𝐸⁰=𝐶; (38) that is, 𝑢^𝑛≤𝐶,𝑢^𝑛_𝑥≤𝐶,𝑢^𝑛_𝑥𝑥≤𝐶. (39) Then the discrete Sobolev inequality (see [30]) shows that 𝑢^𝑛_∞≤𝐶,𝑢^𝑛_𝑥∞≤𝐶. (40)

Theorem 6. Suppose that 𝑢₀∈𝐻²[𝑥_𝐿,𝑥_𝑅], 𝑢(𝑥,𝑡)∈𝐶^5,3. Then the solution to difference scheme (12)–(15) 𝑢^𝑛 converges to the solution of problem (3)–(5) in the sense of norm ‖⋅‖_∞, and the convergent rate is 𝑂(𝜏²+ℎ⁴).

Proof. Subtracting (12)–(15) from (32) and letting 𝑒^𝑛_𝑗=𝑣^𝑛_𝑗−𝑢^𝑛_𝑗, we have 𝑟^𝑛_𝑗=𝑒^𝑛_𝑗𝑡̂−43𝑒^𝑛_𝑗𝑥𝑥𝑡̂+13𝑒^𝑛_𝑗𝑥𝑥𝑡̂+53𝑒^𝑛_{𝑗𝑥𝑥𝑥𝑥}𝑡̂−23𝑒^𝑛_𝑗𝑥𝑥𝑥𝑥𝑡̂+43𝑒^𝑛_𝑗𝑥−13𝑒^𝑛_𝑗̈𝑥+𝜑𝑣^𝑛_𝑗,𝑣^𝑛_𝑗−𝜑𝑢^𝑛_𝑗,𝑢^𝑛_𝑗−𝜅𝑣^𝑛_𝑗,𝑣^𝑛_𝑗+𝜅𝑢^𝑛_𝑗,𝑢^𝑛_𝑗,𝑒𝑗=1,2,…,𝐽−1,𝑛=1,2,…,𝑁−1_0𝑗,𝑟=0𝑗=0,1,2,…,𝐽_0𝑗=𝑒¹_𝑗−43𝑒¹_𝑗𝑥𝑥+13𝑒¹_𝑗𝑥𝑥+53𝑒¹_{𝑗𝑥𝑥𝑥𝑥}−23𝑒¹_𝑗𝑥𝑥𝑥𝑥,𝑒𝑗=1,2,…,𝐽−1_𝑛∈𝑍⁰_ℎ,𝑒^𝑛_0𝑥𝑥=𝑒^𝑛_𝐽𝑥𝑥𝑒=0,_𝑛0𝑥𝑥=𝑒^𝑛_𝐽𝑥𝑥=0(𝑛=0,1,2,…,𝑁). (41)(42)(43)(44) Taking the inner product of (43) and 𝑒¹, and using boundary condition (44), we obtain 𝑒¹²+43𝑒¹_𝑥²−13𝑒𝑥¹²+53𝑒¹_𝑥𝑥²−23𝑒¹_𝑥𝑥²=⟨𝑟⁰,𝑒¹⟩. (45) Noticing that 𝑟⁰_𝑗=𝑂𝜏²+ℎ⁴𝑗=1,2,…,𝐽−1, (46) we can obtain that 𝑒¹²+𝑒¹_𝑥²+𝑒¹_𝑥𝑥²≤𝑂𝜏²+ℎ⁴², (47) from the Cauchy-Schwarz inequality and (30), and (44).
Then taking the inner product of (41) and 2𝑒^𝑛, and combining with boundary condition (44) again, one can get 𝑟^𝑛,2𝑒^𝑛=𝑒𝑡^𝑛2̂+43𝑒𝑡^𝑛𝑥2̂−13𝑒𝑥^𝑛𝑡²̂+53𝑒𝑡^{𝑛𝑥𝑥2}̂−23𝑒^𝑛_𝑥𝑥𝑡²̂+83𝑒𝑥^𝑛,𝑒^𝑛−23𝑒^𝑛_̈𝑥,𝑒^𝑛+2𝜑𝑣^𝑛,𝑣^𝑛−𝜑𝑢^𝑛,𝑢^𝑛,𝑒^𝑛−2𝜅𝑣^𝑛,𝑣^𝑛−𝜅𝑢^𝑛,𝑢^𝑛,𝑒^𝑛. (48)
Similar to (23), we have 𝑒𝑥^𝑛,𝑒^𝑛=0,𝑒^𝑛_̈𝑥,𝑒^𝑛=0. (49) By using Lemma 4, Theorem 5, Lemma 1, and the Cauchy-Schwarz inequality, we can get 𝜑𝑣^𝑛,𝑣^𝑛−𝜑𝑢^𝑛,𝑢^𝑛,𝑒^𝑛=4ℎ9^𝐽−1_𝑗=1𝑣^𝑛_𝑗𝑣^𝑛_𝑗𝑥−𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥𝑒^𝑛_𝑗+4ℎ9^𝐽−1_𝑗=1𝑣^𝑛_𝑗𝑣^𝑛_𝑗𝑥−𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥𝑒^𝑛_𝑗=4ℎ9^𝐽−1_𝑗=1𝑣^𝑛_𝑗𝑣^𝑛_𝑗𝑥−𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑥𝑒^𝑛_𝑗−4ℎ9^𝐽−1_𝑗=1𝑣^𝑛_𝑗𝑣^𝑛_𝑗−𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑒^𝑛_𝑗𝑥=4ℎ9^𝐽−1_𝑗=1𝑒^𝑛_𝑗𝑣^𝑛_𝑗𝑥+𝑢^𝑛_𝑗𝑒^𝑛_𝑗𝑥𝑒^𝑛_𝑗−4ℎ9^𝐽−1_𝑗=1𝑒^𝑛_𝑗𝑣^𝑛_𝑗+𝑢^𝑛_𝑗𝑒^𝑛_𝑗𝑒^𝑛_𝑗𝑥≤𝐶𝑒^𝑛2+𝑒^𝑛2+𝑒𝑥^𝑛2≤𝐶𝑒^𝑛+12+𝑒^𝑛2+𝑒^𝑛−12+𝑒_𝑥^𝑛+12+𝑒_𝑥^𝑛−12,𝜅𝑣^𝑛,𝑣^𝑛−𝜅𝑢^𝑛,𝑢^𝑛,𝑒^𝑛=ℎ9^𝐽−1_𝑗=1𝑣^𝑛_𝑗𝑣^𝑛_𝑗̈𝑥−𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥𝑒^𝑛_𝑗+ℎ9^𝐽−1_𝑗=1𝑣^𝑛_𝑗𝑣^𝑛_𝑗̈𝑥−𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥𝑒^𝑛_𝑗=ℎ9^𝐽−1_𝑗=1𝑣^𝑛_𝑗𝑣^𝑛_𝑗̈𝑥−𝑢^𝑛_𝑗𝑢^𝑛_𝑗̈𝑥𝑒^𝑛_𝑗−ℎ9^𝐽−1_𝑗=1𝑣^𝑛_𝑗𝑣^𝑛_𝑗−𝑢^𝑛_𝑗𝑢^𝑛_𝑗𝑒^𝑛_𝑗̈𝑥=ℎ9^𝐽−1_𝑗=1𝑒^𝑛_𝑗𝑣^𝑛_𝑗̈𝑥+𝑢^𝑛_𝑗𝑒^𝑛_𝑗̈𝑥𝑒^𝑛_𝑗−ℎ9^𝐽−1_𝑗=1𝑒^𝑛_𝑗𝑣^𝑛_𝑗+𝑢^𝑛_𝑗𝑒^𝑛_𝑗𝑒^𝑛_𝑗̈𝑥≤𝐶𝑒^𝑛2+𝑒^𝑛2+𝑒^𝑛_̈𝑥²≤𝐶𝑒^𝑛+12+𝑒^𝑛2+𝑒^𝑛−12+𝑒_𝑥^𝑛+12+𝑒_𝑥^𝑛−12,𝑟^𝑛,2𝑒^𝑛=𝑟^𝑛,𝑒^𝑛+1+𝑒^𝑛−1≤𝑟^𝑛2+𝑒^𝑛+12+𝑒^𝑛−12. (50) Taking (49)-(50) into (48), we obtain 𝑒𝑡^𝑛2̂+43𝑒𝑡^𝑛𝑥2̂−13𝑒𝑥^𝑛𝑡²̂+53𝑒𝑡^{𝑛𝑥𝑥2}̂−23𝑒^𝑛_𝑥𝑥²𝑡̂≤𝑟^𝑛2+𝐶𝑒^𝑛+12+𝑒^𝑛2+𝑒^𝑛−12+𝑒_𝑥^𝑛+12+𝑒_𝑥^𝑛−12. (51) Set 𝐵^𝑛=𝑒^𝑛+12+𝑒^𝑛2+43𝑒_𝑥^𝑛+12+43𝑒^𝑛_𝑥²−13𝑒𝑥^𝑛+12−13𝑒𝑥^𝑛2+53𝑒^𝑛+1_𝑥𝑥²+53𝑒^𝑛_𝑥𝑥²−23𝑒𝑥_𝑥^𝑛+12−23𝑒^𝑛_𝑥𝑥². (52) Summarizing (51) from 1 to 𝑛, we get 𝐵^𝑛≤𝐵⁰+2𝜏^𝑛_𝑙=1𝑟^𝑙2+𝐶𝜏^𝑛+1_𝑙=0𝑒^𝑙2+𝑒^𝑙_𝑥². (53) From (42) and (47), we know that 𝐵⁰≤𝑂𝜏²+ℎ⁴². (54)
Furthermore, 𝜏^𝑛_𝑙=1𝑟^𝑙2≤𝑛𝜏max_{1≤𝑙≤𝑛}𝑟^𝑙2≤𝑇⋅𝑂𝜏²+ℎ⁴². (55) Therefore, 𝑒^𝑛+12+𝑒^𝑛2+𝑒_𝑥^𝑛+12+𝑒^𝑛_𝑥²+𝑒^𝑛+1_𝑥𝑥²+𝑒^𝑛_𝑥𝑥²≤𝐵^𝑛≤𝑂𝜏²+ℎ⁴²+𝐶𝜏^𝑛+1_𝑙=0𝑒^𝑙2+𝑒^𝑙_𝑥²+𝑒^𝑙_𝑥𝑥². (56) The discrete Gronwall inequality (see [30]) implies that 𝑒^𝑛≤𝑂𝜏²+ℎ⁴,𝑒^𝑛_𝑥≤𝑂𝜏²+ℎ⁴,𝑒^𝑛_𝑥𝑥≤𝑂𝜏²+ℎ⁴. (57) Then we can obtain 𝑒^𝑛_∞≤𝑂𝜏²+ℎ⁴, (58) from the discrete Sobolev inequality.

We could prove the following theorems in a similar way of Theorem 6.

Theorem 7. Suppose that 𝑢₀∈𝐻²[𝑥_𝐿,𝑥_𝑅] and 𝑢(𝑥,𝑡)∈𝐶^5,3. The solution to difference scheme (12)–(15) 𝑢^𝑛 is stable in the sense of norm ‖⋅‖_∞.

5. Numerical Simulations

As the difference scheme (12)–(15) is a linear system about 𝑢_𝑗^𝑛+1, it does not need iteration when we solve it numerically.

Let 𝑥_𝐿=−50, 𝑥_𝑅=100, 𝑇=40, and 𝑢₀(𝑥)=1519sech⁴𝑥2√.13 (59) For some different values of 𝜏 and ℎ, we list errors at several times in Table 1 and verify the accuracy of the difference scheme in Table 2. The numerical simulation of two conservative quantities (7) and (8) is listed in Table 3.

Table 1 

The errors estimates of numerical solution with various ℎ and 𝜏.

Table 2 

The numerical verification of theoretical accuracy 𝑂(𝜏²+ℎ⁴).

Table 3 

The numerical simulation for conservative quantities (7) and (8).

The stability and convergence of the scheme are verified by these numerical experiments. And it shows that our proposed algorithm is effective and reliable.

Acknowledgments

The authors are very grateful to the anonymous referees for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by Scientific Research Foundation of Sichuan Provincial Education Department (no. 11ZB009), the Key Scientific Research Foundation of Xihua University (no. Z0912611), and the fund of Key Disciplinary of Computer Software and Theory, Sichuan, China, Grant no. SZD0802-09-1.