Abstract

The generalized regularized long-wave (GRLW) equation is studied by finite difference method. A new fourth-order energy conservative compact finite difference scheme was proposed. It is proved by the discrete energy method that the compact scheme is solvable, the convergence and stability of the difference schemes are obtained, and its numerical convergence order is in the -norm. Further, the compact schemes are conservative. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

1. Introduction

In this paper, we consider the following generalized regularized long-wave equation:with the boundary conditions and the initial condition where is a real-valued function defined on , , is a positive integer, and is a given function with Dirichlet boundary condition.

The GRLW equation was first put forward as a model for small-amplitude long waves on the surface of water in a channel by Peregrine [1, 2]. A special case of (1), that is, is usually called the regularized long-wave (RLW) equation. The RLW equation is a representation form of nonlinear long wave and can describe a lot of important physical phenomena, such as shallow waves and ionic waves. The GRLW equation can also describe that wave motion to the same order of approximation as the KDV equation, so it plays a major role in the study of nonlinear dispersive waves [3]. It is difficult to find the analytical solution for (1), which has been studied by many researchers. The finite difference method for the initial-boundary value problem of the GRLW equation had been studied in [4–8]. Other mathematical theory and numerical methods for GRLW equation were considered in [9–11]. Reference [12] solved the GRLW equation by the Petrov-Galerkin method. Numerical solution of GRLW equation used Sinc-collocation method in [13]. In [14], a time-linearization method that uses a Crank-Nicolson procedure in time and three-point, fourth-order accurate in space, compact difference equations, is presented and used to determine the solutions of the generalized regularized-long wave (GRLW) equation. Recently, there has been growing interest in high-order compact methods for solving partial differential equations [15–17].

In this paper, we consider problem (1)–(3); it has the following conservation law: Using a customary designation, we will refer to the functional as the energy integral, although it is not necessarily identifiable with energy in the original physical problem.

We aim to present a conservative finite difference scheme for problem (1)–(3), which simulates conservation law (5) that the differential equation (1) possesses, and prove convergence and stability of the scheme. This paper is organized as follows. In Section 2, some notations are given and some useful lemmas are proposed. In Section 3, we present a nonlinear compact conservative difference scheme, discuss its discrete conservative law, prove the existence of difference solution by Brouwer fixed point theorem, give some a priori estimates, and then prove by discrete energy method that the difference scheme is uniquely solvable, unconditionally stable and that convergence of the difference solutions with order is based on some a priori estimates. In Section 4, numerical results are provided to test the theoretical results.

2. Notations and Lemmas

Let and be the spatial and temporal step sizes, respectively. Denote . Let denote the approximation of , and let

As usual, the following notations will be used:

We now introduce the discrete -inner product and the associated norm

The discrete -norm is defined as

Let . It is convenient to let denote the normed vector space as . The corresponding matrices are defined, respectively, as

For a simple notation, the discrete function is defined by where . Obviously, , , , are symmetric positive definite matrices. To obtain some important results, we introduce the following lemmas.

Lemma 1 (see [18]). For , one has

Lemma 2. For any real symmetric positive definite matrices and for , one can get where is obtained by Cholesky decomposition of , denoted as .

Proof. For , we have

Lemma 3 (see [16]). On the matrices , . The eigenvalues of the matrices and are, respectively, as follows:

Lemma 4. For , we can get where are obtained by Cholesky decomposition of , denoted as .

Proof. It follows from Lemma 3 that the eigenvalues of and satisfy This gives the spectral radius ,  , and consequently Thus

Lemma 5. For , one has

Proof. For , we have

Lemma 6 (see [18]). For any discrete function and for any given , there exists a constant , depending only on and , such that

Lemma 7 (see [19]). Let be a finite-dimensional inner product space, let be the associated norm, and let be continuous. Assume, moreover, that , , , . Then, there exists a such that and .

Lemma 8 (see [18]). Suppose that the discrete function satisfies recurrence formula where , , and are nonnegative constants. Then where is sufficiently small, such that .

3. A Nonlinear-Implicit Conservative Scheme

In this section, we propose a nonlinear-implicit conservative scheme for the initial-boundary value problem (1)–(3) and give its numerical analysis.

3.1. The Nonlinear-Implicit Scheme and Its Conservative Law

Next we consider the compact finite difference scheme for problem (1)–(3) as follows: where weight coefficient . Note that we need another finite difference scheme to calculate , so the following scheme will be used:The matrix form of the difference scheme (25) can be written as

Theorem 9. Suppose that ; then the finite difference scheme (25) is conservative for discrete energy; that is, where are obtained by Cholesky decomposition of , denoted as .

Proof. Taking an inner product of (27) with , from Lemma 5, we obtain Noting that from (31)-(32), we obtain Let Then, from (33), we get . This completes the proof.

3.2. Existence and Prior Estimates of Difference Solution

Theorem 10. Suppose that ; then the finite difference scheme (25) is solvable.

Proof. Assume that there exist which satisfy (25) as ; now we try to prove that satisfy (25). We define the mapping as follows: is obviously continuous. Taking in (35) the inner product with , from Lemmas 2, 4, and 5, we obtainThus for , there exists . The existence of follows from Lemma 7 and consequently the existence of is obtained. This completes the proof.

Next we will give some a priori estimates of difference solutions.

Lemma 11. Suppose that ; then there exists the estimation for the solution of problem (1)–(3):

Proof. It follows from (5) that Hence, it follows from the Sobolev inequality that

Lemma 12. Suppose that ; then there exists the estimation for the solution of the difference scheme (25): where ,  .

Proof. From Theorem 9, we obtain then from Lemma 4, we have That is let be small, such that ; then we can get where It follows from Lemma 6 that where .
This completes the proof.