A Fourth-Order Conservative Compact Finite Difference Scheme for the Generalized RLW Equation

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 isO(τ + h4) in the L-norm. Further, the compact schemes are conservative. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

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][5][6][7][8].Other mathematical theory and numerical methods for GRLW equation were considered in [9][10][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][16][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.

Mathematical Problems in Engineering
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 ( 2 + ℎ 4 ) order is based on some a priori estimates.In Section 4, numerical results are provided to test the theoretical results.
Lemma 1 (see [18]).For , V ∈ R  0 , one has Lemma 2. For any real symmetric positive definite matrices  and for , V ∈ R  0 , one can get where  is obtained by Cholesky decomposition of , denoted as  =   .

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.

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  ∈ [0, 1].Note that we need another finite difference scheme to calculate  1 , so the following scheme will be used: The matrix form of the difference scheme (25) can be written as Theorem 9. Suppose that  0 ∈  1 0 (Ω); then the finite difference scheme ( 25) is conservative for discrete energy; that is, where   are obtained by Cholesky decomposition of   , denoted as   =      , ( = 1, 2).
Next we will give some a priori estimates of difference solutions.

Convergence and Stability of Difference Solution.
First, we consider the truncation error of the finite difference scheme (25).Suppose that V   = (  ,   ), which is the solution of problem ( 1)-( 3).Then we have according to Taylor's expansion,    = ( 2 + ℎ 4 ) can be easily obtained.Next, we consider convergence and stability of the finite difference scheme (25).
Below, we can similarly prove stability of the difference solution.
Theorem 14.Under the conditions of Theorem 13, the solution of conservative finite difference scheme (25) is stable by  ∞ norm.