A new conservative finite difference scheme for the Rosenau equation
Introduction
In recent years, a vast amount of work and computation has been devoted to the initial value problem for the Korteweg-de Vries (KdV) equation, which describes the vibrations of a unidimensional anharmonic lattice associated with the birth of the soliton. To overcome the shortcomings of KdV equation, Rosenau [1] has developed a model called the Rosenau equation, to describe the dynamics of dense discrete systems.
The Rosenau equation is given bywith the boundary conditionsand an initial conditionwhere and
The following conservation law is known as
The theoretical results on existence, uniqueness and regularity of solutions to (1.1a), (1.1b), (1.1c) have been investigated by Park [2]. Using Faedo–Galerkin method, Chung and Pani in [3] have established well-posedness and some regularity results of the above problem. The numerical approximation to the problem (1.1a), (1.1b), (1.1c) in one dimensional space has been developed by Chung and Ha [4] using finite element Galerkin method, where they obtained optimal -estimates and suboptimal order error estimates in and -norms.
However, Chung and Pani [3] have considered the Rosenau equation in several space variables and have derived optimal error estimates in and quasi-optimal estimates in -norm. For both these articles, finite element spaces consisting of -piecewise polynomials of degree are used. Subsequently, in [5] using the splitting method, -piecewise linear spaces are employed in conjunction with the lumped mass finite element method and a priori bounds are derived. Manickam et al. [6] have developed the numerical solution of (1.1a), (1.1b), (1.1c) using an orthogonal cubic spline collocation method. Convergence of the semi discrete solution of the Rosenau equation and the error analysis of the modified Galerkin–Crank–Nicolson method has been studied in [7], [8].
In spite of computational convenience of finite difference methods, there are few studies on finite difference methods compared to large amount of studies on finite element approximate solutions for (1.1a), (1.1b), (1.1c) because of uneasy control of nonlinear term . In [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] the conservation finite difference schemes were used for a system of the generalized nonlinear Schrödinger equation, regularized long wave equation, sine-Gordon equation, Klein–Gordon equation, Zakharov equation and Cahn–Hilliard equation, respectively.
In [27], Chung has proposed a nonlinear conservative scheme for the initial boundary value problem (1.1a), (1.1b), (1.1c). In this paper, we propose a three-level finite difference scheme for the Rosenau equation (1.1a), (1.1b), (1.1c) which is conservative and unconditionally stable, and we prove error estimates of second-order. The remainder of this article is organized as follows; in Section 2 we describe a finite difference for the Rosenau equation. In Section 3, we show that the scheme is energy conserving. In Section 4, we prove that the difference scheme is uniquely solvable. In Section 5, we derive a priori error estimates for numerical solution. In Section 6, we prove the convergence and stability for our difference scheme.
Section snippets
Finite difference scheme
Let be the uniform step size in the spatial direction for a positive integer M and denotes the uniform step size in the temporal direction for a positive integer N. Denote for , and .
We define the difference operators, for a function asFurther, we define operators and respectively, asWe
Discrete conservative law
To obtain conservative law, we introduce the following lemma: Lemma 1 For , we have Proof (1) In view of difference properties and the boundary conditions (2.1d), we obtain(2) For , we haveNoting that
Solvability
Below, we are going to prove the solvability of the finite difference scheme (2.1a), (2.1b), (2.1c), (2.1d). Theorem 2 The difference scheme (2.1a), (2.1b), (2.1c), (2.1d) is uniquely solvable. Proof It is obvious that and are uniquely determined by (2.1b), (2.1c). Now suppose be solved uniquely. Consider the equation of (2.1a) for Computing the inner product of Eq. (4.1) with and using (3.1), we obtain
Some priori estimates for the difference solution
The following Lemma can be verified by summation by parts and using minimum eigenvalue of symmetric matrix. For a proof, see Agarwall [28]. Lemma 2 For function V defined on , the following identity and inequality hold:
Next, we shall use the following Lemma [29]. Lemma 3 Discrete Sobolev’s inequality Suppose that is mesh functions. Given there exists a constant C dependent on such that Lemma 4 Suppose that is
Convergence
Define the net function and let and Lemma 5 Assume that the solution of (1.1a), (1.1b), (1.1c) is sufficiently regular, there exists a positive constant C independent of h and k such that Proof It follows from (2.2), (5.13) thatUsing Theorem 3,
Conclusion
In this paper, we have proposed a new conservative finite difference scheme for the Rosenau equation, which has a wide range of applications in physics.
A second-order error estimates as well as conservation of energy, and stability of the finite difference approximate solutions were discussed in detail.
Acknowledgements
The authors thank the referees for their valuable comments which improved this article.
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