Two Conservative Difference Schemes for the Generalized Rosenau Equation
 Jinsong Hu^{1} and
 Kelong Zheng^{2}Email author
DOI: 10.1155/2010/543503
© J. Hu and K. Zheng. 2010
Received: 31 October 2009
Accepted: 26 January 2010
Published: 8 March 2010
Abstract
Numerical solutions for generalized Rosenau equation are considered and two energy conservative finite difference schemes are proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, and priori error estimate of the scheme are proved using energy method. Numerical results demonstrate that two schemes are efficient and reliable.
1. Introduction
Consider the following initialboundary value problem for generalized Rosenau equation:
with an initial condition
and boundary conditions
where is a integer.
When , (1.1) is called as usual Rosenau equation proposed by Rosenau [1] for treating the dynamics of dense discrete systems. Since then, the Cauchy problem of the Rosenau equation was investigated by Park [2]. Many numerical schemes have been proposed, such as conforming finite element method by Chung and Pani [3], discontinuous Galerkin method by Choo et al. [4], orthogonal cubic spline collocation method by Manickam [5], and finite difference method by Chung [6] and Omrani et al. [7]. As for the generalized case, however, there are few studies on theoretical analysis and numerical methods.
It can be proved easily that the problem (1.1)–(1.3) has the following conservative law:
Hence, we propose two conservative difference schemes which simulate conservative law (1.4). The outline of the paper is as follows. In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, a linearized difference scheme is proposed and theoretical results are obtained. In Section 4, some numerical experiments are shown.
2. Nonlinear Finite Difference Scheme
Let and be the uniform step size in the spatial and temporal direction, respectively. Denote , , and . Define
and in the paper, denotes a general positive constant which may have different values in different occurrences.
Since , then the following finite difference scheme is considered:
Lemma 2.1 (see [8]).
Theorem 2.2.
Proof.
According to
By the definition of , (2.7) holds.
To prove the existence of solution for scheme (2.2)–(2.4), the following Browder fixed point Theorem should be introduced. For the proof, see [9].
Lemma 2.3 (Browder fixed point Theorem).
Let be a finite dimensional inner product space. Suppose that is continuous and there exists an such that for all with . Then there exists such that and .
Theorem 2.4.
There exists satisfying the difference scheme (2.2)–(2.4).
Proof.
By the mathematical induction, for , assume that satisfy (2.2)–(2.4). Next we prove that there exists satisfying (2.2)–(2.4).
Define a operator on as follows:
Obviously, for all , with . It follows from Lemma 2.3 that there exists which satisfies . Let , it can be proved that is the solution of the scheme (2.2)–(2.4).
Next, we discuss the convergence and stability of the scheme (2.2)–(2.4). Let be the solution of problem (1.1)–(1.3), , then the truncation of the scheme (2.2)–(2.4) is
Using Taylor expansion, we know that holds if .
Lemma 2.5.
Proof.
Hence, . According to Sobolev inequality, we have .
Lemma 2.6 (Discrete Sobolev's inequality [10]).
Lemma 2.7 (Discrete Gronwall inequality [10]).
Theorem 2.8.
Suppose , then the solution of (2.2) satisfies , which yield .
Proof.
According to Lemma 2.6, we have .
Theorem 2.9.
Suppose , then the solution of the scheme (2.2)–(2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is .
Proof.
This completes the proof of Theorem 2.9.
Similarly, the following theorem can be proved.
Theorem 2.10.
Under the conditions of Theorem 2.9, the solution of the scheme (2.2)–(2.4) is stable by .
3. Linearized Finite Difference Scheme
In this section, we propose a linearimplicit finite difference scheme as follows:
Theorem 3.1.
Proof.
By the definition of , (3.2) holds.
Theorem 3.2.
The difference scheme (3.1) is uniquely solvable.
Proof.
That is, there uniquely exists trivial solution satisfying (3.8). Hence, in (3.1) is uniquely solvable.
To discuss the convergence and stability of the scheme (3.1), we denote the truncation of the scheme (3.1):
Using Taylor expansion, we know that holds if .
Theorem 3.3.
Suppose , then the solution of (3.1) satisfies , which yield .
Proof.
Theorem 3.4.
Suppose , then the solution of the schemes (3.1), (2.3), and (2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is .
Proof.
This completes the proof of Theorem 3.4.
Similarly, the following theorem can be proved that.
Theorem 3.5.
Under the conditions of Theorem 3.4, the solution of the schemes (3.1), (2.3), and (2.4) are stable by .
4. Numerical Experiments
Consider the generalized Rosenau equation:
with an initial condition
and boundary conditions
The errors estimates in the sense of , when and .


 

Scheme (2.2)  Scheme (3.1)  Scheme (2.2)  Scheme (3.1)  Scheme (2.2)  Scheme (3.1)  



































The errors estimates in the sense of , when and .


 

Scheme (2.2)  Scheme (3.1)  Scheme (2.2)  Scheme (3.1)  Scheme (2.2)  Scheme (3.1)  



































The errors estimates in the sense of , when and .


 

Scheme (2.2)  Scheme (3.1)  Scheme (2.2)  Scheme (3.1)  Scheme (2.2)  Scheme (3.1)  



































From the numerical results, two finite difference schemes of this paper are efficient.
Declarations
Acknowledgment
This work was supported by the Youth Research Foundation of SWUST (no. 08zx3125).
Authors’ Affiliations
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