- Research Article
- Open Access
Two Conservative Difference Schemes for the Generalized Rosenau Equation
© J. Hu and K. Zheng. 2010
Received: 31 October 2009
Accepted: 26 January 2010
Published: 8 March 2010
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.
Consider the following initial-boundary value problem for generalized Rosenau equation:
with an initial condition
and boundary conditions
When , (1.1) is called as usual Rosenau equation proposed by Rosenau  for treating the dynamics of dense discrete systems. Since then, the Cauchy problem of the Rosenau equation was investigated by Park . Many numerical schemes have been proposed, such as -conforming finite element method by Chung and Pani , discontinuous Galerkin method by Choo et al. , orthogonal cubic spline collocation method by Manickam , and finite difference method by Chung  and Omrani et al. . 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
Lemma 2.1 (see ).
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 .
Lemma 2.3 (Browder fixed point Theorem).
Lemma 2.6 (Discrete Sobolev's inequality ).
Lemma 2.7 (Discrete Gronwall inequality ).
This completes the proof of Theorem 2.9.
Similarly, the following theorem can be proved.
3. Linearized Finite Difference Scheme
In this section, we propose a linear-implicit finite difference scheme as follows:
The difference scheme (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):
This completes the proof of Theorem 3.4.
Similarly, the following theorem can be proved that.
4. Numerical Experiments
Consider the generalized Rosenau equation:
with an initial condition
and boundary conditions
From the numerical results, two finite difference schemes of this paper are efficient.
This work was supported by the Youth Research Foundation of SWUST (no. 08zx3125).
- Rosenau P: Dynamics of dense discrete systems. Progress of Theoretical Physics 1988, 79: 1028-1042. 10.1143/PTP.79.1028View ArticleGoogle Scholar
- Park MA: On the rosenau equation. Matematica Aplicada e Computacional 1990, 9: 145-152.MathSciNetMATHGoogle Scholar
- Chung SK, Pani AK: Numerical methods for the rosenau equation. Applicable Analysis 2001,77(3-4):351-369. 10.1080/00036810108840914MathSciNetView ArticleMATHGoogle Scholar
- Choo SM, Chung SK, Kim KI: A discontinuous galerkin method for the rosenau equation. Applied Numerical Mathematics 2008,58(6):783-799. 10.1016/j.apnum.2007.02.008MathSciNetView ArticleMATHGoogle Scholar
- Manickam SA, Pani AK, Chung SK: A second-order splitting combined with orthogonal cubic spline collocation method for the rosenau equation. Numerical Methods for Partial Differential Equations 1998,14(6):695-716. 10.1002/(SICI)1098-2426(199811)14:6<695::AID-NUM1>3.0.CO;2-LMathSciNetView ArticleMATHGoogle Scholar
- Chung SK: Finite difference approximate solutions for the rosenau equation. Applicable Analysis 1998,69(1-2):149-156. 10.1080/00036819808840652MathSciNetView ArticleMATHGoogle Scholar
- Omrani K, Abidi F, Achouri T, Khiari N: A new conservative finite difference scheme for the rosenau equation. Applied Mathematics and Computation 2008,201(1-2):35-43. 10.1016/j.amc.2007.11.039MathSciNetView ArticleMATHGoogle Scholar
- Hu B, Xu Y, Hu J: Crank-nicolson finite difference scheme for the rosenau-burgers equation. Applied Mathematics and Computation 2008,204(1):311-316. 10.1016/j.amc.2008.06.051MathSciNetView ArticleMATHGoogle Scholar
- Browder FE: Existence and uniqueness theorems for solutions of nonlinear boundary value problems. Proceedings of Symposia in Applied Mathematics 1965, 17: 24-49.View ArticleMathSciNetGoogle Scholar
- Zhou Y: Applications of Discrete Functional Analysis to the Finite Difference Method. International Academic Publishers, Beijing, China; 1991:vi+260.MATHGoogle Scholar
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