Two Conservative Difference Schemes for the Generalized Rosenau Equation
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Research Article Two Conservative Difference Schemes for the Generalized Rosenau Equation Jinsong Hu1 and Kelong Zheng2 1 2
School of Mathematics and Computer Engineering, Xihua University, Chengdu, Sichuan 610039, China School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China
Correspondence should be addressed to Kelong Zheng, kl [email protected] Received 31 October 2009; Accepted 26 January 2010 Academic Editor: Sandro Salsa Copyright q 2010 J. Hu and K. Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 initial-boundary value problem for generalized Rosenau equation: ut uxxxxt ux up x 0,
x ∈ 0, L , t ∈ 0, T ,
1.1
x ∈ 0, L ,
1.2
with an initial condition ux, 0 u0 x, and boundary conditions u0, t uL, t 0,
uxx 0, t uxx L, t 0,
t ∈ 0, T ,
1.3
where p ≥ 2 is a integer. When p 2, 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 C1 -conforming finite element method by Chung and Pani 3 ,
2
Boundary Value Problems
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: 1.4
Et u2L2 uxx 2L2 E0.
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 h and τ be the uniform step size in the spatial and temporal direction, respectively. Denote xj jh 0 ≤ j ≤ J, tn nτ 0 ≤ n ≤ N, unj ≈ uxj , tn , and Zh0 {u uj | u0 uj 0, j 0, 1, 2, . . . , J}. Define
unj
unj
t
x
unj1 − unj h
− unj un1 j τ
,
,
unj
unj
unj
t
x
unj − unj−1 h
− un−1 un1 j j
un−1 un1 j j 2
J−1 un , vn h
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