A high-accuracy conservative difference approximation for Rosenau-KdV equation
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Authors
Jinsong Hu
- School of Science, Xihua University, Chengdu 610039, China.
Jun Zhou
- School of Mathematics and Statistics, Yangtze Normal University, Chongqing 408100, China.
Ru Zhuo
- School of Science, Xihua University, Chengdu 610039, China.
Abstract
In this paper, we study the initial-boundary value problem of Rosenau-KdV equation. A conservative two level nonlinear
Crank-Nicolson difference scheme, which has the theoretical accuracy \(O(\tau^2 + h^4)\), is proposed. The scheme simulates two
conservative properties of the initial boundary value problem. Existence, uniqueness, and priori estimates of difference solution
are obtained. Furthermore, we analyze the convergence and unconditional stability of the scheme by the energy method.
Numerical experiments demonstrate the theoretical results.
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ISRP Style
Jinsong Hu, Jun Zhou, Ru Zhuo, A high-accuracy conservative difference approximation for Rosenau-KdV equation, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 3013--3022
AMA Style
Hu Jinsong, Zhou Jun, Zhuo Ru, A high-accuracy conservative difference approximation for Rosenau-KdV equation. J. Nonlinear Sci. Appl. (2017); 10(6):3013--3022
Chicago/Turabian Style
Hu, Jinsong, Zhou, Jun, Zhuo, Ru. "A high-accuracy conservative difference approximation for Rosenau-KdV equation." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 3013--3022
Keywords
- Rosenau-KdV equation
- finite difference scheme
- conservative
- convergence
- stability.
MSC
References
-
[1]
F. E. Browder , Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Proc. Sympos. Appl. Math., Amer. Math. Soc., Providence, R.I., 17 (1965), 24–49.
-
[2]
Q. S. Chang, B. L. Guo, H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comp., 64 (1995), 537–553.
-
[3]
K.-L. Cheng, W.-Q. Feng, S. Gottlieb, C. Wang, A Fourier pseudospectral method for the ”good” Boussinesq equation with second-order temporal accuracy, Numer. Methods Partial Differential Equations, 31 (2015), 202–224.
-
[4]
K.-L. Cheng, C. Wang, S. M. Wise, X.-Y. Yue, A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn- Hilliard equation and its solution by the homogeneous linear iteration method, J. Sci. Comput., 69 (2016), 1083–1114.
-
[5]
S. K. Chung, Finite difference approximate solutions for the Rosenau equation, Appl. Anal., 69 (1998), 149–156.
-
[6]
S. K. Chung, S. N. Ha, Finite element Galerkin solutions for the Rosenau equation, Appl. Anal., 54 (1994), 39–56.
-
[7]
S. K. Chung, A. K. Pani, Numerical methods for the Rosenau equation, Appl. Anal., 77 (2001), 351–369.
-
[8]
G. Ebadi, A. Mojaver, H. Triki, A. Yildirim, A. Biswas, Topological solitons and other solutions of the Rosenau-KdV equation with power law nonlinearity, Romanian J. Phys., 58 (2013), 3–14.
-
[9]
A. Esfahani, Solitary wave solutions for generalized Rosenau-KdV equation, Commun. Theor. Phys. (Beijing), 55 (2011), 396–398.
-
[10]
F. Gao, X.-J. Yang, Local fractional Euler’s method for the steady heat-conduction problem, Therm. Sci., 20 (2016), S735– S738.
-
[11]
J.-S. Hu, B. Hu, Y.-C. Xu, C-N difference schemes for dissipative symmetric regularized long wave equations with damping term, Math. Probl. Eng., 2011 (2011), 16 pages.
-
[12]
J.-S. Hu, Y.-C. Xu, B. Hu, Conservative linear difference scheme for Rosenau-KdV equation, Adv. Math. Phys., 2013 (2013), 7 pages.
-
[13]
J.-S. Hu, K.-L. Zheng, Two conservative difference schemes for the generalized Rosenau equation, Bound. Value Probl., 2010 (2010), 18 pages.
-
[14]
Y. D. Kim, H. Y. Lee, The convergence of finite element Galerkin solution for the Roseneau equation, Korean J. Comput. Appl. Math., 5 (1998), 171–180.
-
[15]
S. Li, L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 32 (1995), 1839–1875.
-
[16]
S. A. V. Manickam, A. K. Pani, S. K. Chung, A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation, Numer. Methods Partial Differential Equations, 14 (1998), 695–716.
-
[17]
T. Nie, A decoupled and conservative difference scheme with fourth-order accuracy for the Symmetric Regularized Long Wave equations, Appl. Math. Comput., 219 (2013), 9461–9468.
-
[18]
K. Omrani, F. Abidi, T. Achouri, N. Khiari, A new conservative finite difference scheme for the Rosenau equation, Appl. Math. Comput., 201 (2008), 35–43.
-
[19]
X.-T. Pan, L.-M. Zhang, Numerical simulation for general Rosenau-RLW equation: an average linearized conservative scheme, Math. Probl. Eng., 2012 (2012), 15 pages.
-
[20]
X.-T. Pan, L.-M. Zhang, On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation, Appl. Math. Model., 36 (2012), 3371–3378.
-
[21]
M. A. Park, On the Rosenau equation, Mat. Apl. Comput., 9 (1990), 145–152.
-
[22]
P. Rosenau, A quasi-continuous description of a nonlinear transmission line, Phys. Scripta, 34 (1986), 827–829.
-
[23]
P. Rosenau, Dynamics of dense discrete systems high order effects, Progr. Theoret. Phys., 79 (1988), 1028–1042.
-
[24]
T.-C. Wang, B.-L. Guo, L.-M. Zhang, New conservative difference schemes for a coupled nonlinear Schrödinger system, Appl. Math. Comput., 217 (2010), 1604–1619.
-
[25]
X.-J. Yang, F. Gao, A new technology for solving diffusion and heat equations, Therm. Sci., 21 (2017), 133–140.
-
[26]
X.-J. Yang, F. Gao, H. M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Math. Appl., 73 (2017), 203–210.
-
[27]
L.-M. Zhang, A finite difference scheme for generalized regularized long-wave equation, Appl. Math. Comput., 168 (2005), 962–972.
-
[28]
Y. L. Zhou, Applications of discrete functional analysis to the finite difference method, International Academic Publishers, Beijing (1991)
-
[29]
J. Zhou, M. B. Zheng, R.-X. Jiang, The conservative difference scheme for the generalized Rosenau-KDV equation, Therm. Sci., 20 (2016), 903–910.
-
[30]
J.-M. Zuo, Solitons and periodic solutions for the Rosenau-KdV and Rosenau-Kawahara equations, Appl. Math. Comput., 215 (2009), 835–840.