zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Conservative linear difference scheme for Rosenau-KdV equation. (English) Zbl 1282.35332
Summary: A conservative three-level linear finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-KdV equation is proposed. The difference scheme simulates two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference scheme is of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
WorldCat.org
Full Text: DOI
References:
[1] Y. Cui and D.-k. Mao, “Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation,” Journal of Computational Physics, vol. 227, no. 1, pp. 376-399, 2007. · Zbl 1131.65073 · doi:10.1016/j.jcp.2007.07.031
[2] S. Zhu and J. Zhao, “The alternating segment explicit-implicit scheme for the dispersive equation,” Applied Mathematics Letters, vol. 14, no. 6, pp. 657-662, 2001. · Zbl 0996.65083 · doi:10.1016/S0893-9659(01)80022-7
[3] A. R. Bahadır, “Exponential finite-difference method applied to Korteweg-de Vries equation for small times,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 675-682, 2005. · Zbl 1062.65087 · doi:10.1016/j.amc.2003.11.025
[4] S. Özer and S. Kutluay, “An analytical-numerical method for solving the Korteweg-de Vries equation,” Applied Mathematics and Computation, vol. 164, no. 3, pp. 789-797, 2005. · Zbl 1070.65077 · doi:10.1016/j.amc.2004.06.011
[5] P. Rosenau, “A quasi-continuous description of a nonlinear transmission line,” Physica Scripta, vol. 34, pp. 827-829, 1986. · doi:10.1088/0031-8949/34/6B/020
[6] P. Rosenau, “Dynamics of dense discrete systems,” Progress of Theoretical Physics, vol. 79, pp. 1028-1042, 1988. · doi:10.1143/PTP.79.1028
[7] M. A. Park, “On the Rosenau equation,” Matemática Aplicada e Computacional, vol. 9, no. 2, pp. 145-152, 1990. · Zbl 0723.35071
[8] S. K. Chung and S. N. Ha, “Finite element Galerkin solutions for the Rosenau equation,” Applicable Analysis, vol. 54, no. 1-2, pp. 39-56, 1994. · Zbl 0830.65097 · doi:10.1080/00036819408840267
[9] K. Omrani, F. Abidi, T. Achouri, and N. Khiari, “A new conservative finite difference scheme for the Rosenau equation,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 35-43, 2008. · Zbl 1156.65078 · doi:10.1016/j.amc.2007.11.039
[10] S. K. Chung, “Finite difference approximate solutions for the Rosenau equation,” Applicable Analysis, vol. 69, no. 1-2, pp. 149-156, 1998. · Zbl 0904.65093 · doi:10.1080/00036819808840652
[11] S. K. Chung and A. K. Pani, “Numerical methods for the Rosenau equation,” Applicable Analysis, vol. 77, no. 3-4, pp. 351-369, 2001. · Zbl 1021.65048 · doi:10.1080/00036810108840914
[12] S. A. V. Manickam, A. K. Pani, and S. K. Chung, “A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation,” Numerical Methods for Partial Differential Equations, vol. 14, no. 6, pp. 695-716, 1998. · Zbl 0930.65111 · doi:10.1002/(SICI)1098-2426(199811)14:6<695::AID-NUM1>3.0.CO;2-L
[13] Y. D. Kim and H. Y. Lee, “The convergence of finite element Galerkin solution for the Roseneau equation,” The Korean Journal of Computational & Applied Mathematics, vol. 5, no. 1, pp. 171-180, 1998. · Zbl 0977.65080
[14] J.-M. Zuo, “Solitons and periodic solutions for the Rosenau-KdV and Rosenau-Kawahara equations,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 835-840, 2009. · Zbl 1175.65124 · doi:10.1016/j.amc.2009.06.011
[15] A. Esfahani, “Solitary wave solutions for generalized Rosenau-KdV equation,” Communications in Theoretical Physics, vol. 55, no. 3, pp. 396-398, 2011. · Zbl 1264.35192 · doi:10.1088/0253-6102/55/3/04
[16] P. Razborova, H. Triki, and A. Biswas, “Perturbation of dispersive shallow water waves,” Ocean Engineering, vol. 63, pp. 1-7, 2013. · doi:10.1016/j.oceaneng.2013.01.014
[17] G. Ebadi, A. Mojaver, H. Triki, A. Yildirim, and A. Biswas, “Topological solitons and other solutions of the Rosenau-KdV equation with power law nonlinearity,” Romanian Journal of Physics, vol. 58, no. 1-2, pp. 1-10, 2013.
[18] T. Wang, L. Zhang, and F. Chen, “Conservative schemes for the symmetric regularized long wave equations,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1063-1080, 2007. · Zbl 1124.65080 · doi:10.1016/j.amc.2007.01.105
[19] S. Li and L. Vu-Quoc, “Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation,” SIAM Journal on Numerical Analysis, vol. 32, no. 6, pp. 1839-1875, 1995. · Zbl 0847.65062 · doi:10.1137/0732083
[20] Q. Chang, E. Jia, and W. Sun, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” Journal of Computational Physics, vol. 148, no. 2, pp. 397-415, 1999. · Zbl 0923.65059 · doi:10.1006/jcph.1998.6120
[21] T.-C. Wang and L.-M. Zhang, “Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1780-1794, 2006. · Zbl 1161.65349 · doi:10.1016/j.amc.2006.06.015
[22] T. Wang, B. Guo, and L. Zhang, “New conservative difference schemes for a coupled nonlinear Schrödinger system,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1604-1619, 2010. · Zbl 1205.65242 · doi:10.1016/j.amc.2009.07.040
[23] L. Zhang, “A finite difference scheme for generalized regularized long-wave equation,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 962-972, 2005. · Zbl 1080.65079 · doi:10.1016/j.amc.2004.09.027
[24] Z. Fei and L. Vázquez, “Two energy conserving numerical schemes for the sine-Gordon equation,” Applied Mathematics and Computation, vol. 45, no. 1, pp. 17-30, 1991. · Zbl 0732.65107 · doi:10.1016/0096-3003(91)90087-4
[25] Y. S. Wong, Q. Chang, and L. Gong, “An initial-boundary value problem of a nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 84, no. 1, pp. 77-93, 1997. · Zbl 0884.65091 · doi:10.1016/S0096-3003(96)00065-3
[26] Q. S. Chang, B. L. Guo, and H. Jiang, “Finite difference method for generalized Zakharov equations,” Mathematics of Computation, vol. 64, no. 210, pp. 537-553, 1995. · Zbl 0827.65138 · doi:10.2307/2153438
[27] J. Hu and K. Zheng, “Two conservative difference schemes for the generalized Rosenau equation,” Boundary Value Problems, Article ID 543503, 18 pages, 2010. · Zbl 1187.65090 · doi:10.1155/2010/543503 · eudml:224305
[28] X. Pan and L. Zhang, “On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation,” Applied Mathematical Modelling, vol. 36, no. 8, pp. 3371-3378, 2012. · Zbl 1252.65144 · doi:10.1016/j.apm.2011.08.022
[29] X. Pan and L. Zhang, “Numerical simulation for general Rosenau-RLW equation: an average linearized conservative scheme,” Mathematical Problems in Engineering, vol. 2012, Article ID 517818, 15 pages, 2012. · Zbl 1264.65140
[30] T. Wang and B. Guo, “A robust semi-explicit difference scheme for the Kuramoto-Tsuzuki equation,” Journal of Computational and Applied Mathematics, vol. 233, no. 4, pp. 878-888, 2009. · Zbl 1181.65117 · doi:10.1016/j.cam.2009.07.058
[31] B. Hu, J. Hu, and Y. Xu, “C-N difference schemes for dissipative symmetric regularized long wave equations with damping term,” Mathematical Problems in Engineering, vol. 2011, Article ID 651642, 16 pages, 2011. · Zbl 1213.76125 · doi:10.1155/2011/651642 · eudml:229908