A new conservative high-order accurate difference scheme for the Rosenau equation

References

• Rosenau P. Dynamics of dense discrete systems. Progr. Theor. Phys. 1988;79:1028–1042.
   Web of Science ®Google Scholar
• Park MA. On the Rosenau equation in multidimensional space. Nonlinear Anal. Theory Methods Appl. 1993;21:77–85.
   Web of Science ®Google Scholar
• Chung SK, Pani AK. Numerical methods for the Rosenau equation. Appl. Anal. 2001;77:351–369.
   Google Scholar
• Chung SK, Ha SN. Finite element Galerkin solutions for the Rosenau equation. Appl. Anal. 1994;54:39–56.
   Google Scholar
• Chung SK, Pani AK. A second order splitting lumped mass finite element method for the Rosenau equation. Differen. Equat. Dyn. Syst. 2004;12:331–351.
   Google Scholar
• Manickam SA, Pani AK, Chung SK. A second order splitting combined with cubic spline collacation method for the Rosenau equation. Numer. Methods Partial Differ. Equ. 1998;14:695–716.
   Web of Science ®Google Scholar
• Kim YD, Lee HY. The convergence of finite element Galerkin solution for the Rosenau equation. Korean J. Comput. Appl. Math. 1998;5:171–180.
   Google Scholar
• Lee HY, Ahn MJ. The convergence of the fully discrete solution for the Rosenau equation. Comput. Math. Appl. 1996;32:15–22.
   Web of Science ®Google Scholar
• Atouani N, Omrani K. Galerkin finite element method for the Rosenau-RLW equation. Comut. Math. App. 2013;66:289–303.
   Web of Science ®Google Scholar
• Zhang L, Chang Q. A conservative numerical scheme for nonlinear Schrödinger with wave operator. Appl. Math. Comput. 2003;145:603–612.
   Web of Science ®Google Scholar
• Zhang L. A finite difference scheme for generalized regularized long-wave equation. Appl. Math. Comput. 2005;168:472–962.
   Web of Science ®Google Scholar
• Wang T, Zhang L. Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator. Appl. Math. Comput. 2006;182:1780–1794.
   Web of Science ®Google Scholar
• Chang Q, Xu L. A numerical method for a system of generalized nonlinear Schrödinger equations. J. Comput. Math. 1986;4:191–199.
   Web of Science ®Google Scholar
• Chang Q, Jia E, Sun W. Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 1999;148:397–415.
   Web of Science ®Google Scholar
• Chang Q, Wan G, Guo B. Conserving scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary notion. J. Comput. Phys. 1991;93:360–375.
   Web of Science ®Google Scholar
• Zhang L, Chang Q. A new finite difference method for regularized long-wave equation. Chin. J. Numer. Method Comput. Appl. 2001;23:58–66.
   Google Scholar
• Achouri T, Khiari N, Omrani K. On the convergence of difference schemes for the Benjamin-Bona-Mahony (BBM) equation. Appl. Math. Comput. 2006;182:999–1005.
   Web of Science ®Google Scholar
• Wang T, Zhang L, Chen F. Conservative schemes for the symmetric regularized long wave equations. Appl. Math. Comput. 2007;190:1062–1080.
   Web of Science ®Google Scholar
• Zhang F, Va’zquez L. Two energy conserving numerical schemes for the Sine-Gordon equation. Appl. Math. Comput. 1991;45:17–30.
   Web of Science ®Google Scholar
• Ben-Yu Guo. P.J. Pascual, M.J. Rodriguez, L. Va’zquez, Numerical solution of the sine-Gordon equation. Appl. Math. Comput. 1986;18:1–14.
   Web of Science ®Google Scholar
• Wong YS, Chang Q, Gong L. A initial-boundary value problem of a nonlinear Klein-Gordon equation. Appl. Math. Comput. 1997;84:77–93.
   Web of Science ®Google Scholar
• Chang Q, Jiang H. A conservative difference scheme for the Zakharov equations. J. Comput. Phys. 1994;113:309–319.
   Web of Science ®Google Scholar
• Chang Q, Guo B, Jiang H. Finite difference method for the generalized Zakharov equations. Math. Comput. 1995;64:537–553.
   Web of Science ®Google Scholar
• Choo SM, Chung SK. Conservative nonlinear difference scheme for the Cahn-Hilliard equation. Comput. Math. Appl. 1998;36:31–39.
   Web of Science ®Google Scholar
• Choo SM, Chung SK, Kim KI. Conservative nonlinear difference scheme for the Cahn-Hilliard equation II. Comput. Math. Appl. 2000;39:229–243.
   Web of Science ®Google Scholar
• Furihata D. A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 2001;87:675–699.
   Web of Science ®Google Scholar
• Furihata D, Mastuo T. A stable convergent, conservative and linear finite difference scheme for the Cahn-Hilliard equation. Jpn J. Indust. Appl. Math. 2003;20:65–85.
   Web of Science ®Google Scholar
• Dehghan M, Manafian J. Application of semi-analytical methods for solving the Rosenau-Hyman equation arising in the pattern formation in liquid drops. Int. J. Numer. Methods Heat Fluid Flow. 2012;23:777–790.
   Web of Science ®Google Scholar
• Dehghan M, Salehi R. The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas. Comput. Phys. Comm. 2011;182:2540–2549.
   Web of Science ®Google Scholar
• Dehghan M. Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simulat. 2006;71:16–30.
   Web of Science ®Google Scholar
• Shokri A, Dehghan M. A meshless method the using radial basis functions for numerical solution of the regularized long wave equation. Numer. Methods Partial Differ. Equ. 2010;26:807–825.
   Web of Science ®Google Scholar
• Omrani K, Abidi F, Achouri T, Khiari N. A new conservative finite difference scheme for the Rosenau equation. Appl. Math. Comput. 2008;201:35–43.
   Web of Science ®Google Scholar
• Chung SK. Finite difference approximate solutions for the Rosenau equation. Appl. Anal. 1998;69:149–156.
   Google Scholar
• Berikelashvili G, Gupta MM, Mirianashvili M. Convergence of fourth order compact difference schemes for three-dimensional convection-diffusion equations. SIAMJ. Numer. Anal. 2007;45:443–455.
   Web of Science ®Google Scholar
• Cohen G. High-Order Numerical Methods for Transient Wave Equations. New York (NY): Springer; 2002.
   Google Scholar
• Dai W. An improved compact finite difference scheme for solving an N-carrier system with Neumann boundary conditions. Numer. Methods Partial Differ. Equ. 2011;27:436–446.
   Web of Science ®Google Scholar
• Dai W, Tzou DY. A fourth-order compact finite difference scheme for solving an N-carrier system with Neumann boundary conditions. Numer. Methods Partial Differ. Equ. 2010;25:274–289.
   Web of Science ®Google Scholar
• Dehghan M, Taleei A. A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput. Phys. Commun. 2010;181:43–51.
   Web of Science ®Google Scholar
• Dehghan M, Mohebbi A, Asgari Z. Fourth-order compact solution of the nonlinear Klein-Gordon equation. Numer. Algorithms. 2009;52:523–540.
   Web of Science ®Google Scholar
• Gao Z, Xie S. Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations. Appl. Numer. Math. 2011;61:593–614.
   Web of Science ®Google Scholar
• Gustafsson B, Mossberg E. Time compact high order difference methods for wave propagation. SIAM J. Sci. Comput. 2004;26:259–271.
   Web of Science ®Google Scholar
• Gustafsson B, Wahlund P. Time compact difference methods for wave propagation in discontinuous media. SIAM J. Sci. Comput. 2004;26:272–293.
   Web of Science ®Google Scholar
• Liao H, Sun Z. Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Partial Differ. Equ. 2010;26:37–60.
   Web of Science ®Google Scholar
• Liao H, Sun Z. Error estimate of fourth-order compact scheme for linear Schrödinger equations. SIAM J. Numer. Anal. 2010;47:4381–4401.
   Web of Science ®Google Scholar
• Mohebbi A, Dehghan M. High-order solution of one-dimensional Sine-Gordon equation using compact finite difference and DIRKN methods. Math. Comput. Model. 2010;51:537–549.
   Web of Science ®Google Scholar
• Mohebbi A, Dehghan M. High-order compact solution of the one-dimensional heat and advection-diffusion equations. Appl. Math. Model. 2010;34:3071–3084.
   Web of Science ®Google Scholar
• Sun Z. Compact difference schemes for heat equation with Neumann boundary conditions. Numer. Methods Partial Differ. Equ. 2009;25:1320–1341.
   Web of Science ®Google Scholar
• Xie S, Li G, Yi S. Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödnger equation. Comput. Methods Appl. Mech. Eng. 2009;198:1052–1060.
   Web of Science ®Google Scholar
• Wang T, Guo B, Xu Q. Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. Comput. Phys. 2013;243:382–399.
   Google Scholar
• Karaa S, Zhang J. High-order ADI method for solving unsteady convection-diffusion problems. J. Comput. Phys. 2004;198:1–9.
   Web of Science ®Google Scholar
• Karaa S. A high-order compact ADI method for solving three-dimentional unsteady convection-diffusion problems. Numer. Methods Partial Differ. Equ. 2006;22:983–993.
   Web of Science ®Google Scholar
• Karaa S. High-order difference schemes for 2D elliptic and parabolic problems with mixed derivatives. Numer. Methods Partial Differ. Equ. 2007;23:366–378.
   Web of Science ®Google Scholar
• Karaa S. High-order ADI method for stream-function vorticity equations. Proc. Appl. Math. Mech. 2007;7:1025601–1025602.
   Google Scholar
• Zhou Y. Application of Discrete Functional Analysis to the Finite Difference Methods. Beijing: International Academic Publishers; 1990.
   Google Scholar
• Browder FE. Existence and uniqueness theorems for solutions of nonlinear boundary value problems. Applications of nonlinear partial differential equation. In: Finn R, editor. Proceedings of Symposia Applied Mathematics. Vol. 17, Providence: AMS; 1965; p. 24–49.
   Google Scholar