Finite difference discretization of the Rosenau-RLW equation

Abstract In this paper, numerical solutions of the Rosenau-RLW equation are considered using Crank–Nicolson type finite difference method. Existence of the numerical solutions is derived by Brouwer fixed point theorem. A priori bound and the error estimates as well as conservation of discrete mass and discrete energy for the finite difference solutions are discussed. Unconditional stability, second-order convergence and uniqueness of the scheme are proved using discrete energy method. Some numerical experiments have been conducted in order to verify the theoretical results.

[1]  İdris Daǧ.  Least-squares quadratic B-spline finite element method for the regularised long wave equation , 2000 .

[2]  Qianshun Chang,et al.  Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion , 1991 .

[3]  Ting-chun Wang,et al.  Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator , 2006, Appl. Math. Comput..

[4]  Qianshun Chang,et al.  Difference Schemes for Solving the Generalized Nonlinear Schrödinger Equation , 1999 .

[5]  Abdulkadir Dogan,et al.  Numerical solution of RLW equation using linear finite elements within Galerkin's method , 2002 .

[6]  Kelong Zheng,et al.  Two Conservative Difference Schemes for the Generalized Rosenau Equation , 2010 .

[7]  S. K. Chung,et al.  FINITE DIFFERENCE APPROXIMATE SOLUTIONS FOR THE ROSENAU EQUATION , 1998 .

[8]  Philip Rosenau,et al.  Dynamics of Dense Discrete Systems High Order Effects , 1988 .

[9]  Amiya K. Pani,et al.  Numerical methods for the rosenau equation , 2001 .

[10]  D. Peregrine.  Calculations of the development of an undular bore , 1966, Journal of Fluid Mechanics.

[11]  Luming Zhang,et al.  A finite difference scheme for generalized regularized long-wave equation , 2005, Appl. Math. Comput..

[12]  Qianshun Chang,et al.  A Conservative Difference Scheme for the Zakharov Equations , 1994 .

[13]  Felix E. Browder,et al.  Existence and uniqueness theorems for solutions of nonlinear boundary value problems , 1965 .

[14]  Luming Zhang.  Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space dimension , 2005, Appl. Math. Comput..

[15]  Luming Zhang,et al.  New conservative difference schemes for a coupled nonlinear Schrödinger system , 2010, Appl. Math. Comput..

[16]  Khaled Omrani,et al.  A new conservative finite difference scheme for the Rosenau equation , 2008, Appl. Math. Comput..

[17]  Bülent Saka,et al.  Galerkin method for the numerical solution of the RLW equation using quintic B-splines , 2006 .

[18]  S. M. Choo,et al.  A discontinuous Galerkin method for the Rosenau equation , 2008 .

[19]  D. Peregrine.  Long waves on a beach , 1967, Journal of Fluid Mechanics.

[20]  Edward H. Twizell,et al.  A linearized implicit pseudo‐spectral method for some model equations: the regularized long wave equations , 2003 .

[21]  Amiya K. Pani,et al.  A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation , 1998 .

[22]  Z. Fei,et al.  Numerical simulation of nonlinear Schro¨dinger systems: a new conservative scheme , 1995 .

[23]  S. M. Choo,et al.  Conservative nonlinear difference scheme for the Cahn-Hilliard equation—II , 1998 .

[24]  K. R. Raslan,et al.  A finite difference scheme for the MRLW and solitary wave interactions , 2007, Appl. Math. Comput..

[25]  Luming Zhang,et al.  On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation , 2012 .