Energy-preserving finite volume element method for the improved Boussinesq equation

Abstract In this paper, we design an energy-preserving finite volume element scheme for solving the initial boundary problems of the improved Boussinesq equation. Theoretical analysis shows that the proposed numerical schemes can conserve the energy and mass. Numerical experiments are performed to illustrate the efficiency of the scheme and theoretical analysis. While the results demonstrate that the proposed finite volume element scheme is second-order accuracy in space and time. Moreover, the new scheme can conserve mass and energy.

[1]  A. G. Bratsos.  A predictor–corrector scheme for the improved Boussinesq equation , 2009 .

[2]  Labib Iskandar,et al.  Numerical solutions of the improved Boussinesq equation , 1980 .

[3]  N. Zabusky,et al.  Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States , 1965 .

[4]  Zhiyue Zhang.  Error estimates of finite volume element method for the pollution in groundwater flow , 2009 .

[5]  Jinchao Xu,et al.  Analysis of linear and quadratic simplicial finite volume methods for elliptic equations , 2009, Numerische Mathematik.

[7]  Takayasu Matsuo,et al.  New conservative schemes with discrete variational derivatives for nonlinear wave equations , 2007 .

[8]  Yuto Miyatake,et al.  Conservative finite difference schemes for the Degasperis-Procesi equation , 2012, J. Comput. Appl. Math..

[9]  Guang Lin,et al.  Numerical studies of the stochastic Korteweg-de Vries equation , 2006, J. Comput. Phys..

[10]  Quanxiang Wang,et al.  A Fourier finite volume element method for solving two-dimensional quasi-geostrophic equations on a sphere , 2013 .

[11]  Qun Lin,et al.  Linear B-spline finite element method for the improved Boussinesq equation , 2009 .

[12]  H. El-Zoheiry,et al.  Numerical study of the improved Boussinesq equation , 2002 .

[13]  Numerical simulation of the stochastic damped improved Boussinesq equation , 2013 .

[14]  Zhiyue Zhang,et al.  Quadratic finite volume element method for the improved Boussinesq equation , 2012 .

[15]  A. G. Bratsos.  A second order numerical scheme for the improved Boussinesq equation , 2007 .

[16]  I. Bogolubsky,et al.  Some examples of inelastic soliton interaction , 1977 .

[17]  Daisuke Furihata,et al.  Nonlinear and linear conservative finite difference schemes for regularized long wave equation , 2009 .

[18]  Daisuke Furihata,et al.  A stable and conservative finite difference scheme for the Cahn-Hilliard equation , 2001, Numerische Mathematik.

[19]  Idris Dag,et al.  Numerical simulations of the improved Boussinesq equation , 2010 .

[20]  Miki Wadati.  Stochastic Korteweg-de Vries Equation , 1983 .

[21]  P. Christiansen,et al.  Solitary wave solutions to a system of Boussinesq-like equations , 1992 .

[22]  Anjan Biswas,et al.  Solitary waves of Boussinesq equation in a power law media , 2009 .