Numerical analysis of a conservative linear compact difference scheme for the coupled Schrödinger–Boussinesq equations

ABSTRACT In this article, a decoupled and linearized compact difference scheme is investigated to solve the coupled Schrödinger–Boussinesq equations numerically. We establish the convergence rates for the error at the order of in the -norm with the time step τ and mesh size h. The linear scheme is proved to conserve the total energy which is defined as a recursion relationship. Due to the difficulty in obtaining the priori estimate from the discrete energy, we utilize cut-off function technique to prove the convergence. The numerical results are reported to verify the theoretical analysis, and the numerical comparison between our scheme with previous methods are conducted to show the efficiency of our scheme.

[1]  Xiaomin Wang,et al.  Application of the extended simplest equation method to the coupled Schrödinger-Boussinesq equation , 2013, Appl. Math. Comput..

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

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

[4]  Ruoxia Yao,et al.  EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHRÖDINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION , 2003 .

[5]  L. Vu-Quoc,et al.  Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation , 1995 .

[6]  Tingchun Wang,et al.  Optimal Point-Wise Error Estimate of a Compact Difference Scheme for the Coupled Gross–Pitaevskii Equations in One Dimension , 2014, J. Sci. Comput..

[7]  Adem Kilicman,et al.  Travelling Wave Solutions of the Schrödinger-Boussinesq System , 2012 .

[8]  Hossein Jafari,et al.  Numerical Solution for Partial Differential Equations , 2018 .

[9]  Wenbin Guo,et al.  Criteria of existence of Hall subgroups in non-soluble finite groups , 2010 .

[10]  Ademir Pastor,et al.  On the periodic Schrödinger–Boussinesq system , 2009, 0905.4072.

[11]  Tingchun Wang,et al.  Optimal l∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions , 2014 .

[12]  Boling Guo,et al.  Existence of the Periodic Solution for the Weakly Damped Schrödinger–Boussinesq Equation , 2001 .

[13]  Junkichi Satsuma,et al.  Soliton Solutions in a Diatomic Lattice System , 1979 .

[14]  N. N. Rao,et al.  Exact solutions of coupled scalar field equations , 1989 .

[15]  Luming Zhang,et al.  The quadratic B-spline finite-element method for the coupled Schrödinger–Boussinesq equations , 2011, Int. J. Comput. Math..

[16]  Lang-Yang Huang,et al.  Multi-symplectic scheme for the coupled Schrödinger—Boussinesq equations , 2013 .

[17]  N. N. Rao.  Coupled scalar field equations for nonlinear wave modulations in dispersive media , 1996 .

[18]  Boling Guo,et al.  The behavior of attractors for damped Schrödinger-Boussinesq equation , 2001 .

[19]  Luming Zhang,et al.  Numerical analysis for a conservative difference scheme to solve the Schrödinger-Boussinesq equation , 2011, J. Comput. Appl. Math..

[20]  Weizhu Bao,et al.  Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation , 2012, Math. Comput..

[21]  Dongmei Bai,et al.  The time-splitting Fourier spectral method for the coupled Schrödinger-Boussinesq equations , 2012 .

[22]  Weizhu Bao,et al.  Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator , 2012, SIAM J. Numer. Anal..

[23]  Feng Liao,et al.  Conservative compact finite difference scheme for the coupled Schrödinger–Boussinesq equation , 2016 .

[24]  V. G. Makhankov,et al.  On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying boussinesq's equation , 1974 .

[25]  Wu Zhuan-bao.  The Numerical Solution of A Partial Differential Equation , 2007 .

[26]  Yongsheng Li,et al.  Finite Dimensional Global Attractor for Dissipative Schrödinger–Boussinesq Equations , 1997 .