Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

In this paper, we study two compact finite difference schemes for the Schrödinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H 1 error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O ( τ 2 + h 4 ) with the time step τ and mesh size h . In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

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

[2]  G. Akrivis,et al.  On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation , 1991 .

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

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

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

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

[8]  Qi Wang,et al.  A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation , 2017, J. Comput. Phys..

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

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

[11]  Chun Zhang,et al.  Efficient mass- and energy-preserving schemes for the coupled nonlinear Schrödinger-Boussinesq system , 2019, Appl. Math. Lett..

[12]  Wang Ting,et al.  Unconditional convergence of two conservative compact difference schemes for non-linear Schrdinger equation in one dimension , 2011 .

[13]  Xiaofei Zhao,et al.  Unconditional and optimal H2-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions , 2018, Adv. Comput. Math..

[14]  Shanshan Wang,et al.  Time-splitting combined with exponential wave integrator fourier pseudospectral method for Schrödinger-Boussinesq system , 2018, Commun. Nonlinear Sci. Numer. Simul..

[15]  Zhi-Zhong Sun,et al.  Error Estimate of Fourth-Order Compact Scheme for Linear Schrödinger Equations , 2010, SIAM J. Numer. Anal..

[16]  Shusen Xie,et al.  Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations , 2011 .

[17]  Tingchun Wang,et al.  Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions , 2013, J. Comput. Phys..

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

[19]  Feng Liao,et al.  Numerical analysis of cubic orthogonal spline collocation methods for the coupled Schrödinger–Boussinesq equations , 2017 .

[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]  Weizhu Bao,et al.  Uniform and Optimal Error Estimates of an Exponential Wave Integrator Sine Pseudospectral Method for the Nonlinear Schrödinger Equation with Wave Operator , 2013, SIAM J. Numer. Anal..

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

[23]  Tingchun Wang,et al.  Unconditional and optimal H1 error estimate of a Crank–Nicolson finite difference scheme for the Gross–Pitaevskii equation with an angular momentum rotation term , 2018 .

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

[25]  Luming Zhang,et al.  Numerical analysis of a conservative linear compact difference scheme for the coupled Schrödinger–Boussinesq equations , 2018, Int. J. Comput. Math..

[26]  Qianshun Chang,et al.  Finite difference method for generalized Zakharov equations , 1995 .

[27]  Weizhu Bao,et al.  Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime , 2011, Numerische Mathematik.