Analysis of a Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger equation

ABSTRACT In this paper, we focus on constructing and analysing a new Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger (KGS) equation. After rewriting the KGS equation as an infinite-dimensional Hamiltonian system, we use a Fourier pseudo-spectral method to discrete the system in space to obtain a semi-discrete system, which can be cast into a canonical finite-dimensional Hamiltonian form. Then, an energy-preserving and charge-preserving scheme is constructed by using the symmetric discrete gradient method. Based on the discrete conservation laws and the equivalence of the semi-norm between the Fourier pseudo-spectral method and the finite difference method, the pseudo-spectral solution of the proposed scheme is proved to be bounded in the discrete norm. The proposed scheme is shown to be convergent with the convergence order of in the discrete norm afterwards, where J is the number of nodes and τ is the time step size. Numerical experiments are conducted to verify the theoretical analysis.

[1]  M. Qin,et al.  MULTI-SYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHR ¨ ODINGER EQUATION , 2001 .

[2]  Xiang Xinmin,et al.  Spectral method for solving the system of equations of Schro¨dinger-Klein-Gordon field , 1988 .

[3]  Ameneh Taleei,et al.  A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions , 2014 .

[4]  Yushun Wang,et al.  A conservative Fourier pseudospectral method for the nonlinear Schrödinger equation , 2016 .

[5]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

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

[7]  Wolf von Wahl,et al.  On the global strong solutions of coupled Klein-Gordon-Schrödinger equations , 1987 .

[8]  M. Dehghan,et al.  Numerical solution of the Yukawa-coupled Klein–Gordon–Schrödinger equations via a Chebyshev pseudospectral multidomain method , 2012 .

[9]  Li Yang,et al.  Efficient and accurate numerical methods for the Klein-Gordon-Schrödinger equations , 2007, J. Comput. Phys..

[10]  Ameneh Taleei,et al.  Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations , 2014, Comput. Phys. Commun..

[11]  O. Gonzalez.  Time integration and discrete Hamiltonian systems , 1996 .

[12]  Ruxun Liu,et al.  Numerical simulation of interaction between Schrödinger field and Klein-Gordon field by multisymplectic method , 2006, Appl. Math. Comput..

[13]  L. Kong,et al.  New energy-preserving schemes for Klein–Gordon–Schrödinger equations , 2016 .

[14]  Bin Wang,et al.  Local structure-preserving algorithms for partial differential equations , 2008 .

[15]  Tingchun Wang,et al.  Point-wise errors of two conservative difference schemes for the Klein–Gordon–Schrödinger equation , 2012 .

[16]  Yushun Wang,et al.  Multi-symplectic Fourier pseudospectral method for the Kawahara equation , 2014 .

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

[19]  Mingliang Wang,et al.  The periodic wave solutions for the Klein–Gordon–Schrödinger equations , 2003 .

[20]  Mehdi Dehghan,et al.  Two numerical meshless techniques based on radial basis functions (RBFs) and the method of generalized moving least squares (GMLS) for simulation of coupled Klein-Gordon-Schrödinger (KGS) equations , 2016, Comput. Math. Appl..

[21]  Tingchun Wang,et al.  Optimal point-wise error estimate of a compact difference scheme for the Klein–Gordon–Schrödinger equation , 2014 .

[22]  M. Qin,et al.  Symplectic Geometric Algorithms for Hamiltonian Systems , 2010 .

[23]  Jie Shen,et al.  Spectral and High-Order Methods with Applications , 2006 .

[24]  Foek T Hioe,et al.  Periodic solitary waves for two coupled nonlinear Klein–Gordon and Schrödinger equations , 2003 .

[25]  Mehdi Dehghan,et al.  Numerical Solution of System of N–Coupled NonlinearSchrödinger Equations via Two Variants of the MeshlessLocal Petrov–Galerkin (MLPG) Method , 2014 .

[26]  Luming Zhang,et al.  A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schrödinger equations , 2008, Appl. Math. Comput..

[27]  Masahito Ohta,et al.  Stability of stationary states for the coupled Klein-Gordon-Schro¨dinger equations , 1996 .

[28]  V. G. Makhankov,et al.  Dynamics of classical solitons (in non-integrable systems) , 1978 .

[29]  Ashraf Darwish,et al.  A series of new explicit exact solutions for the coupled Klein–Gordon–Schrödinger equations , 2004 .

[30]  Shanshan Jiang,et al.  Explicit multi-symplectic methods for Klein-Gordon-Schrödinger equations , 2009, J. Comput. Phys..

[31]  S. Reich,et al.  Multi-symplectic spectral discretizations for the Zakharov–Kuznetsov and shallow water equations , 2001 .

[32]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[34]  Yushun Wang,et al.  A local energy-preserving scheme for Klein–Gordon–Schrödinger equations* , 2015 .

[35]  Masayoshi Tsutsumi,et al.  On coupled Klein-Gordon-Schrödinger equations, II , 1978 .

[36]  S. Reich,et al.  Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity , 2001 .

[37]  A. Quarteroni,et al.  Approximation results for orthogonal polynomials in Sobolev spaces , 1982 .

[38]  Jialin Hong,et al.  Numerical comparison of five difference schemes for coupled Klein -Gordon -Schrödinger equations in quantum physics , 2007 .

[39]  Mehdi Dehghan,et al.  The solitary wave solution of coupled Klein-Gordon-Zakharov equations via two different numerical methods , 2013, Comput. Phys. Commun..

[40]  Tohru Ozawa,et al.  Asymptotic Behavior of Solutions for the Coupled Klein–Gordon–Schrödinger Equations , 1994 .

[41]  Zhi-zhong Sun,et al.  On Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation , 1998 .