Conservative Fourier pseudo-spectral schemes for general Klein–Gordon–Schrödinger equations

References

• G. Akrivis, V. Dougalis and O. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math. 59 (1991), pp. 31–53. doi: 10.1007/BF01385769
   Web of Science ®Google Scholar
• Z. Asgari and S. Hosseini, Numerical solution of two-dimensional sine-Gordon and MBE models using Fourier spectral and high order explicit time stepping methods, Comput. Phys. Commun. 184 (2013), pp. 565–572. doi: 10.1016/j.cpc.2012.10.009
   Web of Science ®Google Scholar
• W. Bao and Y. Cai, Optimal error estimates of finite difference methods for the Gross–Pitaevskii equation with angular momentum rotation, Math. Comput. 82 (281) (2013), pp. 99–128. doi: 10.1090/S0025-5718-2012-02617-2
   Web of Science ®Google Scholar
• C. Bardos and E. Tadmor, Stability and spectral convergence of Fourier method for nonlinear problems: On the shortcomings of the 2/3 dealiasing method, Numer. Math. 129 (2015), pp. 749–782. doi: 10.1007/s00211-014-0652-y
   Web of Science ®Google Scholar
• D. Bai and J. Wang, The time-splitting Fourier spectral method for the coupled Schrödinger–Boussinesq equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), pp. 1201–1210. doi: 10.1016/j.cnsns.2011.08.012
   Web of Science ®Google Scholar
• W. Bao and L. Yang, Efficient and accurate numerical methods for the Klein–Gordon–Schrödinger equations, J. Comput. Phys. 225 (2007), pp. 1863–1893. doi: 10.1016/j.jcp.2007.02.018
   Web of Science ®Google Scholar
• W. Bao and X. Zhao, A uniformly accurate (UA) multiscale time integrator Fourier pseudospectral method for the Klein–Gordon–Schrödinger equations in the nonrelativistic limit regime a UA method for Klein–Gordon–Schrödinger equation, Numer. Math. 135 (2017), pp. 833–873. doi: 10.1007/s00211-016-0818-x
   Web of Science ®Google Scholar
• P. Biler, Attractors for the system of Schrödinger and Klein–Gordon equations with Yukawa coupling, SIAM J. Math. Anal. 21 (5) (1990), pp. 1190–1212. doi: 10.1137/0521065
   Web of Science ®Google Scholar
• M. Cavalcanti and V. Cavalcanti, Global existence and uniform Klein–Gordon–Schrödinger equations, Nonlinear Differ. Equ. Appl. 7 (2000), pp. 285–307. doi: 10.1007/PL00001426
   Web of Science ®Google Scholar
• M. Dehghan and V. Mohammadi, 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, Comput. Math. Appl. 71 (2016), pp. 892–921. doi: 10.1016/j.camwa.2015.12.033
   Web of Science ®Google Scholar
• I. Fukuda and M. Tsutsumi, On coupled Klein–Gordon–Schrödinger equations II, J. Math. Anal. Appl. 66 (2) (1973), pp. 358–378. doi: 10.1016/0022-247X(78)90239-1
   Google Scholar
• I. Fukuda and M. Tsutsumi, On coupled Klein–Gordon–Schrödinger equation. III. Higher order interaction, decay and blow-up, Jpn. Math. 24 (80) (1979), pp. 307–321.
   Google Scholar
• Y. Gong, Q. Wang, Y. Wang and J. Cai, A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation, J. Comput. Phys. 328 (2017), pp. 354–370. doi: 10.1016/j.jcp.2016.10.022
   Web of Science ®Google Scholar
• B. Guo and Y. Li, Attractor for dissipative Klein–Gordon–Schrödinger equations in R3, J. Differ. Equ. 136 (1997), pp. 356–377. doi: 10.1006/jdeq.1996.3242
   Web of Science ®Google Scholar
• N. Hayashi, Global strong solution of coupled Klein–Gordon–Schrödinger equations, Funkcialaj Ekvacioj 29 (1986), pp. 299–307.
   Google Scholar
• J. Hong, S. Jiang and C. Li, Explicit multi-symplectic methods for Klein–Gordon–Schrödinger equations, J. Comput. Phys. 228 (9) (2009), pp. 3517–3532. doi: 10.1016/j.jcp.2009.02.006
   Web of Science ®Google Scholar
• L. Kong, J. Hong and R. Liu, Long-term numerical simulation of the interaction between a neutron field and a neutral meson field by a symplectic-preserving scheme, J. Phys. A – Math. Theor. 41 (2008), pp. 1–17. doi: 10.1088/1751-8113/41/25/255207
   Web of Science ®Google Scholar
• L. Kong, L. Wang, S. Jiang and Y. Duan, Multi-symplectic Fourier pseudo-spectral integrators for Klein–Gordon–Schrödinger equations, Sci. China Math. 56 (5) (2013), pp. 915–932. doi: 10.1007/s11425-013-4575-3
   Web of Science ®Google Scholar
• L. Kong, J. Zhang, Y. Cao, Y. Duan and H. Huang, Semi-explicit symplectic partitioned Runge–Kutta Fourier pseudo-spectral scheme for Klein–Gordon–Schrödinger equations, Comput. Phys. Commun. 181 (2010), pp. 1369–1377. doi: 10.1016/j.cpc.2010.04.003
   Web of Science ®Google Scholar
• H. Lee and J. Lee, A semi-analytical Fourier spectral method for the Allen–Cahn equation, Comput. Math. Appl. 68 (2014), pp. 174–184. doi: 10.1016/j.camwa.2014.05.015
   Web of Science ®Google Scholar
• O. Masahito, Stability of stationary states for the coupled Klein–Gordon–Schrödinger equations, Nonlinear Anal. Theory Method Appl. 27 (4) (1996), pp. 455–461. doi: 10.1016/0362-546X(95)00017-P
   Web of Science ®Google Scholar
• C. Miao and G. Xu, Global solution of the Klein–Gordon–Schrödinger system with rough data in R2+1, J. Differ. Equ. 227 (2006), pp. 365–405. doi: 10.1016/j.jde.2005.10.012
   Web of Science ®Google Scholar
• C. Miao and Y. Zhu, Low regularity global well-posedness for the Klein–Gordon–Schrödinger system with the higher-order Yukawa coupling, Differ. Integr. Equ. 20 (2007), pp. 643–656.
   Web of Science ®Google Scholar
• F. Natali and A. Pastor, Stability properties of periodic standing wave for Klein–Gordon–Schrödinger system, Commun. Pure Appl. Anal. 9 (2) (2017), pp. 413–430. doi: 10.3934/cpaa.2010.9.413
   Google Scholar
• X. Pan and L. Zhang, High-order linear compact conservative method for the nonlinear Schrödinger equation coupled with the nonlinear Klein–Gordon equation, Nonlinear Anal. 92 (2013), pp. 108–118. doi: 10.1016/j.na.2013.07.003
   Web of Science ®Google Scholar
• E. Pindza and K. Owolabi, Fourier spectral method for higher order space fractional reaction–diffusion equations, Commun. Nonlinear Sci. Numer. Simul. 40 (2016), pp. 112–128. doi: 10.1016/j.cnsns.2016.04.020
   Web of Science ®Google Scholar
• J. Shen, T. Tang and L. Wang, Spectral Methods Algorithms, Analysis and Applications, Springer, 2011.
   Google Scholar
• Q. Shi, W. Li and S. Wang, Well posedness in energy space for the nonlinear Klein–Gordon–Schrödinger system, Appl. Math. Comput. 251 (2015), pp. 55–64.
   Web of Science ®Google Scholar
• Q. Shi, S. Wang and Y. Li, Existence and uniqueness of energy solution to Klein–Gordon–Schrödinger equations, J. Differ. Equ. 252 (1) (2012), pp. 168–180. doi: 10.1016/j.jde.2011.09.025
   Web of Science ®Google Scholar
• Z. Sun, Numerical Methods of the Partial Differential Equation, Science Press, China, 2005.
   Google Scholar
• Z. Sun and Q. Zhu, On Tsertsvadze's difference scheme for the Kuramoto–Tsuzuki equation, J. Comput. Appl. Math. 98 (2) (1998), pp. 289–300. doi: 10.1016/S0377-0427(98)00135-6
   Web of Science ®Google Scholar
• L. Wang, Y. Ma, L. Kong and Y. Duan, Symplectic Fourier pseudo-spectral schemes for Klein–Gordon–Schrödinger equation, China J. Comput. Phys. 28 (2) (2011), pp. 275–282.
   Google Scholar
• J. Wang, Y. Wang and D. Liang, Analysis of a Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger equation, Int. J. Comput. Math. 95 (2018), pp. 36–60. doi: 10.1080/00207160.2017.1366460
   Web of Science ®Google Scholar
• T. Wang, X. Zhao and J. Jiang, Unconditional and optimal H2-error estimates of two linear and conservative finite difference schemes for the Klein–Gordon–Schrödinger equation in high dimensions, Adv. Comput. Math. 44 (2017), pp. 477–503. doi: 10.1007/s10444-017-9557-5
   Web of Science ®Google Scholar
• S. Wang and L. Zhang, A class of conservative orthogonal spline collocation schemes for solving coupled Klein–Gordon–Schrödinger equations, Appl. Math. Comput. 203 (2008), pp. 799–812.
   Web of Science ®Google Scholar
• T. Wang and X. Zhao, Optimal l∞ error estimates of finite difference methods for the coupled Gross–Pitaevskii equations in high dimensions, Sci. China Math. 57 (2014), pp. 2189–2214. doi: 10.1007/s11425-014-4773-7
   Web of Science ®Google Scholar
• M. Wang and Y. Zhou, The periodic wave solutions for the Klein–Gordon–Schrödinger equations, Phys. Lett. A 318 (1) (2003), pp. 84–92. doi: 10.1016/j.physleta.2003.07.026
   Web of Science ®Google Scholar
• B. Wang, Classical global solution for non-linear Klein–Gordon–Schrödinger equation, Math. Methods Appl. Sci. 20 (1997), pp. 599–616. doi: 10.1002/(SICI)1099-1476(19970510)20:7<599::AID-MMA866>3.0.CO;2-7
   Web of Science ®Google Scholar
• T. Wang, Optimal point-wise error estimate of a compact difference scheme for the Klein–Gordon–Schrödinger equation, J. Math. Anal. Appl. 412 (2014), pp. 155–167. doi: 10.1016/j.jmaa.2013.10.038
   Web of Science ®Google Scholar
• J. Wang, Conservative Fourier spectral scheme for the coupled Schrödinger–Boussinesq equations, Adv. Differ. Equ. 2018 (2018), pp. 1–19. doi: 10.1186/s13662-017-1452-3
   Google Scholar
• P. Xanthopoulos and E. Georgios, A linearly implicit finite difference method for a Klein–Gordon–Schrödinger system modeling electron ion plasma waves, Discrete Contin. Dyn. Syst. Ser. B 10 (1) (2008), pp. 239–263. doi: 10.3934/dcdsb.2008.10.239
   Web of Science ®Google Scholar
• J. Xia, S. Han and M. Wang, The exact solitary wave solution for the Klein–Gordon–Schrödinger equations, Appl. Math. Mech. 23 (1) (2002), pp. 52–58.
   Google Scholar
• Y. Xia, Fourier spectral methods for Degasperis–Procesi equation with discontinuous solutions, J. Sci. Comput. 61 (2014), pp. 584–603. doi: 10.1007/s10915-014-9839-8
   Web of Science ®Google Scholar
• J. Zhang and L. Kong, New energy-preserving schemes for Klein–Gordon–Schrödinger equations, Appl. Math. Model. 40 (2016), pp. 6969–6982. doi: 10.1016/j.apm.2016.02.026
   Web of Science ®Google Scholar
• L. Zhang, Convergence of a conservative difference scheme for a class of Klein–Gordon–Schrödinger equations in one space dimension, Appl. Math. Comput. 163 (2005), pp. 343–355.
   Web of Science ®Google Scholar