Journal of the Korean Society for Industrial and Applied Mathematics

  • 제24권1호
  • /
  • Pages.39-78
  • /
  • 2020
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

OPTIMAL ERROR ESTIMATE OF A DECOUPLED CONSERVATIVE LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE KLEIN-GORDON-SCHRÖDINGER EQUATIONS

  • YANG, HE (DEPARTMENT OF MATHEMATICS, AUGUSTA UNIVERSITY)
  • 투고 : 2020.01.29
  • 심사 : 2020.03.20
  • 발행 : 2020.03.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we propose a decoupled local discontinuous Galerkin method for solving the Klein-Gordon-Schrödinger (KGS) equations. The KGS equations is a model of the Yukawa interaction of complex scalar nucleons and real scalar mesons. The advantage of our scheme is that the computation of the nucleon and meson field is fully decoupled, so that it is especially suitable for parallel computing. We present the conservation property of our fully discrete scheme, including the energy and Hamiltonian conservation, and establish the optimal error estimate.

키워드

과제정보

연구 과제 주관 기관 : Augusta University

The author acknowledges the valuable comments from anonymous reviewers and the support from Augusta University.

참고문헌

  1. W. Bao and C. Su, Uniform error estimates of a finite difference method for the Klein-Gordon-Schrodinger system in the nonrelativistic and massless limit regimes, Kinetic and Related Models, 11 (2018), 1037-1062. https://doi.org/10.3934/krm.2018040
  2. J. Cai, J. Hong and Y.Wang, Local energy and momentum-preserving schemes for Klein-Gordon-Schrodinger equations and convergence analysis, Numerical Methods for Partial Differential Equations, 33 (2017), 1329-1351. https://doi.org/10.1002/num.22145
  3. L. Kong, M. Chen and X. Yin, A novel kind of efficient symplectic scheme for Klein-Gordon-Schrodinger equation, Applied Numerical Mathematics, 135 (2019), 481-496. https://doi.org/10.1016/j.apnum.2018.09.005
  4. T. Wang, X. Zhao and J. Jiang, Unconditional and optimal $H^2$-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrodinger equation in high dimensions, Advances in Computational Mathematics, 44 (2018), 477-503. https://doi.org/10.1007/s10444-017-9557-5
  5. W. Bao and L. Yang, Efficient and accurate numerical methods for the Klein-Gordon-Schrodinger equations, Journal of Computational Physics, 225 (2007), 1863-1893. https://doi.org/10.1016/j.jcp.2007.02.018
  6. W. Bao and X. Zhao, A uniformly accurate multiscale time integrator Fourier pseudospectral method for the Klein-Gordon-Schrodinger equations in the nonrelativistic limit regime, Numerische Mathematik, 135 (2017), 833-873. https://doi.org/10.1007/s00211-016-0818-x
  7. M. Dehghan and A. Taleei, Numerical solution of the Yukawa-coupled Klein-Gordon-Schrodinger equations via a Chebyshev pseudospectral multidomain method, Applied Mathematical Modelling, 36 (2012), 2340-2349. https://doi.org/10.1016/j.apm.2011.08.030
  8. Q. Hong, Y. Wang and J. Wang, Optimal error estimate of a linear Fourier pseudo-spectral scheme for two dimensional Klein-Gordon-Schrodinger equations, Journal of Mathematical Analysis and Applications, 468 (2018), 817-838. https://doi.org/10.1016/j.jmaa.2018.08.045
  9. L. Kong, J. Zhang, Y. Cao, Y. Duan and H. Huang, Semi-explicit symplectic partitioned Runge-Kutta Fourier pseudo-spectral scheme for Klein-Gordon-Schrodinger equations, Computer Physics Communications, 181 (2010), 1369-1377. https://doi.org/10.1016/j.cpc.2010.04.003
  10. H. Liang, Linearly implicit conservative schemes for long-term numerical simulation of Klein-Gordon-Schrodinger equations, Applied Mathematics and Computation, 238 (2014), 475-484. https://doi.org/10.1016/j.amc.2014.04.032
  11. J. Wang, Y. Wang and D. Liang, Analysis of a Fourier pseudo-spectral conservative scheme for the Klein-Gordon-Schrodinger equation, International Journal of Computer Mathematics, 95 (2018), 36-60. https://doi.org/10.1080/00207160.2017.1366460
  12. J. Hong, S. Jiang and C. Li, Explicit multi-symplectic methods for Klein-Gordon-Schrodinger equations, Journal of Computational Physics, 228 (2009), 3517-3532. https://doi.org/10.1016/j.jcp.2009.02.006
  13. J. Zhang and L. Kong, New energy-preserving schemes for Klein-Gordon-Schrodinger equations, Applied Mathematical Modelling, 40 (2016), 6969-6982. https://doi.org/10.1016/j.apm.2016.02.026
  14. 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-Schrodinger (KGS) equations, Computer and Mathematics with Applications, 71 (2016), 892-921. https://doi.org/10.1016/j.camwa.2015.12.033
  15. S. Wang and L. Zhang, A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schrodinger equations, Applied Mathematics and Computation, 203 (2008), 799-812. https://doi.org/10.1016/j.amc.2008.05.089
  16. H. Yang, Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein-Gordon-Schrodinger equations, Journal of Applied Mathematics and Computing, 62 (2020), 377-424. https://doi.org/10.1007/s12190-019-01289-4
  17. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463. https://doi.org/10.1137/S0036142997316712
  18. J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV-type equations, SIAM Journal on Numerical Analysis, 40 (2002), 769-791. https://doi.org/10.1137/S0036142901390378
  19. J. Yan and C.-W. Shu, Local discontinuous Galerkin methods for partial differential equations with higher order Derivatives, Journal of Scientific Computing, 17 (2002), 27-47. https://doi.org/10.1023/A:1015132126817
  20. C.-S. Chou, W. Sun, Y. Xing and H. Yang, Local discontinuous Galerkin methods for the Khokhlov-Zabolotskaya-Kuznetzov equation, Journal of Scientific Computing, 73 (2017), 593-616. https://doi.org/10.1007/s10915-017-0502-z
  21. Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations, Physica D, 208 (2005), 21-58. https://doi.org/10.1016/j.physd.2005.06.007
  22. Y. Xing, C.-S. Chou and C.-W. Shu, Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Problems and Imaging, 7 (2013), 967-986. https://doi.org/10.3934/ipi.2013.7.967
  23. R. Zhang, X. Yu, M. Li and X. Li, A conservative local discontinuous Galerkin method for the solution of nonlinear Schrodinger equation in two dimensions, Science China Mathematics, 60 (2017), 2515-2530. https://doi.org/10.1007/s11425-016-9118-x
  24. H. Yang, High-order energy and linear momentum conserving methods for the Klein-Gordon equation, Mathematics, 6 (2018), 200. https://doi.org/10.3390/math6100200
  25. H. Yang and F. Li, Error estimates of Runge-Kutta discontinuous Galerkin methods for the Vlasov-Maxwell system, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), 69-99. https://doi.org/10.1051/m2an/2014025
  26. H. Yang, F. Li and J. Qiu, Dispersion and dissipation errors of two fully discrete discontinuous Galerkin methods, Journal of Scientific Computing, 55 (2013), 552-574. https://doi.org/10.1007/s10915-012-9647-y