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Energy Conserving Local Discontinuous Galerkin Methods for the Nonlinear Schrödinger Equation with Wave Operator

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Abstract

In this paper, we present a fully discrete scheme by discretizing the space with the local discontinuous Galerkin method and the time with the Crank–Nicholson scheme to simulate the multi-dimensional Schrödinger equation with wave operator. The scheme can preserve the energy conservation which is an important property of the nonlinear Schrödinger equation with wave operator. The energy conservation is also a crucial property for long time simulations which will be demonstrated in the numerical experiment. The optimal error estimates of the semi-discrete scheme can be obtained for the linear case. Some numerical experiments in multi-dimensional spaces are shown to demonstrate the accuracy and capability of this scheme.

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References

  1. Bao, W.Z., Cai, Y.Y.: Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 20, 492–521 (2012)

    Article  Google Scholar 

  2. Bao, W.Z., Dong, X.C., Xin, J.: Comparisons between sine-Gordon equation and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse. Phys. D 239, 1120–1134 (2010)

    Article  MATH  Google Scholar 

  3. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131, 267–279 (1997)

    Article  MATH  Google Scholar 

  4. Bergé, L., Colin, T.: A singular perturbation problem for an envelope equation in plasma physics. Phys. D 84, 437–459 (1995)

    Article  Google Scholar 

  5. Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    Article  MATH  Google Scholar 

  6. Cockburn, B., Hou, S., Shu, C.-W.: The Runge–Kutta local projection discontinuous Galerkin finnite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)

    MATH  Google Scholar 

  7. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MATH  Google Scholar 

  8. Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    Article  MATH  Google Scholar 

  9. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MATH  Google Scholar 

  10. Dong, B., Shu, C.-W.: Analysis of a local discontinuous Galerkin method for fourth-order time-dependent problems. SIAM J. Numer. Anal. 47, 3240–3268 (2009)

    Article  MATH  Google Scholar 

  11. Fan, K., Cai, W., Ji, X.: A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions. J. Comput. Phys. 227, 2387–2410 (2008)

    Article  MATH  Google Scholar 

  12. Guo, B.L., Liang, H.X.: On the problem of numerical calculation for a class of the system of nonlinear Schrödinger equations with wave operator. J. Numer. Methods Comput. Appl. 4, 176–182 (1983)

    Google Scholar 

  13. Hairer, E., Lubich, C., Wanner, G.: Numerical geometric integration. Unpublished Lecture Notes (1999)

  14. Li, X., Zhang, L.M., Wang, S.S.: A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator. Appl. Math. Comput. 219, 3187–3197 (2012)

    Article  MATH  Google Scholar 

  15. Lu, T., Cai, W., Zhang, P.W.: Conservative local discontinuous Galerkin methods for time dependent Schrödinger equation. Int. J. Anal. Mod. 2, 75–84 (2005)

    MATH  Google Scholar 

  16. Lu, T., Cai, W.: A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schröinger–Poisson equations with discontinuous potentials. J. Comput. Appl. Math. 220, 588–614 (2008)

    Article  MATH  Google Scholar 

  17. Machihara, S., Nakanishi, K., Ozawa, T.: Nonrelativistic limit in the energy space for nonlinear Klein–Gordon equations. Math. Ann. 322, 603–621 (2002)

    Article  MATH  Google Scholar 

  18. Reed, W.H., Hill, T.R.: Triangular mesh method for the neutron transport equation, Technical report LA-UR-73-479. Los Alamos Scientific Laboratory, Los Alamos, NM (1973)

  19. Schoene, A.Y.: On the nonrelativistic limits of the Klein–Gordon and Dirac equations. J. Math. Anal. Appl. 71, 36–47 (1979)

    Article  MATH  Google Scholar 

  20. Shampine, L.F.: Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. 12B, 1287–1296 (1986)

    Article  Google Scholar 

  21. Tsutumi, M.: Nonrelativistic approximation of nonlinear Klein–Gordon equations in two space dimensions. Nonlinear Anal. 8, 637–643 (1984)

    Article  Google Scholar 

  22. Wang, J.: Multisymplectic Fourier pseudospectral method for the nonlinear Schröinger equations with wave operator. J. Comput. Math. 25, 31–48 (2007)

    MATH  Google Scholar 

  23. Wang, S.S., Zhang, L.M., Fan, R.: Discrete-time orthogonal spline collocation methods for the nonlinear Schröinger equation with wave operator. J. Comput. Appl. Math. 235, 1993–2005 (2011)

    Article  MATH  Google Scholar 

  24. Wang, T.C., Zhang, L.M.: Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator. Appl. Math. Comput. 182, 1780–1794 (2006)

    Article  MATH  Google Scholar 

  25. Xin, J.: Modeling light bullets with the two-dimensional sine-Gordon equation. Phys. D 135, 345–368 (2000)

    Article  MATH  Google Scholar 

  26. Xing, Y., Chou, C.-S., Shu, C.-W.: Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Probl. Imaging 7, 967–986 (2013)

    Article  MATH  Google Scholar 

  27. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205, 72–97 (2005)

    Article  MATH  Google Scholar 

  28. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1–46 (2010)

    Google Scholar 

  29. Xu, Y., Shu, C.-W.: Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50, 79–104 (2012)

    Article  MATH  Google Scholar 

  30. Zhang, F., Peréz-Ggarcía, V.M., Vázquez, L.: Numerical simulation of nonlinear Schrödinger equation system: a new conservative scheme. Appl. Math. Comput. 71, 165–177 (1995)

    Article  MATH  Google Scholar 

  31. Zhang, L.M., Li, X.G.: A conservative finite difference scheme for a class of nonlinear Schröinger equation with wave operator. Acta Math. Sci. 22A, 258–263 (2002)

    Google Scholar 

  32. Zhang, L.M., Chang, Q.S.: A conservative numerical scheme for a class of nonlinear Schrödinger with wave operator. Appl. Math. Comput. 145, 603–612 (2003)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Li Guo & Yan Xu

Authors
  1. Li Guo
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  2. Yan Xu
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Correspondence to Yan Xu.

Additional information

Research supported by NSFC Grant Nos. 11371342, 11031007, Fok Ying Tung Education Foundation No. 131003.

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Guo, L., Xu, Y. Energy Conserving Local Discontinuous Galerkin Methods for the Nonlinear Schrödinger Equation with Wave Operator. J Sci Comput 65, 622–647 (2015). https://doi.org/10.1007/s10915-014-9977-z

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  • DOI: https://doi.org/10.1007/s10915-014-9977-z

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