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Optimal l ∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

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Abstract

Due to the difficulty in obtaining the a priori estimate, it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions (2D or 3D). We here propose and analyze finite difference methods for solving the coupled Gross-Pitaevskii equations in two dimensions, which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction. The methods which we considered include two conservative type schemes and two non-conservative type schemes. Discrete conservation laws and solvability of the schemes are analyzed. For the four proposed finite difference methods, we establish the optimal convergence rates for the error at the order of O(h 2 +τ 2) in the l ∞-norm (i.e., the point-wise error estimates) with the time step τ and the mesh size h. Besides the standard techniques of the energy method, the key techniques in the analysis is to use the cut-off function technique, transformation between the time and space direction and the method of order reduction. All the methods and results here are also valid and can be easily extended to the three-dimensional case. Finally, numerical results are reported to confirm our theoretical error estimates for the numerical methods.

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References

  1. Akrivis G, Dougalis V, Karakashian O. On fully discrete Galerkin methods of secondorder temporal accuracy for the nonlinear Schrödinger equation. Numer Math, 1991, 59: 31–53

    Article  MATH  MathSciNet  Google Scholar 

  2. Bao W, Cai Y. Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction. East Asian J Appl Math, 2011, 1: 49–81

    MATH  MathSciNet  Google Scholar 

  3. Bao W, Cai Y. Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J Numer Anal, 2012, 50: 492–521

    Article  MATH  MathSciNet  Google Scholar 

  4. Bao W, Cai Y. Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation. Math Comput, 2013, 82: 99–128

    Article  MATH  MathSciNet  Google Scholar 

  5. Bao W, Cai Y. Mathematical theorey and numerical methods for Bose-Einstein condensation. Kinet Relat Mod, 2013, 6: 1–135

    Article  MATH  MathSciNet  Google Scholar 

  6. Browder F E. Existence and uniqueness theorems for solutions of nonlinear boundary value problems. In: Finn R, ed. Application of Nonlinear Partial Differential Equations. Proceedings of Symposia in Applied Mathematics, vol. 17. Providence, RI: Amer Math Soc, 1965, 24–49

    Google Scholar 

  7. Chippada S, Dawson C N, Martínez M L, et al. Finite element approximations to the system of shallow water equations, Part II: Discrete time a priori error estimates. SIAM J Numer Anal, 1999, 36: 226–250

    Article  MathSciNet  Google Scholar 

  8. Dawson CN, Martínez M L. A characteristic-Galerkin approximation to a system of shallow water equations. Numer Math, 2000, 86: 239–256

    Article  MATH  MathSciNet  Google Scholar 

  9. Ismail M S. Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method. Math Comp Simul, 2008, 78: 532–547

    Article  MATH  Google Scholar 

  10. Liao H, Sun Z. Error estimate of fourth-order compact scheme for linear Schrödinger equations. SIAM J Numer Anal, 2010, 47: 4381–4401

    Article  MATH  MathSciNet  Google Scholar 

  11. Liao H, Sun Z, Shi H. Maximum norm error analysis of explicit schemes for two-dimensional nonlinear Schrödinger equations (in Chinese). Sci Sin Math, 2010, 40: 827–842

    Google Scholar 

  12. Menyuk C R. Stability of solitons in birefringent optical fibers. J Opt Soc Am B, 1998, 5: 392–402

    Article  Google Scholar 

  13. Pitaevskii L P, Stringary S. Bose-Einstein condensation. Clarendon: Clarendon Press, 2003

    MATH  Google Scholar 

  14. Sepúlveda M, Vera O. Numerical methods for a coupled nonlinear Schrödinger system. Bol Soc Esp Mat Apl, 2008, 43: 95–102

    Google Scholar 

  15. Sonnier W J, Christov C I. Strong coupling of Schrödinger equations: Conservative scheme approach. Math Comp Simul, 2005, 69: 514–525

    Article  MATH  MathSciNet  Google Scholar 

  16. Sun Z. The Method of Order Reduction and its Application to the Numerical Solutions of Partial Differential Equations. Beijing: Science Press, 2009

    Google Scholar 

  17. Sun Z, Zhao D. On the L ∞ convergence of a difference secheme for coupled nonlinear Schrödinger equations. Comp Math Appl, 2010, 59: 3286–3300

    Article  MATH  MathSciNet  Google Scholar 

  18. Thomée V. Galerkin Finite Element Methods for Parabolic Problems. Berlin: Springer-Verlag, 1997

    MATH  Google Scholar 

  19. Wang H. A time-splitting spectral method for coupled Gross-Pitaevskii equations with applications to rotating Bose-Einstein condensates. J Comput Appl Math, 2007, 205: 88–104

    Article  MATH  MathSciNet  Google Scholar 

  20. Wang T. Maximum norm error bound of a linearized difference scheme for a coupled nonlinear Schrödinger equations. J Comput Appl Math, 2011, 235: 4237–4250

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang T. Optimal point-wise error estimate of a compact difference scheme for the coupled Gross-Pitaevskii equations in one dimension. J Sci Comput, doi: 10.1007/s10915-013-9757-1, 2013

    Google Scholar 

  22. Wang T, Guo B, Xu Q. Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. J Comput Phys, 2013, 243: 382–399

    Article  MathSciNet  Google Scholar 

  23. Williams J, Walser R, Cooper J, et al. Nonlinear Josephson-type oscillations of a driven two-component Bose-Einstein condensate. Phys Rev A, 1999, 59: R31–R34

    Article  Google Scholar 

  24. Yang J. Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics. Phys Rev E, 1999, 59: 2393–2405

    MathSciNet  Google Scholar 

  25. Zhang Y, Bao W, Li H. Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation. Physica D, 2007, 234: 49–69

    MATH  MathSciNet  Google Scholar 

  26. Zhou Y. Application of Discrete Functional Analysis to the Finite Difference Methods. Beijing: International Academic Publishers, 1990

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    TingChun Wang

  2. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    XiaoFei Zhao

Authors
  1. TingChun Wang
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  2. XiaoFei Zhao
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Correspondence to TingChun Wang.

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Wang, T., Zhao, X. Optimal l ∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions. Sci. China Math. 57, 2189–2214 (2014). https://doi.org/10.1007/s11425-014-4773-7

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  • DOI: https://doi.org/10.1007/s11425-014-4773-7

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