Skip to main content
SpringerLink
Log in

Error Estimates of Finite Difference Methods for the Biharmonic Nonlinear Schrödinger Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We present two finite difference time domain methods for the biharmonic nonlinear Schrödinger equation (BNLS) by reformulating it into a system of second-order partial differential equations instead of a direct discretization, including a second-order conservative Crank–Nicolson finite difference (CNFD) method and a second-order semi-implicit finite difference (SIFD) method. The CNFD method conserves the mass and energy in the discretized level, and the SIFD method only needs to solve a linear system at each time step, which is more efficient. By energy method, we establish optimal error bounds at the order of \( O(h^2+\tau ^2) \) in both \( L^2 \) and \( H^2 \) norms for both CNFD and SIFD methods, with mesh size h and time step \(\tau \). The proof of the error bounds are mainly based on the discrete Gronwall’s inequality and mathematical induction. Finally, numerical results are reported to confirm our error bounds and to demonstrate the properties of our schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Agrawal G.P.: Nonlinear fiber optics. In: Nonlinear Science at the Dawn of the 21st Century, pp. 195–211. Springer (2000)

  2. Aksas, B., Rebiai, S.-E.: Uniform stabilization of the fourth order Schrödinger equation. J. Math. Anal. Appl. 2, 1794–1813 (2017)

    MATH  Google Scholar 

  3. Antoine, X., Bao, W.Z., Besse, C.: Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations. Comput. Phys. Commun. 184, 2621–2633 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Antoine, X., Tang, Q.L., Zhang, Y.: On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions. J. Comput. Phys. 325, 74–97 (2016)

    MathSciNet  MATH  Google Scholar 

  5. 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. 50, 492–521 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Bao, W.Z., Cai, Y.Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Models 6, 1–135 (2013)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Bao, W.Z., Shen, J.: A fourth-order time-splitting Laguerre–Hermite pseudo-spectral method for Bose–Einstein condensates. SIAM J. Sci. Comput. 26, 2010–2028 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Bao, W.Z., Su, C.M.: Uniform error bounds of a finite difference method for the Zakharov system in the subsonic limit regime via an asymptotic consistent formulation. Multiscale Model. Simul. 15, 977–1002 (2017)

    MathSciNet  Google Scholar 

  10. Bao, W.Z., Wang, H.: An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates. J. Comput. Phys. 217, 612–626 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Bao, W.Z., Jin, S., Markowich, P.A.: On time-splitting spectral approximation for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175, 487–524 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Bao, W.Z., Jaksch, D., Markowich, P.A.: Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation. J. Comput. Phys. 187, 318–342 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Bao, W.Z., Du, Q., Zhang, Y.: Dynamics of rotating Bose–Einstein condensates and its efficient and accurate numerical computation. SIAM J. Appl. Math. 66, 758–786 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Bao, W.Z., Li, H., Shen, J.: A generalized-Laguerre–Fourier–Hermite pseudospectral method for computing the dynamics of rotating Bose–Einstein condensates. SIAM J. Sci. Comput. 31, 3685–3711 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Baruch, G., Fibich, G.: Singular solutions of the \(L^2\)-supercritical biharmonic nonlinear Schrödinger equation. Nonlinearity 24, 1843–1859 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Baruch, G., Fibich, G., Mandelbaum, E.: Singular solutions of the biharmonic nonlinear Schrödinger equation. SIAM J. Appl. Math. 70, 3319–3341 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Ben-Artzi, M., Koch, H., Saut, J.-C.: Dispersion estimates for fourth order Schrödinger equations. C. R. Math. Acad. Sci. Paris 330, 87–92 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Bonheure, D., Casteras, J.-B., Dos Santos, E.M., Nascimento, R.: Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J. Math. Anal. 50, 5027–5071 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Bourgain, J.: Global Solutions of Nonlinear Schrödinger Equations. American Mathematical Society, Providence (1999)

    MATH  Google Scholar 

  20. Burq, N., Thomann, L., Tzvetkov, N.: Long time dynamics for the one dimensional nonlinear Schrödinger equation. Ann. Inst. Four. (Grenoble) 63, 2137–2198 (2013)

    MATH  Google Scholar 

  21. Capistrano-Filho, R.D.A., Cavalcante, M.: Stabilization and control for the biharmonic Schrödinger equation. Appl. Math. Optim. 84, 103–144 (2021)

    MathSciNet  MATH  Google Scholar 

  22. Capistrano-Filho, R.D.A., Cavalcante, M., Gallego, F.A.: Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane. Pac. J. Math. 309, 35–70 (2020)

    MATH  Google Scholar 

  23. Clark, D.S.: Short proof of a discrete Gronwall inequality. Discrete Appl. Math. 16, 279–281 (1987)

    MathSciNet  MATH  Google Scholar 

  24. Cui, S., Guo, C.: Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces \( H^s (R^n) \) and applications. Nonlinear Anal. 67, 687–707 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Debussche, A., Faou, E.: Modified energy for split-step methods applied to the linear Schrödinger equations. SIAM J. Numer. Anal. 47, 3705–3719 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Ehrlich, L.W.: Solving the biharmonic equation as coupled finite difference equations. SIAM J. Numer. Anal. 8, 278–287 (1971)

    MathSciNet  MATH  Google Scholar 

  27. Feng, Y.: Improved error bounds of the Strang splitting method for the highly oscillatory fractional nonlinear Schrödinger equation. J. Sci. Comput. 88, 1–24 (2021)

    MATH  Google Scholar 

  28. Fibich, G., Ilan, B., Papanicolaou, G.: Self-focusing with fourth-order dispersion. SIAM J. Appl. Math. 62, 1437–1462 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Gao, P.: Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evol. Equ. Control Theory 7, 465–499 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Gardner, L.R.T., Gardner, G.A., Zaki, S.I., El Sahrawi, Z.: B-spline finite element studies of the non-linear Schrödinger equation. Comput. Methods Appl. Mech. Eng. 108, 303–318 (1993)

    MATH  Google Scholar 

  31. Karakashian, O., Makridakis, C.: A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method. Math. Comput. 67, 479–499 (1998)

    MATH  Google Scholar 

  32. Karakashian, O., Akrivis, G., Dougalis, V.: On optimal order error estimates for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 30, 377–400 (1993)

    MathSciNet  MATH  Google Scholar 

  33. Karpman, V.I.: Stabilization of soliton instabilities by higher-order dispersion: fourth-order nonlinear Schrödinger-type equations. Phys. Rev. E 53, 1336–1339 (1996)

    Google Scholar 

  34. Karpman, V.I., Shagalov, A.G.: Influence of high-order dispersion on self-focusing. II. Numerical investigation. Phys. Lett. A 160, 538–540 (1991)

    Google Scholar 

  35. Karpman, V.I., Shagalov, A.G.: Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion. Phys. D 144, 194–210 (2000)

    MathSciNet  MATH  Google Scholar 

  36. Kelley, C.T.: Numerical methods for nonlinear equations. Acta Numer. 27, 207–287 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Kwak, C.: Periodic fourth-order cubic NLS: local well-posedness and non-squeezing property. J. Math. Anal. Appl. 461, 1327–1364 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Landes, R.: On Galerkin’s method in the existence theory of quasilinear elliptic equations. J. Funct. Anal. 39, 123–148 (1980)

    MathSciNet  MATH  Google Scholar 

  39. Makhankov, V.G.: Dynamics of classical solitons (in non-integrable systems). Phys. Lett. C 35, 1–128 (1978)

    MathSciNet  Google Scholar 

  40. Markowich, P.A., Pietra, P., Pohl, C.: Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit. Numer. Math. 81, 595–630 (1999)

    MathSciNet  MATH  Google Scholar 

  41. Natali, F., Pastor, A.: The fourth-order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave. SIAM J. Appl. Dyn. Syst. 14, 1326–1347 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Neuhauser, C., Thalhammer, M.: On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT 49, 199–215 (2009)

    MathSciNet  MATH  Google Scholar 

  43. Özsari, T., Yolcu, N.: The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Commun. Pure Appl. Anal. 18, 3285–3316 (2019)

    MathSciNet  MATH  Google Scholar 

  44. Pausader, B.: Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Part. Differ. Equ. 4, 197–225 (2007)

    MathSciNet  MATH  Google Scholar 

  45. Pausader, B.: The cubic fourth-order Schrödinger equation. J. Funct. Anal. 256, 2473–2517 (2009)

    MathSciNet  MATH  Google Scholar 

  46. Planchon, F., Raphaël, P.: Existence and stability of the log-log blow-up dynamics for the \(L^2\)-critical nonlinear Schrödinger equation in a domain. Ann. Henri Poincaré 8, 1177–1219 (2007)

    MathSciNet  MATH  Google Scholar 

  47. Raphaël, P.: Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation. Math. Ann. 331, 577–609 (2005)

    MathSciNet  MATH  Google Scholar 

  48. Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press (1985)

  49. Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999)

    MATH  Google Scholar 

  50. Tadahiro, O., Tzvetkov, N.: Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation. Probab. Theory Relat. Fields 169, 1121–1168 (2016)

    MATH  Google Scholar 

  51. Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46, 2022–2038 (2008)

    MathSciNet  MATH  Google Scholar 

  52. Wang, T.C., Wang, J.L., Guo, B.L.: Two completely explicit and unconditionally convergent Fourier pseudo-spectral methods for solving the nonlinear Schrödinger equation. J. Comput. Phys. 404, 109116 (2019)

    MATH  Google Scholar 

  53. Wang, T.C., Guo, B.L.: Unconditional convergence of two conservative compact difference schemes for non-linear Schrödinger equation in one dimension (in Chinese). Sci. Sin. Math. 41, 207–233 (2011)

    MATH  Google Scholar 

  54. Wang, P.D., Huang, C.M.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)

    MathSciNet  MATH  Google Scholar 

  55. Wang, T.C., Zhao, X.F.: Optimal \(l^\infty \) error estimates of finite difference methods for the coupled Gross–Pitaevskii equations in high dimensions. Sci. China Math. 57, 2189–2214 (2014)

    MathSciNet  MATH  Google Scholar 

  56. Wen, R., Chai, S., Guo, B.-Z.: Well-posedness and exact controllability of fourth order Schrödinger equation with boundary control and collocated observation. SIAM J. Control. Optim. 52, 365–396 (2014)

    MathSciNet  MATH  Google Scholar 

  57. Zhang, T., Wang, T.C.: Optimal error estimates of fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation in the nonrelativistic regime. Numer. Methods Part. Differ. Equ. 37, 2089–2108 (2021)

    MathSciNet  Google Scholar 

  58. Zheng, C.: Inverse problems for the fourth order Schrödinger equation on a finite domain. Math. Control Relat. Fields 5, 177–189 (2015)

    MathSciNet  MATH  Google Scholar 

  59. Zhou, Y.L.: Applications of Discrete Functional Analysis to Finite Difference Method. International Academic Publishers, Beijing (1990)

    Google Scholar 

Download references

Funding

This work was supported by the Research Foundation for Beijing University of Technology New Faculty Grant No. 006000514122521 (Y. Ma), and the National Natural Science Foundation of China Grant No. U2230402 (T. Zhang).

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Beijing University of Technology, Beijing, 100124, China

    Ying Ma

  2. Beijing Computational Science Research Center, Beijing, 100193, China

    Teng Zhang

Authors
  1. Ying Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Teng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Teng Zhang.

Ethics declarations

Conflict of interest

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the Research Foundation for Beijing University of Technology New Faculty Grant No. 006000514122521 (Y. Ma), and the National Natural Science Foundation of China Grant No. U2230402 (T. Zhang).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, Y., Zhang, T. Error Estimates of Finite Difference Methods for the Biharmonic Nonlinear Schrödinger Equation. J Sci Comput 95, 24 (2023). https://doi.org/10.1007/s10915-023-02124-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02124-6

Keywords

Mathematics Subject Classification

Search

Navigation