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A fast second-order absorbing boundary condition for the linearized Benjamin-Bona-Mahony equation

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Abstract

In this paper, we present a fully discrete finite difference scheme with efficient convolution of artificial boundary conditions for solving the Cauchy problem associated with the one-dimensional linearized Benjamin-Bona-Mahony equation. The scheme utilizes the Padé expansion of the square root function in the complex plane to implement the fast convolution, resulting in significant reduction of computational costs involved in the time convolution process. Moreover, the introduction of a constant dam** term in the governing equations allows for convergence analysis under specific conditions. The theoretical analysis is complemented by numerical examples that illustrate the performance of the proposed numerical method.

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References

  1. Achouri, T., Khiari, N., Omrani, K.: On the convergence of difference schemes for the Benjamin Bona Mahony BBM equation. Appl. Math. Comput. 182, 999–1005 (2006)

    MathSciNet  Google Scholar 

  2. Alazman, A., Albert, J., Bona, J., Chen, M., Wu, J.: Comparisons between the BBM equation and a Boussinesq system. Adv. Differ. Equ. 11, 121–166 (2006)

    MathSciNet  Google Scholar 

  3. Besse, C., Gireau, B., Noble, P.: Artificial boundary conditions for the linearized Benjamin-Bona-Mahony equation. Num. Math. 139, 281–314 (2018)

    Article  MathSciNet  Google Scholar 

  4. Kazakova, M., Noble, P.: Discrete transparent boundary conditions for the linearized Green-Naghdi system of equations. SIAM. J. Num. Anal. 58(1), 657–683 (2020)

    Article  MathSciNet  Google Scholar 

  5. Zheng, C., Du, Q., Ma, X., Zhang, J.: Stability and error analysis for a second-order fast approximation of the local and nonlocal diffusion equations on the real Line. SIAM. J. Num. Anal. 58, 1893–1917 (2020)

    Article  MathSciNet  Google Scholar 

  6. Fevens, T., Jiang, H.: Absorbing boundary conditions for the Schrödinger equation. SIAM. J. Sci. Comput. 21, 255–282 (1999)

    Article  MathSciNet  Google Scholar 

  7. Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential. J. Comput. Phys. 228, 312–335 (2009)

    Article  MathSciNet  Google Scholar 

  8. Antoine, X., Besse, C.: Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation. J. Comput. Phys. 188, 157–175 (2003)

    Article  MathSciNet  Google Scholar 

  9. Wu, X., Sun, Z.: Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions. Appl. Nume. Math. 50, 261–277 (2004)

    Article  MathSciNet  Google Scholar 

  10. Baskakov, V., Popov, A.: Implementation of transparent boundaries for numerical solution of the Schrödinger equation. Wave Motion. 14, 123–128 (1991)

    Article  MathSciNet  Google Scholar 

  11. Han, H., Huang, Z.: Exact and approximating boundary conditions for the parabolic problems on unbounded domains. Comput. Mathe. Appl. 44, 655–666 (2002)

    Article  MathSciNet  Google Scholar 

  12. Han, H., Huang, Z.: Exact artificial boundary conditions for the Schrödinger equation in \(R^2\). Commun. Math. Sci. 2, 79–94 (2004)

    Article  MathSciNet  Google Scholar 

  13. Arnold, A., Ehrhardt, M., Schulte, M., Sofronov, I.: Discrete transparent boundary conditions for the Schrödinger equation on circular domains. Commun. Math. Sci. 10(3), 889–916 (2012)

    Article  MathSciNet  Google Scholar 

  14. Li, H., Wu, X., Zhang, J.: Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary. Comput. Phys. Commun. 185, 1606–1615 (2014)

    Article  MathSciNet  Google Scholar 

  15. Antoine, X., Besse, C., Mouysset, V.: Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions. Math Comput. 73, 1779–1799 (2004)

    Article  Google Scholar 

  16. Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential. Part II: Discretization and Numerical Results. Num. Math. 125, 191–223 (2013)

  17. Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for general nonlinear Schrödinger Equations. SIAM. J. Sci. Comput. 33, 1008–1033 (2011)

    Article  MathSciNet  Google Scholar 

  18. Pang, G., Yang, Y., Antoine, X., Tang, S.: Stability and convergence analysis of artificial boundary conditions for the Schrödinger equation on a rectangular domain, Preprint

  19. Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Schaedle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys. 4, 729–796 (2008)

    MathSciNet  Google Scholar 

  20. Pang, G., Tang, S.: Approximate linear relations for Bessel functions. Commun. Math. Sci. 15, 1967–1986 (2017)

    Article  MathSciNet  Google Scholar 

  21. Pang, G., Bian, L., Tang, S.: ALmost EXact boundary condition for one-dimensional Schrödinger equation. Phys. Rev. E. 86, 066709 (2012)

    Article  Google Scholar 

  22. Ji, S., Yang, Y., Pang, G., Antoine, X.: Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains. Comput. Phys. Commun. 222, 84–93 (2018)

    Article  MathSciNet  Google Scholar 

  23. Arnold, A., Ehrhardt, M., Sofronov, I.: Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability. Commun. Math. Sci. 1, 501–556 (2003)

    Article  MathSciNet  Google Scholar 

  24. Jiang, S., Greengard, L.: Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension. Comput. Math. Appl. 47, 955–966 (2004)

    Article  MathSciNet  Google Scholar 

  25. Zheng, C.: Approximation, stability and fast evaluation of exact artificial boundary condition for one-dimensional heat equation. J. Comput. Math. 25, 730–745 (2007)

    MathSciNet  Google Scholar 

  26. Alpert, B., Greengard, L., Hagstrom, T.: Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM. J. Num. Anal. 37, 1138–1164 (2000)

    Article  MathSciNet  Google Scholar 

  27. Lu, Y.: A Padé approximation method for square roots of symmetric positive definite matrices. SIAM. J. Num. Anal. 19, 833–845 (1998)

    Article  Google Scholar 

  28. Lubich, C., Schädle, A.: Fast convolution for nonreflecting boundary conditions. SIAM. J. Sci. Compt. 24, 161–182 (2002)

    Article  MathSciNet  Google Scholar 

  29. Feng, Y., Wang, X.: Matching boundary conditions for Euler-Bernoulli beam. Shock. Vib. 6685852 (2021)

  30. Tang, S., Karpov, E.: Artificial boundary conditions for Euler-Bernoulli beam equation. Acta. Mech. Sinica-PRC. 30, 687–692 (2014)

    Article  MathSciNet  Google Scholar 

  31. Li, B., Zhang, J., Zheng, C.: An efficient second-order finite difference method for the one-dimensional Schrödinger equation with absorbing boundary conditions. SIAM. J. Num. Anal. 56, 766–791 (2018)

    Article  Google Scholar 

  32. Zhao, X.: Optimal convergence of a second order low-regularity integrator for the KdV equation. IMA J. Num. Anal. to appear

  33. Wang, X., Tang, S.: Matching boundary conditions for diatomic chains. Comput. Mech. 46(6), 813–826 (2010)

    Article  Google Scholar 

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Acknowledgements

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Funding

Dr Zheng is supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202301130) and Dr Liu is supported by NSFC under grant Nos. 12102282.

Author information

Authors and Affiliations

  1. School of Mathematical Science, Beihang University, Bei**g, 102206, China

    Gang Pang

  2. Chongqing University of technology, Chongqing, 400054, China

    Zijun Zheng

  3. Chair of Applied and Computational Mathematics, School of Mathematics and Natural Sciences, University of Wuppertal, Wuppertal, D-42119, Germany

    Matthias Ehrhardt

  4. School of Physics and Electronic Engineering, Centre for Computational Sciences, Sichuan Normal University, Chengdu, 610066, China

    Baiyili Liu

Authors
  1. Zijun Zheng
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  2. Gang Pang
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  3. Matthias Ehrhardt
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  4. Baiyili Liu
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Contributions

Zheng and Liu did the computation, Pang did the numerical analysis. Pang and Ehrhardt wrote the main manuscript text. All authors reviewed the manuscript.

Corresponding author

Correspondence to Gang Pang.

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Zheng, Z., Pang, G., Ehrhardt, M. et al. A fast second-order absorbing boundary condition for the linearized Benjamin-Bona-Mahony equation. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01864-2

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  • DOI: https://doi.org/10.1007/s11075-024-01864-2

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