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A Deuflhard-Type Exponential Integrator Fourier Pseudo-Spectral Method for the “Good” Boussinesq Equation

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Abstract

We propose a Deuflhard-type exponential integrator Fourier pseudo-spectral (DEI-FP) method for solving the “Good” Boussinesq (GB) equation. The numerical scheme is based on a Deuflhard-type exponential integrator and a Fourier pseudo-spectral method for temporal and spatial discretizations, respectively. The scheme is fully explicit and efficient due to the fast Fourier transform. Rigorous error estimates are established for the method without any CFL-type condition constraint. In more details, the method converges quadratically and spectrally in time and space, respectively. Extensive numerical experiments are reported to confirm the theoretical analysis and to demonstrate rich dynamics of the GB equation.

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References

  1. Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108 (1872)

    MathSciNet  MATH  Google Scholar 

  2. Manoranjan, V.S., Ortega, T., Sanz-Serna, J.M.: Soliton and antisoliton interactions in the good Boussinesq equation. J. Math. Phys. 29, 964–1968 (1988)

    Article  MathSciNet  Google Scholar 

  3. Varlamov, V.: Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete Contin. Dyn. Syst. 7, 675–702 (2001)

    Article  MathSciNet  Google Scholar 

  4. Bona, J.L., Smith, R.A.: A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Philos. Soc. 79, 167–182 (1976)

    Article  MathSciNet  Google Scholar 

  5. Bona, J.L., Sachs, R.L.: Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Commun. Math. Phys. 118, 15–29 (1988)

    Article  MathSciNet  Google Scholar 

  6. Farah, L.: Local solutions in Sobolev spaces with negative indices for the good Boussinesq equation. Commun. Partial Differ. Equ. 34, 52–73 (2009)

    Article  MathSciNet  Google Scholar 

  7. Kishimoto, N., Tsugawa, K.: Local well-posedness for quadratic nonlinear Schrödinger equations and the good Boussinesq equation. Differ. Integral Equ. 23, 463–493 (2010)

    MATH  Google Scholar 

  8. Fang, Y., Grillakis, M.: Existence and uniqueness for Boussinesq type equations on a circle. Commun. Partial Differ. Equ. 21, 1253–1277 (1996)

    Article  MathSciNet  Google Scholar 

  9. Farah, L., Scialom, M.: On the periodic good Boussinesq equation. Proc. Amer. Math. Soc. 138, 953–964 (2010)

    Article  MathSciNet  Google Scholar 

  10. Kishimoto, N.: Sharp local well-posedness for the good Boussinesq equation. J. Differ. Equ. 254, 2393–2433 (2013)

    Article  MathSciNet  Google Scholar 

  11. Oh, S., Stefanov, A.: Improved local well-posedness for the periodic good Boussinesq equation. J. Differ. Equ. 254, 4047–4065 (2013)

    Article  MathSciNet  Google Scholar 

  12. Manoranjan, V.S., Mitchell, A., Morris, J.L.: Numerical solutions of the good Boussinesq equation. SIAM J. Sci. Comput. 5, 946–957 (1984)

    Article  MathSciNet  Google Scholar 

  13. Bratsos, A.G.: A second order numerical scheme for the solution of the one-dimensional Boussinesq equation. Numer. Algorithms 46, 45–58 (2007)

    Article  MathSciNet  Google Scholar 

  14. El-Zoheiry, H.: Numerical investigation for the solitary waves interaction of the good Boussinesq equation. Appl. Numer. Math. 45, 161–173 (2003)

    Article  MathSciNet  Google Scholar 

  15. Ortega, T., Sanz-Serna, J.M.: Nonlinear stability and convergence of finite-difference methods for the good Boussinesq equation. Numer. Math. 58, 215–229 (1990)

    Article  MathSciNet  Google Scholar 

  16. Cheng, K., Feng, W., Gottlieb, S., Wang, C.: A Fourier pseudo-spectral method for the good Boussinesq equation with second-order temporal accuracy. Numer. Methods Partial Differ. Equ. 31, 202–224 (2015)

  17. Frutos, J.D., Ortega, T., Sanz-Serna, J.M.: A Hamiltonian explicit algorithm with spectral accuracy for the good Boussinesq equation. Comput. Methods Appl. Mech. Eng. 80, 417–423 (1990)

    Article  Google Scholar 

  18. Frutos, J.D., Ortega, T., Sanz-Serna, J.M.: Pseudo-spectral method for the good Boussinesq equation. Math. Comput. 57, 109–122 (1991)

    MATH  Google Scholar 

  19. Yan, J., Zhang, Z.: New energy-preserving schemes using Hamiltonian boundary value and Fourier pseudo-spectral methods for the numerical solution of the good Boussinesq equation. Comput. Phys. Commun. 201, 33–42 (2016)

    Article  MathSciNet  Google Scholar 

  20. Zhang, C., Wang, H., Huang, J., Wang, C., Yue, X.: A second order operator splitting numerical scheme for the good Boussinesq equation. Appl. Numer. Math. 119, 179–193 (2017)

    Article  MathSciNet  Google Scholar 

  21. Dehghan, M., Salehi, R.: A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation. Appl. Math. Model. 36, 1939–1956 (2012)

    Article  MathSciNet  Google Scholar 

  22. Zhang, C., Huang, J., Wang, C., Yue, X.: On the operator splitting and integral equation preconditioned deferred correction methods for the Good Boussinesq equation. J. Sci. Comput. 75, 687–712 (2018)

    Article  MathSciNet  Google Scholar 

  23. Cai, J., Wang, Y.: Local structure-preserving algorithms for the good Boussinesq equation. J. Comp. Phys. 239, 72–89 (2013)

    Article  MathSciNet  Google Scholar 

  24. Chen, M., Kong, L., Hong, Y.: Efficient structure-preserving schemes for good Boussinesq equation. Math. Meth. Appl. Sci. 41, 1743–1752 (2018)

    Article  MathSciNet  Google Scholar 

  25. Jiang, C., Sun, J., He, X., Zhou, L.: High order energy-preserving method of the good Boussinesq equation. Numer. Math. Theor. Meth. Appl. 9, 111–122 (2016)

    Article  MathSciNet  Google Scholar 

  26. Mohebbi, A., Asgari, Z.: Efficient numerical algorithms for the solution of good Boussinesq equation in water wave propagation. Comput. Phys. Commun. 182, 2464–2470 (2011)

    Article  MathSciNet  Google Scholar 

  27. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  Google Scholar 

  28. Ostermann, A., Su, C.: Two exponential-type integrators for the good Boussinesq equation. Numer. Math. 143, 683–712 (2019)

    Article  MathSciNet  Google Scholar 

  29. Zhao, X.: On error estimates of an exponential wave integrator sine pseudo-spectral method for the Klein–Gordon–Zakharov system. Numer. Methods Partial Differ. Equ. 32, 266–291 (2016)

    Article  Google Scholar 

  30. Shen, J., Tang, T.: Spectral and High-Order Methods With Applications. Science Press, Beijing (2006)

    MATH  Google Scholar 

  31. Deuflhard, P.: A study of extrapolation methods based on multistep schemes without parasitic solutions. ZAMP 30, 177–189 (1979)

    MathSciNet  MATH  Google Scholar 

  32. Gottlieb, S., Tone, F., Wang, C., Wang, X., Wirosoetisno, D.: Long time stability of a classical efficient scheme for two dimensional Navier–Stokes equations. SIAM J. Numer. Anal. 50, 126–150 (2012)

    Article  MathSciNet  Google Scholar 

  33. Gottlieb, S., Wang, C.: Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers’ equation. J. Sci. Comput. 53, 102–128 (2012)

    Article  MathSciNet  Google Scholar 

  34. Cheng, K., Wang, C., Wise, S.M., Yue, X.: A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn–Hilliard equation and its solution by the homogeneous linear iteration method. J. Sci. Comput. 69, 1083–1114 (2016)

    Article  MathSciNet  Google Scholar 

  35. Chartier, Ph, Méhats, F., Thalhammer, M., Zhang, Y.: Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations. Math. Comp. 85, 2863–2885 (2016)

    Article  MathSciNet  Google Scholar 

  36. Adams, R.A., Fournier, J.J.: Sobolev Spaces. Elsevier, New York (2003)

    MATH  Google Scholar 

  37. Su, C., Muslu, G. M.: An exponential integrator sine pseudo-spectral method for the generalized improved Boussinesq equation. preprint (2020)

  38. Ismail, M.S., Mosally, F.: A fourth order finite difference method for the good Boussinesq equation. Abs. Appl. Anal. 2014, 323260 (2014)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, 80290, München, Germany

    Chunmei Su

  2. School of Mathematics, South China University of Technology, Guangzhou, 510641, China

    Wenqi Yao

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  1. Chunmei Su
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  2. Wenqi Yao
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Correspondence to Wenqi Yao.

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This first author was supported by the Alexander von Humboldt Foundation and the second author was supported by NSFC (No. 11801183).

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Su, C., Yao, W. A Deuflhard-Type Exponential Integrator Fourier Pseudo-Spectral Method for the “Good” Boussinesq Equation. J Sci Comput 83, 4 (2020). https://doi.org/10.1007/s10915-020-01192-2

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  • DOI: https://doi.org/10.1007/s10915-020-01192-2

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