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Journal of Scientific Computing

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01-04-2020

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

Authors: Chunmei Su, Wenqi Yao

Published in: Journal of Scientific Computing | Issue 1/2020

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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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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)MathSciNetMATH

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)MathSciNetCrossRef

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

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)MathSciNetCrossRef

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)MathSciNetCrossRef

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

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

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

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

10.

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

11.

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

12.

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

13.

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

14.

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

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)MathSciNetCrossRef

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)CrossRef

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

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)MathSciNetCrossRef

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)MathSciNetCrossRef

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)MathSciNetCrossRef

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)MathSciNetCrossRef

23.

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

24.

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

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)MathSciNetCrossRef

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)MathSciNetCrossRef

27.

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

28.

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

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)CrossRef

30.

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

31.

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

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)MathSciNetCrossRef

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)MathSciNetCrossRef

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)MathSciNetCrossRef

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)MathSciNetCrossRef

36.

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

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)MathSciNetMATH

Title: A Deuflhard-Type Exponential Integrator Fourier Pseudo-Spectral Method for the “Good” Boussinesq Equation
Authors: Chunmei Su
Wenqi Yao
Publication date: 01-04-2020
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2020
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01192-2

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