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September  2020, 25(9): 3749-3763. doi: 10.3934/dcdsb.2020089

A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation

Zeyu Xia †, and  Xiaofeng Yang ‡,

†. 

School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu, Sichuan 611731, China
‡. 

Department of Mathematics, University of South Carolina Columbia, SC 29208

Received  May 2019 Revised  July 2019 Published  April 2020

The nonlinear stability and convergence of a numerical scheme for the "Good" Boussinesq equation is provided in this article, with second order temporal accuracy and Fourier pseudo-spectral approximation in space. Instead of introducing an intermediate variable $ \psi $ to approximate the first order temporal derivative, we apply a direct approximation to the second order temporal derivative, which in turn leads to a reduction of the intermediate numerical variable and improvement in computational efficiency. A careful analysis reveals an unconditional stability and convergence for such a temporal discretization. In addition, by making use of the techniques of aliasing error control, we obtain an $ \ell^\infty (0,T^*; H^2) $ convergence for $ u $ and $ \ell^\infty (0,T^*; \ell^2) $ convergence for the discrete time-derivative of the solution in this paper, in comparison with the $ \ell^\infty (0,T^*; \ell^2) $ convergence for $ u $ and the $ \ell^\infty (0,T^*; H^{-2}) $ convergence for the time-derivative, given in [19].

Keywords: "Good" Boussinesq equation, Fourier pseudospectral method, second-order time-stepping, stability and convergence.
Mathematics Subject Classification: 65M06, 65M15, 65M22.
Citation: Zeyu Xia, Xiaofeng Yang. A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3749-3763. doi: 10.3934/dcdsb.2020089
References:
[1]

B. S. Attili, The Adomian decomposition method for solving the Boussinesq equation arising in water wave propagation, Numer. Methods Partial Differential Equations, 22 (2006), 1337-1347.  doi: 10.1002/num.20155.  Google Scholar

[2]

A. Baskaran, J. S. Lowengrub, C. Wang and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), 2851-2873.  doi: 10.1137/120880677.  Google Scholar

[3]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[4]

A. G. Bratsos, A predictor-corrector scheme for the improved Boussinesq equation, Chaos Solitons Fractals, 40 (2009), 2083-2094.  doi: 10.1016/j.chaos.2007.09.083.  Google Scholar

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A. G. Bratsos, A second order numerical scheme for the improved Boussinesq equation, Phys. Lett. A, 370 (2007), 145-147.  doi: 10.1016/j.physleta.2007.05.050.  Google Scholar

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C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer, Berlin, 2007.  Google Scholar

[7]

C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), 67-86.  doi: 10.1090/S0025-5718-1982-0637287-3.  Google Scholar

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W. Chen, W. Li, Z. Luo, C. Wang and X. Wang, A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection, arXiv: 1907.02234. Google Scholar

[9]

K. Cheng, W. Feng, S. Gottlieb and C. Wang, A Fourier pseudospectral method for the "good" Boussinesq equation with second-order temporal accuracy, Numer. Methods Partial Differential Equations, 31 (2015), 202-224.  doi: 10.1002/num.21899.  Google Scholar

[10]

K. Cheng, W. Feng, C. Wang and S. M. Wise, An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation, J. Comput. Appl. Math., 362 (2019), 574-595.  doi: 10.1016/j.cam.2018.05.039.  Google Scholar

[11]

K. Cheng, Z. Qiao and C. Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, J. Sci. Comput., 81 (2019), 154-185.  doi: 10.1007/s10915-019-01008-y.  Google Scholar

[12]

K. Cheng and C. Wang, Long time stability of high order multistep numerical schemes for two-dimensional incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 54 (2016), 3123-3114.  doi: 10.1137/16M1061588.  Google Scholar

[13]

K. Cheng, C. Wang and S. M. Wise, An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation, Commun. Comput. Phys., 26 (2019), 1335-1364.  doi: 10.4208/cicp.2019.js60.10.  Google Scholar

[14]

K. Cheng, C. Wang, S. M. Wise and X. Yue, 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 (2016), 1083-1114.  doi: 10.1007/s10915-016-0228-3.  Google Scholar

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R. Cienfuegos, E. Barthélemy and P. Bonneton, A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Ⅰ. Model development and analysis, Internat. J. Numer. Methods Fluids, 51 (2006), 1217-1253.  doi: 10.1002/fld.1141.  Google Scholar

[16]

R. Cienfuegos, E. Barthélemy and P. Bonneton, A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Ⅱ. Boundary conditions and validation, Internat. J. Numer. Methods Fluids, 53 (2007), 1423-1455.  doi: 10.1002/fld.1359.  Google Scholar

[17]

T. Dupont, Galerkin methods for first order hyperbolics: An example, SIAM J. Numer. Anal., 10 (1973), 890-899. doi: 10.1137/0710074.  Google Scholar

[18]

L. G. Farah and M. Scialom, On the periodic "good" Boussinesq equation, Proc. Amer. Math. Soc., 138 (2010), 953-964.  doi: 10.1090/S0002-9939-09-10142-9.  Google Scholar

[19]

J. de Frutos, T. Ortega and J. M. Sanz-Serna, Pseudospectral method for the "good" Boussinesq equation, Math. Comp., 57 (1991), 109-122.  doi: 10.2307/2938665.  Google Scholar

[20]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1977.  Google Scholar

[21]

S. Gottlieb, F. Tone, C. Wang, X. Wang and D. Wirosoetisno, Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 50 (2012), 126-150.  doi: 10.1137/110834901.  Google Scholar

[22]

S. Gottlieb and C. Wang, Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers' equation, J. Sci. Comput., 53 (2012), 102-128.  doi: 10.1007/s10915-012-9621-8.  Google Scholar

[23] J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-dependent Problems, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618352.  Google Scholar
[24]

M. Li, P. Guyenne, F. Li and L. Xu, High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model, J. Comput. Phys., 257 (2014), 169-192.  doi: 10.1016/j.jcp.2013.09.050.  Google Scholar

[25]

X. Li, Z. Qiao and C. Wang, Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, preprint, arXiv: 1902.04967. Google Scholar

[26]

Y. Maday and A. Quarteroni, Approximation of Burgers' equation by pseudospectral methods, RAIRO Anal. Numér., 16 (1982), 375-404.  doi: 10.1051/m2an/1982160403751.  Google Scholar

[27]

Y. Maday and A. Quarteroni, Legendre and Chebyshev spectral approximations of Burgers' equation, Numer. Math., 37 (1981), 321-332.  doi: 10.1007/BF01400311.  Google Scholar

[28]

Y. Maday and A. Quarteroni, Spectral and pseudospectral approximations of the Navier-Stokes equations, SIAM J. Numer. Anal., 19 (1982), 761-780.  doi: 10.1137/0719053.  Google Scholar

[29]

V. S. Manoranjan, A. R. Mitchell and J. LL. Morris, Numerical solutions of the "Good" Boussinesq equation, SIAM J. Sci. Statist. Comput., 5 (1984), 946-957.  doi: 10.1137/0905065.  Google Scholar

[30]

V. S. Manoranjan, T. Ortega and J. M. Sanz-Serna, Soliton and antisoliton interactions in the "Good" Boussinesq equation, J. Math. Phys., 29 (1988), 1964-1968.  doi: 10.1063/1.527850.  Google Scholar

[31]

S. Oh and A. Stefanov, Improved local well-posedness for the periodic "good" Boussinesq equation, J. Differential Equations, 254 (2013), 4047-4065.  doi: 10.1016/j.jde.2013.02.006.  Google Scholar

[32]

T. Ortega and J. M. Sanz-Serna, Nonlinear stability and convergence of finite-difference methods for the "good" Boussinesq equation, Numer. Math., 58 (1990), 215-229.  doi: 10.1007/BF01385620.  Google Scholar

[33]

E. Weinan, Convergence of Fourier methods for Navier-Stokes equations, SIAM J. Numer. Anal., 30 (1993), 650-674.  doi: 10.1137/0730032.  Google Scholar

[34]

E. Weinan, Convergence of spectral methods for the Burgers' equation, SIAM J. Numer. Anal., 29 (1992), 1520-1541.  doi: 10.1137/0729088.  Google Scholar

[35]

C. Zhang, J. Huang, C. Wang and X. Yue, On the operator splitting and integral equation preconditioned deferred correction methods for the "Good" Boussinesq equation, J. Sci. Comput., 75 (2018), 687-712.  doi: 10.1007/s10915-017-0552-2.  Google Scholar

[36]

C. Zhang, H. Wang, J. Huang, C. Wang and X. Yue, A second order operator splitting numerical scheme for the "Good" Boussinesq equation, Appl. Numer. Math., 119 (2017), 179-193.  doi: 10.1016/j.apnum.2017.04.006.  Google Scholar

show all references

References:
[1]

B. S. Attili, The Adomian decomposition method for solving the Boussinesq equation arising in water wave propagation, Numer. Methods Partial Differential Equations, 22 (2006), 1337-1347.  doi: 10.1002/num.20155.  Google Scholar

[2]

A. Baskaran, J. S. Lowengrub, C. Wang and S. M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51 (2013), 2851-2873.  doi: 10.1137/120880677.  Google Scholar

[3]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[4]

A. G. Bratsos, A predictor-corrector scheme for the improved Boussinesq equation, Chaos Solitons Fractals, 40 (2009), 2083-2094.  doi: 10.1016/j.chaos.2007.09.083.  Google Scholar

[5]

A. G. Bratsos, A second order numerical scheme for the improved Boussinesq equation, Phys. Lett. A, 370 (2007), 145-147.  doi: 10.1016/j.physleta.2007.05.050.  Google Scholar

[6]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer, Berlin, 2007.  Google Scholar

[7]

C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), 67-86.  doi: 10.1090/S0025-5718-1982-0637287-3.  Google Scholar

[8]

W. Chen, W. Li, Z. Luo, C. Wang and X. Wang, A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection, arXiv: 1907.02234. Google Scholar

[9]

K. Cheng, W. Feng, S. Gottlieb and C. Wang, A Fourier pseudospectral method for the "good" Boussinesq equation with second-order temporal accuracy, Numer. Methods Partial Differential Equations, 31 (2015), 202-224.  doi: 10.1002/num.21899.  Google Scholar

[10]

K. Cheng, W. Feng, C. Wang and S. M. Wise, An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation, J. Comput. Appl. Math., 362 (2019), 574-595.  doi: 10.1016/j.cam.2018.05.039.  Google Scholar

[11]

K. Cheng, Z. Qiao and C. Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, J. Sci. Comput., 81 (2019), 154-185.  doi: 10.1007/s10915-019-01008-y.  Google Scholar

[12]

K. Cheng and C. Wang, Long time stability of high order multistep numerical schemes for two-dimensional incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 54 (2016), 3123-3114.  doi: 10.1137/16M1061588.  Google Scholar

[13]

K. Cheng, C. Wang and S. M. Wise, An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation, Commun. Comput. Phys., 26 (2019), 1335-1364.  doi: 10.4208/cicp.2019.js60.10.  Google Scholar

[14]

K. Cheng, C. Wang, S. M. Wise and X. Yue, 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 (2016), 1083-1114.  doi: 10.1007/s10915-016-0228-3.  Google Scholar

[15]

R. Cienfuegos, E. Barthélemy and P. Bonneton, A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Ⅰ. Model development and analysis, Internat. J. Numer. Methods Fluids, 51 (2006), 1217-1253.  doi: 10.1002/fld.1141.  Google Scholar

[16]

R. Cienfuegos, E. Barthélemy and P. Bonneton, A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Ⅱ. Boundary conditions and validation, Internat. J. Numer. Methods Fluids, 53 (2007), 1423-1455.  doi: 10.1002/fld.1359.  Google Scholar

[17]

T. Dupont, Galerkin methods for first order hyperbolics: An example, SIAM J. Numer. Anal., 10 (1973), 890-899. doi: 10.1137/0710074.  Google Scholar

[18]

L. G. Farah and M. Scialom, On the periodic "good" Boussinesq equation, Proc. Amer. Math. Soc., 138 (2010), 953-964.  doi: 10.1090/S0002-9939-09-10142-9.  Google Scholar

[19]

J. de Frutos, T. Ortega and J. M. Sanz-Serna, Pseudospectral method for the "good" Boussinesq equation, Math. Comp., 57 (1991), 109-122.  doi: 10.2307/2938665.  Google Scholar

[20]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1977.  Google Scholar

[21]

S. Gottlieb, F. Tone, C. Wang, X. Wang and D. Wirosoetisno, Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 50 (2012), 126-150.  doi: 10.1137/110834901.  Google Scholar

[22]

S. Gottlieb and C. Wang, Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers' equation, J. Sci. Comput., 53 (2012), 102-128.  doi: 10.1007/s10915-012-9621-8.  Google Scholar

[23] J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-dependent Problems, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618352.  Google Scholar
[24]

M. Li, P. Guyenne, F. Li and L. Xu, High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model, J. Comput. Phys., 257 (2014), 169-192.  doi: 10.1016/j.jcp.2013.09.050.  Google Scholar

[25]

X. Li, Z. Qiao and C. Wang, Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, preprint, arXiv: 1902.04967. Google Scholar

[26]

Y. Maday and A. Quarteroni, Approximation of Burgers' equation by pseudospectral methods, RAIRO Anal. Numér., 16 (1982), 375-404.  doi: 10.1051/m2an/1982160403751.  Google Scholar

[27]

Y. Maday and A. Quarteroni, Legendre and Chebyshev spectral approximations of Burgers' equation, Numer. Math., 37 (1981), 321-332.  doi: 10.1007/BF01400311.  Google Scholar

[28]

Y. Maday and A. Quarteroni, Spectral and pseudospectral approximations of the Navier-Stokes equations, SIAM J. Numer. Anal., 19 (1982), 761-780.  doi: 10.1137/0719053.  Google Scholar

[29]

V. S. Manoranjan, A. R. Mitchell and J. LL. Morris, Numerical solutions of the "Good" Boussinesq equation, SIAM J. Sci. Statist. Comput., 5 (1984), 946-957.  doi: 10.1137/0905065.  Google Scholar

[30]

V. S. Manoranjan, T. Ortega and J. M. Sanz-Serna, Soliton and antisoliton interactions in the "Good" Boussinesq equation, J. Math. Phys., 29 (1988), 1964-1968.  doi: 10.1063/1.527850.  Google Scholar

[31]

S. Oh and A. Stefanov, Improved local well-posedness for the periodic "good" Boussinesq equation, J. Differential Equations, 254 (2013), 4047-4065.  doi: 10.1016/j.jde.2013.02.006.  Google Scholar

[32]

T. Ortega and J. M. Sanz-Serna, Nonlinear stability and convergence of finite-difference methods for the "good" Boussinesq equation, Numer. Math., 58 (1990), 215-229.  doi: 10.1007/BF01385620.  Google Scholar

[33]

E. Weinan, Convergence of Fourier methods for Navier-Stokes equations, SIAM J. Numer. Anal., 30 (1993), 650-674.  doi: 10.1137/0730032.  Google Scholar

[34]

E. Weinan, Convergence of spectral methods for the Burgers' equation, SIAM J. Numer. Anal., 29 (1992), 1520-1541.  doi: 10.1137/0729088.  Google Scholar

[35]

C. Zhang, J. Huang, C. Wang and X. Yue, On the operator splitting and integral equation preconditioned deferred correction methods for the "Good" Boussinesq equation, J. Sci. Comput., 75 (2018), 687-712.  doi: 10.1007/s10915-017-0552-2.  Google Scholar

[36]

C. Zhang, H. Wang, J. Huang, C. Wang and X. Yue, A second order operator splitting numerical scheme for the "Good" Boussinesq equation, Appl. Numer. Math., 119 (2017), 179-193.  doi: 10.1016/j.apnum.2017.04.006.  Google Scholar

Figure 1.  Discrete $ \ell^2 $ and $ \ell^\infty $ numerical errors for $ D_N^2 u $ at $ T = 4.0 $, plotted versus $ N $, the number of spatial grid point, for the numerical scheme (2.10). The time step size is fixed as $ {\Delta t} = 10^{-4} $. An apparent spatial spectral accuracy is observed for both norms
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Figure 2.  Discrete $ \ell^2 $ and $ \ell^\infty $ numerical errors for $ D_N^2 u $ at $ T = 4.0 $, plotted versus $ N_K $, the number of time steps, for the numerical scheme (2.10). The spatial resolution is fixed as $ N = 512 $. The data lie roughly on curves $ CN_K^{-2} $, for appropriate choices of $ C $, confirming the full second-order temporal accuracy of the scheme
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Figure 3.  The numerical solutions at a sequence of time instants. The outer solid line, the outer dashed line, the inner solid line and the inner dashed line stand for the numerical solutions $ t = 4 $, $ 8 $, $ 12 $ and $ 16 $, respectively. The initial data is given by (5.3), and the physical domain is set as $ \Omega = (-40, 40) $
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