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Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds
A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation
†. | 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 |
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 [
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The Adomian decomposition method for solving the Boussinesq equation arising in water wave propagation, Numer. Methods Partial Differential Equations, 22 (2006), 1337-1347.
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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. |
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J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications, Inc., Mineola, NY, 2001. |
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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. |
[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. |
[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. |
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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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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On the periodic "good" Boussinesq equation, Proc. Amer. Math. Soc., 138 (2010), 953-964.
doi: 10.1090/S0002-9939-09-10142-9. |
[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. |
[20] |
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1977. |
[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. |
[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. |
[23] |
J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-dependent Problems, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9780511618352.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[27] |
Y. Maday and A. Quarteroni,
Legendre and Chebyshev spectral approximations of Burgers' equation, Numer. Math., 37 (1981), 321-332.
doi: 10.1007/BF01400311. |
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
E. Weinan,
Convergence of Fourier methods for Navier-Stokes equations, SIAM J. Numer. Anal., 30 (1993), 650-674.
doi: 10.1137/0730032. |
[34] |
E. Weinan,
Convergence of spectral methods for the Burgers' equation, SIAM J. Numer. Anal., 29 (1992), 1520-1541.
doi: 10.1137/0729088. |
[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. |
[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. |
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. |
[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. |
[3] |
J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications, Inc., Mineola, NY, 2001. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
T. Dupont, Galerkin methods for first order hyperbolics: An example, SIAM J. Numer. Anal., 10 (1973), 890-899.
doi: 10.1137/0710074. |
[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. |
[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. |
[20] |
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1977. |
[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. |
[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. |
[23] |
J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-dependent Problems, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9780511618352.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[27] |
Y. Maday and A. Quarteroni,
Legendre and Chebyshev spectral approximations of Burgers' equation, Numer. Math., 37 (1981), 321-332.
doi: 10.1007/BF01400311. |
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
E. Weinan,
Convergence of Fourier methods for Navier-Stokes equations, SIAM J. Numer. Anal., 30 (1993), 650-674.
doi: 10.1137/0730032. |
[34] |
E. Weinan,
Convergence of spectral methods for the Burgers' equation, SIAM J. Numer. Anal., 29 (1992), 1520-1541.
doi: 10.1137/0729088. |
[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. |
[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. |



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