doi: 10.3934/dcdsb.2021300
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Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

1. 

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

2. 

Institut de Mathématiques de Bourgogne, UMR 5584, Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, 21078 Dijon Cedex, France

3. 

Institut Universitaire de France

*Corresponding author: Vincent Duchêne

Received  March 2021 Revised  August 2021 Early access December 2021

Fund Project: This work is partially supported by the ANR-FWF project ANuI - ANR-17-CE40-0035, the isite BFC project NAANoD, the EIPHI Graduate School (contract ANR-17-EURE-0002) and by the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie RISE 2017 grant agreement no. 778010 IPaDEGAN

We perform numerical experiments on the Serre-Green-Naghdi (SGN) equations and a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the emergence of modulated oscillations and the possibility of a blow-up of solutions in various situations is investigated.

We argue that a simple numerical scheme based on a Fourier spectral method combined with the Krylov subspace iterative technique GMRES to address the elliptic problem and a fourth order explicit Runge-Kutta scheme in time allows to address efficiently even computationally challenging problems.

Citation: Vincent Duchêne, Christian Klein. Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021300
References:
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References:
[1]

S. AbendaT. Grava and C. Klein, Numerical solution of the small dispersion limit of the Camassa-Holm and Whitham equations and multiscale expansions, SIAM J. Appl. Math., 70 (2010), 2797-2821.  doi: 10.1137/090770278.  Google Scholar

[2]

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T. Alazard, N. Burq and C. Zuily, Cauchy theory for the water waves system in an analytic framework, preprint, arXiv: 2007.08329. To appear in Tokyo Journal of Mathematics. Google Scholar

[4]

B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J., 57 (2008), 97-131.  doi: 10.1512/iumj.2008.57.3200.  Google Scholar

[5]

C. J. Amick, Regularity and uniqueness of solutions to the Boussinesq system of equations, J. Differential Equations, 54 (1984), 231-247.  doi: 10.1016/0022-0396(84)90160-8.  Google Scholar

[6]

C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal., 76 (1981), 9-95.  doi: 10.1007/BF00250799.  Google Scholar

[7]

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[8]

H. Bae and R. Granero-Belinchón, Singularity formation for the Serre-Green-Naghdi equations and applications to $abcd$-Boussinesq systems, Monatsh Math, 2021. doi: 10.1007/s00605-021-01623-8.  Google Scholar

[9]

E. Barthélemy, Nonlinear shallow water theories for coastal waves, Surveys in Geophysics, 25 (2004), 315-337.   Google Scholar

[10]

S. Bazdenkov, N. Morozov and O. Pogutse, Dispersive effects in two-dimensional hydrodynamics, Soviet Physics Doklady, 32 (1987), 262, In Russian. Google Scholar

[11]

P. BonnetonF. ChazelD. LannesF. Marche and M. Tissier, A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, J. Comput. Phys., 230 (2011), 1479-1498.  doi: 10.1016/j.jcp.2010.11.015.  Google Scholar

[12]

M. CaliariP. KandolfA. Ostermann and S. Rainer, The Leja method revisited: Backward error analysis for the matrix exponential, SIAM J. Sci. Comput., 38 (2016), 1639-1661.  doi: 10.1137/15M1027620.  Google Scholar

[13]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[14]

R. CamassaD. D. Holm and C. D. Levermore, Long-time effects of bottom topography in shallow water, Phys. D, 98 (1996), 258-286.  doi: 10.1016/0167-2789(96)00117-0.  Google Scholar

[15]

J. D. Carter, Bidirectional Whitham equations as models of waves on shallow water, Wave Motion, 82 (2018), 51-61.  doi: 10.1016/j.wavemoti.2018.07.004.  Google Scholar

[16]

J. D. Carter and R. Cienfuegos, The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations, Eur. J. Mech. B Fluids, 30 (2011), 259-268.  doi: 10.1016/j.euromechflu.2010.12.002.  Google Scholar

[17]

R. CienfuegosE. 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

[18]

D. Clamond and D. Dutykh, Practical use of variational principles for modeling water waves, Phys. D, 241 (2012), 25-36.  doi: 10.1016/j.physd.2011.09.015.  Google Scholar

[19]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[20]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[21]

F. Dias and P. Milewski, On the fully-nonlinear shallow-water generalized Serre equations, Phys. Lett., A, 374 (2010), 1049-1053.  doi: 10.1016/j.physleta.2009.12.043.  Google Scholar

[22]

E. DinvayD. Dutykh and H. Kalisch, A comparative study of bi-directional Whitham systems, Appl. Numer. Math., 141 (2019), 248-262.  doi: 10.1016/j.apnum.2018.09.016.  Google Scholar

[23]

V. A. Dorodnitsyn, E. I. Kaptsov and S. V. Meleshko, Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations, preprint, arXiv: 2008.12852. Google Scholar

[24]

V. Duchêne, Many models for water waves, Open Math Notes, OMN: 202109.111309. Google Scholar

[25]

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Figure 1.  Left: solitary wave for the WGN equations for $ c = 1.1 $ in blue and the SGN equations for the same velocity in red; right: Fourier coefficients for both solitary waves on the left
Figure 2.  Left: solitary wave for the WGN equations for $ c = 2 $ in blue and the SGN equations for the same velocity in red; right: Fourier coefficients for the solitary waves on the left
Figure 3.  Left: solitary wave for the WGN equations for $ c = 20 $ in blue and the SGN equations for the same velocity in red; right: Fourier coefficients on the left
Figure 4.  Left: solitary wave for the WGN equations for $ c = 100 $ in blue and the SGN equations for the same velocity in red; right: Fourier coefficients on the left
Figure 5.  The function $ \zeta $ for the solitary wave for the WGN equations in blue and the SGN equations for the same velocity in red: left $ c = 20 $, right $ c = 100 $
Figure 6.  Left: discrete Fourier transform of the Jacobian for the initial iterate (SGN solitary wave) for $ c = 20 $; right: non diagonal part $ \hat J^{*} $ on the left
Figure 7.  Left: modulus of the Fourier coefficients of the numerical solution to the SGN equations for solitary wave initial data with $ c = 2 $ at t = 1; right: the difference between this solution to the SGN equations and the exact solution
Figure 8.  Left: modulus of the Fourier coefficients of the numerical solution to the WGN equations for solitary wave initial data with $ c = 2 $ at t = 1; right: the difference between this solution to the WGN equations and the exact solution
Figure 9.  Solution to the SGN equations for the initial data $ u(x, t = 0) = 0.99 u_c(x) $ for $ c = 2 $ and $ \zeta(x, t = 0) = \zeta_2(x) $
Figure 10.  $ L^{\infty} $ norms of the solutions to the SGN equations for the initial data $ \zeta(x, t = 0) = \zeta_2(x) $ and $ u(x, t = 0) = \lambda u_2(x) $, on the left for $ \lambda = 0.99 $, on the right for $ \lambda = 1.01 $
Figure 11.  $ L^{\infty} $ norms of the solutions to the SGN equations for initial data $ \zeta(x, t = 0) = \zeta_2(x) $, $ u(x, t = 0) = u_2(x)- 0.01\exp(-x^{2}) $ on the left and $ u(x, t = 0) = u_2(x)+ 0.01\exp(-x^{2}) $ on the right
Figure 12.  $ L^{\infty} $ norms of the solutions to the WGN equations for the initial data $ \zeta(x, t = 0) = \zeta_2(x) $ and $ u(x, t = 0) = \lambda u_2(x) $ in the upper row, on the left for $ \lambda = 0.99 $, on the right for $ \lambda = 1.01 $; and for $ u(x, t = 0) = u_2(x)\pm 0.01\exp(-x^{2}) $ in the lower row, for the minus sign on the left and the plus sign on the right
Figure 13.  $ L^{\infty} $ norms of the solutions to the SGN equations for the initial data $ u(x, t = 0) = \lambda u_4(x) $, on the left for $ \lambda = 0.99 $, on the right for $ \lambda = 1.01 $
Figure 14.  $ L^{\infty} $ norms of the solutions to the SGN equations for the initial data $ \zeta(x, 0) = \zeta_{10}(x) $, and $ u(x, t = 0) = 1.01 u_{10}(x) $ on the left and $ u(x, t = 0) = u_{10}(x)+0.1\exp(-x^{2}) $ on the right
Figure 15.  Solution to the SGN equations for the initial data $ u(x, t = 0) = u_{10}(x+40))+0.1\exp(-((x+40)^{2}) $, on the left $ u $, on the right $ \zeta $
Figure 16.  Solution to the SGN equations (17) with $ \delta = 0.1 $ for $ \zeta(x, t = 0) = \exp(-(x+3)^{2}) $, $ u(x, t = 0) = 2\sqrt{1+\zeta(x, t = 0)} - 2 $ in dependence of time; on the left $ u $, on the right $ \zeta $
Figure 17.  Function $ \zeta $ of Fig. gauss1e1 at $ t = 5 $ on the left, and the Fourier coefficients on the right
Figure 18.  Solution to the SGN equations (17) with $ \delta = 0.01 $ for $ \zeta(x, t = 0) = \exp(-(x+3)^{2}) $, $ u(x, t = 0) = 2\sqrt{1+\zeta(x, t = 0)} - 2 $ at $ t = 1.3 $; on the left $ u $, on the right $ \zeta $
Fig. 18">Figure 19.  Fourier coefficients of the solutions shown in Fig. 18
Figure 20.  Solution to the WGN equations (18) with $ \delta = 0.1 $ for $ \zeta(x, t = 0) = \exp(-(x+3)^{2}) $, $ u(x, t = 0) = 2\sqrt{1+\zeta(x, t = 0)} - 2 $ in dependence of time; on the left $ u $, on the right $ \zeta $
Fig. 20">Figure 21.  Fourier coefficients of the solutions shown in Fig. 20
Figure 22.  Solution to the SGN equations for the initial data (23) with $ \delta = 0.1 $
Figure 23.  Fourier coefficients of the solutions shown in Fig. SGNCH for $ t = 10 $
Figure 24.  Solution to the SGN equations for the initial data (24), on the left $ u $, on the right $ \zeta $
Figure 25.  Solution to the SGN equations for the initial data (24) for $ t = 3 $, on the left $ u $, on the right $ \zeta $
Figure 26.  $ L^{\infty} $ norms of the solution $ \zeta $ to the SGN equations for the initial data (24) on the left and for its gradient on the right
Figure 27.  The Fourier coefficients of the solution to the SGN equations for the initial data (24), on the left $ u $, on the right for $ \zeta $
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