AIMS Mathematics, 2019, 4(2): 254-278. doi: 10.3934/math.2019.2.254.

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On time relaxed schemes and formulations for dispersive wave equations

1 Université de Picardie Jules Verne, LAMFA CNRS UMR 7352, 33, rue Saint-Leu, 80039 Amiens, France
2 Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France

The numerical simulation of nonlinear dispersive waves is a central research topic of many investigations in the nonlinear wave community. Simple and robust solvers are needed for numerical studies of water waves as well. The main diFFIculties arise in the numerical approximation of high order derivatives and in severe stability restrictions on the time step, when explicit schemes are used. In this study we propose new relaxed system formulations which approximate the initial dispersive wave equation. However, the resulting relaxed system involves first order derivatives only and it is written in the form of an evolution problem. Thus, many standard methods can be applied to solve the relaxed problem numerically. In this article we illustrate the application of the new relaxed scheme on the classical Korteweg–de Vries equation as a prototype of stiFF dispersive PDEs.
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Keywords dispersive wave equations; shallow water flows; relaxation; quasi-compressibility

Citation: Jean-Paul Chehab, Denys Dutykh. On time relaxed schemes and formulations for dispersive wave equations. AIMS Mathematics, 2019, 4(2): 254-278. doi: 10.3934/math.2019.2.254

References

  • 1.D. C. Antonopoulos, V. A. Dougalis and D. E. Mitsotakis, Galerkin approximations of the periodic solutions of Boussinesq systems, Bulletin of Greek Math. Soc., 57 (2010), 13-30.
  • 2.M. Antuono, V. Y. Liapidevskii and M. Brocchini, Dispersive Nonlinear Shallow-Water Equations, Stud. Appl. Math., 122 (2009), 1-28.    
  • 3.T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. T. R. Soc. A, 272 (1972), 47-78.    
  • 4. F. Benkhaldoun and M. Seaïd, New finite-volume relaxation methods for the third-order differential equations, Commun. Comput. Phys., 4 (2008), 820-837.
  • 5.J. L. Bona, V. A. Dougalis and D. E. Mitsotakis, Numerical solution of KdV-KdV systems of Boussinesq equations: I. The numerical scheme and generalized solitary waves, Math. Comput. Simulat., 74 (2007), 214-228.    
  • 6.P. Bonneton, F. Chazel, D. Lannes, et al. A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, J. Comput. Phys., 230 (2011), 1479-1498.    
  • 7.J. V. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l'Acad. des Sci. Inst. Nat. France, 1877.
  • 8.H. Chen, M. Chen and N. Nguyen, Cnoidal Wave Solutions to Boussinesq Systems, Nonlinearity, 20 (2007), 1443-1461.    
  • 9.R. Cienfuegos, E. Barthélémy and P. Bonneton, A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: Model development and analysis, Int. J. Numer. Meth. Fl., 51 (2006), 1217-1253.    
  • 10.D. Clamond, Cnoidal-type surface waves in deep water, J. Fluid Mech., 489 (2003), 101-120.    
  • 11.A. Duran, D. Dutykh and D. Mitsotakis, On the Galilean Invariance of Some Nonlinear Dispersive Wave Equations, Stud. Appl. Math., 131 (2013), 359-388.    
  • 12.A. Durán, D. Dutykh and D. Mitsotakis, Peregrine's System Revisited. In N. Abcha, E. N. Pelinovsky, and I. Mutabazi, editors, Nonlinear Waves and Pattern Dynamics, pp. 3-43, Springer International Publishing, Cham, 2018.
  • 13.D. Dutykh, D. Clamond, P. Milewski, et al. Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, Eur. J. Appl. Math., 24 (2013), 761-787.    
  • 14.D. Dutykh, T. Katsaounis and D. Mitsotakis, Finite volume schemes for dispersive wave propagation and runup, J. Comput. Phys., 230 (2011), 3035-3061.    
  • 15.D. Dutykh, T. Katsaounis and D. Mitsotakis, Finite volume methods for unidirectional dispersive wave models, Int. J. Numer. Meth. Fl., 71 (2013), 717-736.    
  • 16.C. Eskilsson and S. J. Sherwin, Discontinuous Galerkin Spectral/hp Element Modelling of Dispersive Shallow Water Systems, J. Sci. Comput., 22 (2005), 269-288.
  • 17.F. Fedele and D. Dutykh, Vortexons in axisymmetric Poiseuille pipe flows, EPL, 101 (2013), 34003.
  • 18.R. Grimshaw, Internal Solitary Waves. In R. Grimshaw, editor, Environmental Stratified Flows, pp. 1-27, Springer US, 2002.
  • 19.M. S. Ismail, A finite difference method for Korteweg-de Vries like equation with nonlinear dispersion, Int. J. Comput. Math., 74 (2000), 185-193.    
  • 20.R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge University Press, Cambridge, 1997.
  • 21.M. Kameyama, A. Kageyama and T. Sato, Multigrid iterative algorithm using pseudo-compressibility for three-dimensional mantle convection with strongly variable viscosity, J. Comput. Phys, 206 (2005), 162-181.    
  • 22.D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443.    
  • 23.S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42.    
  • 24.D. Levy, C.-W. Shu and J. Yan, Local discontinuous Galerkin methods for nonlinear dispersive equations, J. Comput. Phys., 196 (2004), 751-772.    
  • 25.D. Mitsotakis, D. Dutykh and J. Carter, On the nonlinear dynamics of the traveling-wave solutions of the Serre system, Wave Motion, 70 (2017), 166-182.    
  • 26.D. Mitsotakis, B. Ilan and D. Dutykh, On the Galerkin/Finite-Element Method for the Serre Equations, J. Sci. Comput., 61 (2014), 166-195.    
  • 27.D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.    
  • 28.D. H. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827.    
  • 29.A. V. Porubov and G. A. Maugin, Propagation of localized longitudinal strain waves in a plate in the presence of cubic nonlinearity, Phys. Rev. E, 74 (2006), 46617.
  • 30.H. Schamel, A modified Korteweg-de Vries equation for ion acoustic wavess due to resonant electrons, J. Plasma Phys., 9 (1973), 377-387.    
  • 31.F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux, La Houille blanche, 8 (1953), 374-388.
  • 32.J. J. Stoker, Water Waves: The mathematical theory with applications, Interscience, New York, 1957.
  • 33.M. Walkley and M. Berzins, A finite element method for the one-dimensional extended Boussinesq equations, Int. J. Numer. Meth. Fl., 29 (1999), 143-157.    
  • 34.M. Walkley and M. Berzins, A finite element method for the two-dimensional extended Boussinesq equations, Int. J. Numer. Meth. Fl., 39 (2002), 865-885.    
  • 35.M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Commun. Part. Diff. Eq., 12 (1987), 1133-1173.    
  • 36.J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40 (2002), 769-791.    

 

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