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Israwi, Samer; Mourad, Ayman

An explicit solution with correctors for the Green-Naghdi equations. (English) Zbl 1292.35264

Mediterr. J. Math. 11, No. 2, 519-532 (2014).
Summary: In this paper, the water waves problem for uneven bottoms in a highly nonlinear regime is studied. It is well known that, for such regimes, a generalization of the Boussinesq equations called the Green-Naghdi equations can be derived and justified when the bottom is variable [D. Lannes and P. Bonneton, Phys. Fluids 21, No. 1, Paper No. 016601, 9 p. (2009; Zbl 1183.76294)]. Moreover, the Green-Naghdi and Boussinesq equations are fully nonlinear and dispersive systems. We derive here new linear asymptotic models of the Green-Naghdi and Boussinesq equations so that they have the same accuracy as the standard equations. We solve explicitly the new linear models and numerically validate the results.

Cited in 5 Documents

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35L99 Hyperbolic equations and hyperbolic systems
35C05 Solutions to PDEs in closed form
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35B40 Asymptotic behavior of solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Keywords:

water waves; Green-Naghdi and Boussinesq systems; topography effect; asymptotic model; explicit solution; numerical validation

Citations:

Zbl 1183.76294
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Full Text: DOI

References:

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