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Group properties of the extended Green–Naghdi equations
Abstract
The group analysis method is applied to the extended Green–Naghdi equations. The equations are studied in the Eulerian and Lagrangian coordinates. The complete group classification of the equations is provided. The derived Lie symmetries are used to reduce the equations to ordinary differential equations. For solving the ordinary differential equations the Runge–Kutta methods were applied. Comparisons between solutions of the Green–Naghdi equations and the extended Green–Naghdi equations are given.
... The original analytical solution is not periodic and it is stable for small linear perturbations [26]. However, numerical calculations implement a periodic like solution (it was mentioned in Siriwat and Meleshko [41]). Notice that this does not contradict the linear analysis since numerical calculations introduce finite perturbations. ...
... The Serre's solution in Eulerian[41] (left) and mass Lagrangian (center and right) coordinates for R 0 = 0.75 and g = 1. The data for mass Lagrangian coordinates was obtained numerically. ...
The systems of equations of one-dimensional shallow water over uneven bottom in Euler’s and Lagrange’s variables are considered. Intermediate system of equations is introduced. Hydrodynamic conservation laws of intermediate system of equations are used to find all first order conservation laws of shallow water equations in Lagrange’s variables for all bottom profiles. The obtained conservation laws are compared with the hydrodynamic conservation laws of the system of equations of one-dimensional shallow water over uneven bottom in Euler’s variables. Bottom profiles, providing additional conservation laws, are given. The problem of group classification of contact transformations of the
shallow water equation in Lagrange’s variables is solved. First order conservation laws of the shallow water equation in Lagrange’s variables are obtained using Noether’s theorem. In the considered cases, the correspondence between non-divergence symmetries and the original Lagrangian is shown. A similar correspondence is valid for an arbitrary ordinary differential equation of the second order. It is shown that the application of Lagrange’s identity did not allow to find all first order conservation laws of the shallow water equation in Lagrange’s variables.
All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions.
A highly accurate numerical scheme is presented for the Serre system of
partial differential equations, which models the propagation of dispersive
shallow water waves in the fully-nonlinear regime. The fully-discrete scheme
utilizes the Galerkin / finite-element method based on smooth periodic splines
in space, and an explicit fourth-order Runge-Kutta method in time. Computations
compared with exact solitary and cnoidal wave solutions show that the scheme
achieves the optimal orders of accuracy in space and time. These computations
also show that the stability of this scheme does not impose restrictive
conditions on the temporal step size. In addition, solitary, cnoidal, and
dispersive shock waves are studied in detail using this numerical scheme for
the Serre system and compared with the 'classical' Boussinesq system for
small-amplitude shallow water waves. The results show that the interaction of
solitary waves in the Serre system is more inelastic. The efficacy of the
numerical scheme for modeling dispersive shocks is shown by comparison with
asymptotic results. These results have application to the modeling of shallow
water waves of intermediate or large amplitude, such as occurs in the nearshore
zone.
We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for ``depth ratios'' across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system. Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9 figures
A novel method is developed for extending the Green-Naghdi (GN) shallow-water
model equation to the general system which incorporates the arbitrary
higher-order dispersive effects. As an illustrative example, we derive a model
equation which is accurate to the fourth power of the shallowness parameter
while preserving the full nonlinearity of the GN equation, and obtain its
solitary wave solutions by means of a singular perturbation analysis. We show
that the extended GN equations have the same Hamiltonian structure as that of
the GN equation. We also demonstrate that Zakharov's Hamiltonian formulation of
surface gravity waves is equivalent to that of the extended GN system by
rewriting the former system in terms of the momentum density instead of the
velocity potential at the free surface.
The Korteweg‐de Vries equation and the Burgers equation are derived for a wide class of nonlinear Galilean‐invariant systems under the weak‐nonlinearity and long‐wavelength approximations. The former equation is shown to be a limiting form for nonlinear dispersive systems while the latter is a limiting form for nonlinear dissipative systems.
Within the scope of the three-dimensional theory of homogeneous incompressible inviscid fluids, this paper contains a derivation of a system of equations for propagation of waves in water of variable depth. The derivation is effected by means of the incompressibility condition, the energy equation, the invariance requirements under superposed rigid-body motions, together with a single approximation for the (three-dimensional) velocity field.