Communications in Mathematical Sciences
Volume 16 (2018)
A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows
Pages: 1169 – 1202
In geophysics, the shallow water model is a good approximation of the incompressible Navier–Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green–Naghdi models); the second one by proposing a multilayer version of the shallow water model.
In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to approximate the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity.
free surface flows, semi-discretisation in space, dispersive models, energy estimates, linear dispersion relations
2010 Mathematics Subject Classification
35B45, 35Q31, 65M08, 76B15
Received 25 July 2017
Received revised 27 January 2018
Accepted 27 January 2018
Published 19 December 2018