Hamiltonian regularisation of shallow water equations with uneven bottom

Abstract

The regularisation of nonlinear hyperbolic conservation laws has been a problem of great importance for achieving uniqueness of weak solutions and also for accurate numerical simulations. In a recent work, the first two authors proposed a so-called Hamiltonian regularisation for nonlinear shallow water and isentropic Euler equations. The characteristic property of this method is that the regularisation of solutions is achieved without adding any artificial dissipation or dispersion. The regularised system possesses a Hamiltonian structure and, thus, formally preserves the corresponding energy functional. In the present article we generalise this approach to shallow water waves over general, possibly time-dependent, bottoms. The proposed system is solved numerically with continuous Galerkin method and its solutions are compared with the analogous solutions of the classical shallow water and dispersive Serre-Green-Naghdi equations. The numerical results confirm the absence of dispersive and dissipative effects in presence of bathymetry variations.

Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom

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We prove in this note the local (in time) well-posedness of a broad class of 2 × 2 symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond et al [2]. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.

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“…Inspired by [1], similar regularisations have been proposed for the inviscid Burgers equation [8], the scalar conservation laws [6] and the barotropic Euler system [7]. A regularisation of the Saint-Venant equations with uneven bottom (rSVub) has also been proposed by Clamond et al [2]. The latter equations, for the conservation of mass and momentum, can be written in the conservative form…”

Hamiltonian regularisation of the unidimensional barotropic Euler equations

“…Recently, inspired by [14] and with the same properties as (2), a similar regularisation has been proposed for the inviscid Burgers equation in [24] and for general scalar conservation laws in [23], where solutions exist globally (in time) in H 1 , those solutions converging to solutions of the classical equation when ε → 0 at least for a short time [23,24]. The regularised Saint-Venant system (2) has been also generalised for shallow water equations with uneven bottoms [15].…”