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Hamiltonian regularisation of shallow water equations with uneven bottom

journal contribution
posted on 17.06.2020 by D Clamond, D Dutykh, Dimitrios Mitsotakis
© 2019 IOP Publishing Ltd. 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.

History

Preferred citation

Clamond, D., Dutykh, D. & Mitsotakis, D. (2019). Hamiltonian regularisation of shallow water equations with uneven bottom. Journal of Physics A: Mathematical and Theoretical, 52(42), 42LT01-42LT01. https://doi.org/10.1088/1751-8121/ab3eb2

Journal title

Journal of Physics A: Mathematical and Theoretical

Volume

52

Issue

42

Publication date

23/09/2019

Pagination

42LT01-42LT01

Publisher

IOP Publishing

Publication status

Published

Online publication date

23/09/2019

ISSN

1751-8113

eISSN

1751-8121

Exports