A novel energy-bounded Boussinesq model and a well balanced and stable numerical discretisation

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Many Boussinesq models suffer from nonlinear instabilities, especially in the context of rapid variations in the bed topography. In this work, a Boussinesq system is put forward which is derived in such a way as to be both linearly and nonlinearly energy-stable.The proposed system is designed to be robust for coastal simulations with sharply varying bathymetric features while maintaining the dispersive accuracy at any constant depth. For constant bathymetries, the system has the same linear dispersion relation as Peregrine's system ([22]). Furthermore, the system transitions smoothly to the shallow-water system as the depth goes to zero.In the one-dimensional case, we design a stable finite-volume scheme and demonstrate its robustness, accuracy and stability under grid refinement in a suite of test problems including Dingemans's wave experiment.Finally, we generalise the system to the two-dimensional case.