Résumé
This paper contains several new contributions to the numerical approximation of shallow-water equations. We introduce a high-order local subcell monolithic DG/FV discretization method for the approximation of nonlinear free-surface shallow-water equations with source terms. This encompasses both nonlinear hyperbolic shallow-water equations (also known as Saint-Venant equations) and weakly-dispersive fully-nonlinear equations (known as Green-Naghdi equations), approximating the general free-surface water-wave equations in the shallow-water flow regime. The method is grounded on a classical Discontinuous Galerkin (DG) formulation, written as a particular Finite Volume (FV) discretization on a sub-partition, involving reconstructed high-order interface numerical fluxes. In the present work, we investigate the important issue of source terms discretization in the framework of such local subcell monolithic DG/FV strategies, focusing on the bathymetry source terms, and also higher-order differential source terms coming from dispersive corrections for higher-order asymptotic models. The originality of the method is the introduction of a priori blended fluxes, which are convex combinations of the so-called high-order reconstructed fluxes and low-order FV fluxes. The blending coefficients are adaptively computed for each subcell interface to enforce relevant convex properties at the discrete level, including water-height positivity. Starting from the DG piecewise polynomial approximation, the bathymetry source term is also discretized at the subcell level, and we prove that a suitable well-balanced property, ensuring the preservation of motionless steady-states, can be achieved at the subcell-scale relying on some finely tuned combinations of various hydrostatic-like reconstructed states at lower/higher accuracy levels, simultaneously occurring at local and global scales. We also introduce an original and efficient combination of this new wellbalanced local subcell monolithic DG/FV with a high-order Internal Penalty discontinuous-Galerkin method, in order to approximate the higher-order contributions appearing in the Green-Naghdi equations in a consistent and accurate way. We then conduct extensive evaluations of the resulting numerical methods, providing insights about the observed numerical convergence and the overall robustness, including the ability to preserve steadystates, to prevent nonphysical oscillations near discontinuities and to ensure the positivity of the water-height at the discrete level.