A posteriori Finite-Volume local subcell correction of high-order discontinuous Galerkin schemes for the nonlinear shallow-water equations

•New subcell DG-FV formulation for nonlinear Shallow water equation.•The resulting algorithm accurately handles strong shocks with no robustness issues.•It ensures the preservation of the water height positivity at the subcell level.•It preserves the class of motionless steady states (well-balancing).•It retains the highly accurate subcell resolution of DG schemes. We design and analyze a new discretization method for the nonlinear shallow water equations, which is based on an equivalent representation of arbitrary high-order Discontinuous Galerkin (DG) schemes through piecewise constant modes on a sub-grid, together with a selective a posteriori local correction of the sub-interface reconstructed flux. This new approach, based on F. Vilar (2019) [101], allows to combine at the subcell scale the excellent robustness properties of the Finite-Volume (FV) lowest-order method and the high-order accuracy of the DG method. For any order of polynomial approximation, the resulting algorithm is shown to: (i) accurately handle strong shocks with no robustness issues; (ii) ensure the preservation of the water height positivity at the subcell level; (iii) preserve the class of motionless steady states (well-balancing); (iv) retain the highly accurate subcell resolution of DG schemes. These assets are numerically illustrated through an extensive set of test-cases, with a particular emphasize put on the use of very-high order polynomial approximations on coarse grids.