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

Abstract

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.