A fully implicit Discontinuous Galerkin finite element scheme for the 2D vertically averaged and moment equations

In this work, we introduce a novel fully implicit numerical scheme for the two-dimensional Vertically Averaged and Moment (VAM) system of equations. The method combines a Discontinuous Galerkin (DG) discretization of the homogeneous system with a local Taylor-based reconstruction of the non-conservative terms, ensuring stability without the need for empirical tuning parameters. The full set of equations is advanced in time, following a third-order accurate linear implicit Runge-Kutte (LIRK) method, through a single implicit step, where the fluxes and source terms are linearized via a Taylor-series expansion, thus avoiding computationally expensive iterative solvers. The effectiveness of the approach is demonstrated against experimental benchmarks, showing excellent agreement in both steady and unsteady flow regimes. Notably, the scheme remains robust for Courant-Friedrichs-Lewy (CFL) numbers up to 10, underscoring its potential for efficient large-scale simulations. Most importantly, the proposed formulation enables the simulation of non-hydrostatic pressure flows within a two-dimensional grid, thereby capturing essential three-dimensional effects without the prohibitive cost of fully 3D solvers.