A High-Order Velocity-Based Discontinuous Galerkin Scheme for the Shallow Water Equations: Local Conservation, Entropy Stability, Well-Balanced Property, and Positivity Preservation

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady flows in rivers and coastal areas. In this work, we present a novel class of locally conservative, entropy stable well-balanced discontinuous Galerkin (DG) methods for equation with non-flat bottom topography. major novelty our work is use velocity field as an independent solution unknown DG scheme, which closely related variable approach schemes system conservation laws proposed by Tadmor (in: Tezduyar, Hughes T (eds) Proceedings winter annual meeting American Society Mechanical Engineering 1986) back 1986, where recall that part equations. Due unknown, no specific numerical quadrature rules needed achieve stability scheme on general unstructured meshes two dimensions. semi-discretization then carefully combined classical explicit strong preserving Runge–Kutta (SSP–RK) time integrators (Gottlieb et al. SIAM Rev. 43, 89–112, 2001) yield well-balanced, positivity fully discrete scheme. Here preservation property enforced help simple scaling limiter. re-introduce discharge auxiliary variable. doing so, standard slope limiting procedures can be applied conservative variables (water height discharge) without violating local property. apply characteristic-wise TVB limiter (Cockburn Shu J Comput Phys 141:199–224, 1998) using Fu-Shu troubled cell indicator (Fu 347:305–327, 2017) each inner stage stepping suppress oscillations. This readily various SWEs simulations dry areas close zero. case need further special attention, approximation unphysically large near cells small height, may eventually crashes simulation if treatment these cells. propose wetting/drying update enhance robustness overall One- two-dimensional experiments presented demonstrate performance methods.