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

Abstract

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