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We present a novel class of locally conservative, entropy stable andwell-balanced discontinuous Galerkin (DG) methods for the nonlinear shallowwater equation with  a non-flat bottom topography. The major novelty of our work is the use ofvelocity field as an independent solution unknown in the DG scheme, which isclosely related to the entropy variable approach to entropy stable schemes forsystem of conservation laws proposed by Tadmor [22] back in 1986, where recallthat velocity is part of the entropy variable for the shallow water equations.Due to the use of velocity as an independent solution unknown, no specificnumerical quadrature rules are needed to achieve entropy stability of ourscheme on general unstructured meshes in two dimensions.  The proposed DG semi-discretization is then carefully combined with theclassical explicit strong stability preserving Runge-Kutta (SSP-RK) timeintegrators [13] to yield a locally conservative, well-balanced, and positivitypreserving fully discrete scheme. Here the positivity preservation property isenforced with the help of a simple scaling limiter. In the fully discretescheme, we re-introduce discharge as an auxiliary unknown variable. In doingso, standard slope limiting procedures can be applied on the conservativevariables (water height and discharge) without violating the local conservationproperty. Here we apply a characteristic-wise TVB limiter [5] on theconservative variables using the Fu-Shu troubled cell indicator [10] in eachinner stage of the Runge-Kutta time stepping to suppress numericaloscillations.

A high-order velocity-based discontinuous Galerkin scheme for the  shallow water equations: local conservation, entropy stability, well-balanced  property, and positivity preservation