Modeling one- and two-dimensional shallow water flows with discontinuous Galerkin method

Numerical models for one- and two-dimensional shallow water flows are developed using discontinuous Galerkin method. Formulation and characteristics of shallow water equations are discussed. The well-balanced property and wetting/drying treatment are provided in the numerical models. The shock-capturing property is achieved by the approximate Riemann solvers in the schemes. Effects of different approximate Riemann solvers are also investigated. The Total Variation Diminishing property is achieved by adoption of slope limiters. Different slope limiters and their effects are compared through numerical tests. Numerical tests are performed to validate the models. These tests include dam-break flows, hydraulic jump and shocks in channels, and flows in natural rivers. Results show that the numerical models developed in present work are robust, accurate, and efficient for modeling shallow water flows.;The one-dimensional model shows that the area based slope limiter provided the best solution in natural channels. The slope limiter based on the water depth or water surface elevation performs progressively poorer as the cross-section shape deviates from rectangular. In the approximate Riemann solver, the wave speeds are based on the original form of the equations, although the pressure force and the gravity force terms are combined for solving the shallow water equations with discontinuous Galerkin method. The combined term is discretized, in one- and two-dimensional models, such that the stationarity property is preserved. Different wetting and drying procedures are evaluated for the one- and two-dimensional models. Analytical, laboratory, and field tests are conducted to verify the accuracy of the wetting and drying procedures.