A Discontinuous Galerkin Method for Two-Dimensional Shock Wave Modeling

A numerical scheme based on discontinuous Galerkin method is proposed for the two-dimensional shallow water flows. The scheme is applied to model flows with shock waves. The form of shallow water equations that can eliminate numerical imbalance between flux term and source term and simplify computation is adopted here. The HLL approximate Riemann solver is employed to calculate the mass and momentum flux. A slope limiting procedure that is suitable for incompressible two-dimensional flows is presented. A simple method is adapted for flow over initially dry bed. A new formulation is introduced for modeling the net pressure force and gravity terms in discontinuous Galerkin method. To validate the scheme, numerical tests are performed to model steady and unsteady shock waves. Applications include circular dam break with shock, shock waves in channel contraction, and dam break in channel with bend. Numerical results show that the scheme is accurate and efficient to model two-dimensional shallow water flows with shock waves.