A discontinuous Galerkin method for three-dimensional shallow water flows with free surface

Abstract

Numerical modeling of shallow-water flows, especially in case of systems involving discontinuities, shock waves, and greatly varying flow regimes, poses a number of challenges. Most existing numerical methods, with their well known limitations with regard to stability, local mass and momentum conservation, ability to accommodate hp-adaptivity, and parallel implementation, fail to address these challenges in an adequate way. The local discontinuous Galerkin (LDG) method, on the contrary, offers unique advantages in terms of flexibility in approximation space choice, cheap parallel implementation, and stability control. Though demonstrated to work well for the 2D shallow-water equations as well as for the compressible Navier-Stokes equations in 3D, formulation of the LDG method for the 3D shallow-water equations with a free surface requires dealing with a number of nontrivial issues quite different from anything encountered in the 2D case. Preliminary results obtained from application of the proposed scheme to several test problems indicate excellent stability and accuracy properties, provided we use a special numerical flux formulation on the inter-element boundaries.