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Although the discontinuous Galerkin (DG) methods have been widely applied as an effective numerical tool for hyperbolic conservation equations (e.g., shallow water equations (SWEs) and compressible Navier-Stokes equations), one of the well-known drawbacks is the inconvenience in the treatment of second or higher derivative terms. For this reason, since the beginning of DG in the 1970s, many researchers have made efforts in devising accurate and consistent schemes to incorporate the possible jumps in solutions for higher-order derivatives. The turbulent stress terms of the SWEs are expressed as second-order derivatives and cannot be neglected due to their practical importance. So far, as a traditional approach, a flux formulation has been applied in DG SWE modeling. However, it is often criticized for being inefficient in terms of memory and computation time. In this study, the BR2 scheme, a well-known primal formulation in the DG computational fluid dynamics community, was employed and combined with the implicit Euler backward difference scheme for SWEs. The developed model was applied to four benchmark problems (including channel contraction and diverging, partial dam-break flow, and curved channel flow), and good agreements were observed.

One important issue in the approach with shallow water equations, which is not restricted to discontinuous Galerkin formulation, is the limitation to step geometries (discontinuous bathymetry), due to the hydrostatic assumption employed for the derivation of shallow water equations from the Navier-Stokes equations. In addition, the explicit Runge-Kutta time-stepping schemes do not come without any drawbacks even though the majority of discontinuous Galerkin applications have employed explicit ones due to simplicity. In this study, the recently developed implicit discontinuous Galerkin scheme is combined with the surface gradient method for steps (SGMS). The developed scheme is verified with flows over discontinuous bathymetry, i.e., vertical steps and weirs. For one-dimensional problems, the flows over a step and over a rectangular weir are simulated. As for two-dimensional problems, the flow over a weir and the dam-break flow over a step followed by the L-shaped and 45°-bend channels are simulated. The numerical solutions are compared with the experimental data. In all cases, good agreements were observed and the effectiveness of the developed scheme was verified.

The numerical solutions of shallow water equations are presented with implicit discontinuous Galerkin (IDG) method. The motivations for the development of an implicit scheme is stated. Details of the developed model is described including the spatial and temporal discretization, the approximate Riemann solver, and the slope limiter. For the validation of the model, the channel transition flow of contraction and expansion are carried out. As the last case study, the classical dam-break flow is simulated. In all cases, linear triangular meshes are employed and the implicit backward Euler time integration scheme is used. As an approximate Riemann solver, the Roe numerical flux is employed and a van Albada type gradient-reconstruction type slope limiter was applied. Good agreement was observed with experimental observations and exact solutions in all case studies.

We present a numerical method for free surface flows that couples the incompressible Navier-Stokes equations with the level set method in the finite element framework. The implicit characteristic-Galerkin approximation together with the fractional four-step algorithm is employed to discretize the governing equations. The schemes for solving the level set evolution and reinitialization equations are verified with several benchmark cases, including stationary circle, rotation of a slotted disk and stretching of a circular fluid element. The results are compared with those calculated from the level set finite volume method of Yue et al. , which employed the third-order essentially non-oscillatory (ENO) schemes for advection of the level set function in a generalized curvilinear coordinate system