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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.

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.

Recent development of computing power and theoretical advances in computational fluid dynamics have made possible numerical simulations of water waves with full three-dimensional Navier-Stokes equations. In this study, an internal wave maker using the mass source function approach was combined with the level set finite element method for generation and propagation of water waves. The model is first applied to the two-dimensional linear wave generation and propagation. Then, it is applied to the three-dimensional simulation of the wave generation and the problem of wave force evaluation on the vertical wall. To effectively utilize computational resources and enhance the speed of execution, parallel algorithms are developed and applied for the three-dimensional problems. The results of numerical simulations are compared with theoretical values and good agreements are observed.