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 wet-dry scheme for moving boundary treatment is implemented in the framework of discontinuous Galerkin shallow water equations. As a formulation of approximate Riemann solver, the HLL (Harten-Lax-van Leer) numerical fluxes are employed and the TVB (Total Variation Bounded) slope limiter is used for the removal of unnecessary oscillations. As benchmark test problems, the dam-break problems and the classical problem of periodic oscillation in the parabolic bowl are solved with linear triangular elements and second-order Runge-Kutta scheme. The results are compared with exact solutions and the numerical solutions of previous study. For a more practical application, the implicit Runge-Kutta scheme is employed for the bottom friction terms and the moving shoreline in a rectangular basin of varying slopes is simulated. In all case studies, good agreement is observed with exact solutions or other well-known numerical solutions.

Numerical solutions of 1D and 2D shallow water equations are presented with Runge-Kutta Discontinuous Galerkin (RKDG) finite element method. For 1D problems, the transcritical flows such as dam-break flows and a flow over a bump with hydraulic jump were simulated and the numerical solutions were compared with the exact solutions. As a formulation of approximate Riemann solver, the Local Lax-Wendroff (LLF) fluxes were employed and minmod slope limiter was used for 1D flows. For 2D applications, the classical problems of lateral transition were simulated. The HLL (Harten — Lax — van Leer) fluxes were adopted and a van Albada type gradient-reconstruction type slope limiter was applied. For the time integration, 3^rd-order and 2^nd-order TVD Runge-Kutta schemes were used for 1D and 2D simulations, respectively. In all case studies, good agreement was observed.

Combined sewer overflows (CSOs) contain a mixture of untreated, contaminated sanitary, and storm water. CSOs include both precipitated and suspended matters. In this study, a novel system is proposed to treat CSOs using a combined hydrocyclone and dissolved air flotation (DAF) system for removal of both precipitated and suspended matter. The optimum operating conditions were determined by changing the internal flow rate and the split ratio (R_f) in the hydrocyclone and by changing the angle of the nozzle to determine the floc size, flow rate, and recycle ratio in the DAF system. The types of organic matter contained in prepared synthetic water samples are categorized as soluble COD_cr (SCOD) and particulated COD_cr (PCOD) in the characterization of the removal rate of the precipitated and suspended matter. The suspended solids (SS) were classified by their specific gravity to determine the removal rate of both precipitated and suspended matter. In a pretreatment process, the system employed a cationic polymer to obtain a low gradient time (G × t) value. The results showed that poly-aluminum chloride (PAC) used with the cationic polymer had a 9.2% higher turbidity removal rate than PAC alone, due to a stable floc being achieved in the hydrocyclone. The optimum operating conditions of the hydrocyclone showed the highest efficiency at R_f = 10% at a flow rate of 2.8 m/s; the DAF-treated solutions showed a higher removal rate for SCOD solutions vs. PCOD solutions, with the highest turbidity removal rate of 65%. In contrast, when the hydrocyclone-treated solutions had a higher PCOD than SCOD, they showed a higher removal efficiency. The removal rates of SS, COD_cr, turbidity, T-P, and T-N of the total process were 77, 80, 98, 98, and 45%, respectively.