Wencong Lai

Discontinuous Galerkin Method for 1D Shallow Water Flows in Natural Rivers

Abstract A numerical model is proposed for the solution of one-dimensional shallow water flow equations for natural rivers. This model is based on the total variation diminishing Runge-Kutta discontinuous Galerkin finite element method. In natural rivers, the cross-section shape and bed slope can be quite irregular, which requires a compatible discretization scheme for the bed slope term and net pressure force term. Therefore, in this model, the hydrostatic pressure force term and the wall pressure force term are combined and a new discretization for the resulting term is introduced. This formulation is shown to prevent unphysical flow due to improper treatment of bottom slope term. The mass and momentum flux term are calculated by HLL Riemann solver. A scheme is presented to model flow over dry bed. To evaluate the numerical scheme, tests are conducted for idealized dambreak problems in parabolic (with wet and dry beds), and rectangular channels, hydraulic jump in a rectangular channel, dambreak in the Teton River (Idaho, USA) and the Toce River (Northern Alps, Italy), and flooding event in the East Fork River (Wyoming, USA). The comparison of the computational results with analytical and laboratory results of dam break flows shows that the model is capable of handling flow over dry areas. The simulation results for hydraulic jump show the discharge conservation property and shock prediction capability of the model. The dambreak and flood simulations in natural channels show that the model is capable of handling flows in highly varying bed topography and channel geometry.

Discontinuous Galerkin Method for 1D Shallow Water Flow in Nonrectangular and Nonprismatic Channels

A total variation diminishing Runge-Kutta discontinuous Galerkin finite element method for the solution of one-dimensional (1D) shallow water flow equations for natural channels is presented. The hydrostatic pressure force and the wall pressure force terms are combined to simplify the calculations and prevent unphysical flow attributable to improper treatment of the bottom slope term. The treatment of the combined term that appropriately accounts for the momentum flux is given. HLL and Roe Riemann solvers are assessed for the mass and momentum flux terms. Numerical tests are conducted using prismatic rectangular and nonrectangular channels as well as non prismatic channels and natural channel for dam break, supercritical flow, transcritical flow, and dry-bed problems. Slope limiters based on flow cross section area, water surface, and water depth are evaluated. The tests show that HLL and Roe solvers provide similar accuracy. However, the slope limiter based on flow area provides more accurate solutions f...

Modeling dam-break flood over natural rivers using Discontinuous Galerkin method

A well-balanced numerical model is presented for two-dimensional, depth-averaged, shallow water flows based on the Discontinuous Galerkin (DG) method. The model is applied to simulate dam-break flood in natural rivers with wet/dry bed and complex topography. To eliminate numerical imbalance, the pressure force and bed slope terms are combined in the shallow water flow equations. For partially wet/dry elements, a treatment of the source term that preserves the well-balanced property is presented. A treatment for modeling flow over initially dry bed is presented. Numerical results show that the time step used is related to the dry bed criterion. The intercell numerical flux in the DG method is computed by the Harten-Lax-van Contact (HLLC) approximate Riemann solver. A two-dimensional slope limiting procedure is employed to prevent spurious oscillation. The robustness and accuracy of the model are demonstrated through several test cases, including dam-break flow in a channel with three bumps, laboratory dam-break tests over a triangular bump and an L-shape bend, dam-break flood in the Paute River, and the Malpasset dam-break case. Numerical results show that the model is robust and accurate to simulate dam-break flood over natural rivers with complex geometry and wet/dry beds.

Time stepping in discontinuous Galerkin method

The time discretization in the Discontinuous Galerkin (DG) scheme has been traditionally based on the Total Variation Diminishing (TVD) second-order Runge-Kutta (RK2) scheme. Computational efficiency and accuracy with the Euler Forward (EF) and the TVD second-order RK2 time stepping schemes in the DG method are investigated in this work. Numerical tests are conducted with the scalar Burgers equation, 1-D and 2-D shallow water flow equations. The maximum Courant number or time step size required for stability for the EF scheme and RK2 scheme with different slope limiters are compared. Numerical results show that the slope limiters affect the stability requirement in the DG method. The RK2 scheme is generally more diffusive than the EF scheme, and the RK2 scheme allows larger time step sizes. The EF scheme is found to be more efficient and accurate than the RK2 scheme in the DG method in computation.

A Parallel Two-Dimensional Discontinuous Galerkin Method for Shallow-Water Flows Using High-Resolution Unstructured Meshes

AbstractFlood modeling and forecasting using hydraulic models are computationally expensive for high-resolution, large-scale problems. With the advent of high-performance computing, numerical models can obtain promising speedup using parallel computing resources. In this paper, an message passing interface (MPI)-based parallel shallow-water flow solver using the discontinuous Galerkin method is presented. Parallelization is implemented with static domain decomposition using the single program–multiple data design. Data transfer between subdomains are achieved using the MPI implementation. The parallel solver is tested using idealized dam-break tests and field tests, for both fully wet and partially wet domains. The performance of parallel speedup and efficiency are compared for different mesh resolution and cluster architecture. A statistical index (maximum ratio of halo cells to total number of elements) is introduced to evaluate the parallel efficiency and performance, and this index can be used as a gu...