A Godunov‐type fractional semi‐implicit method based on staggered grid for dam‐break modeling

Abstract

A semi-implicit finite volume model based upon staggered grid is presented for solving shallow water equation. The model employs a time-splitting scheme that uses a predictor-corrector method for the advection term. The fluxes are calculated based on a Riemann solver in the prediction step and a downwind scheme in the correction step. A simple TVD scheme is employed for shock capturing purposes in which the Minmond limiter is used for flux functions. As a consequence of using staggered grid, an ADI method is adopted for solving the discretized equations for 2-D problems. Several 1-D and 2-D flows have been modeled with satisfactory results when compared with analytical and experimental test cases. The model is also capable of simulating supercritical as well as subcritical flow.is the TVD-MacCormack method [5,9]. Most numerical methods belonging to these two classes utilize non-staggered grids [13].Staggered grids are often used for modeling subcritical 1-D, 2-D and 3-D hydrostatic currents [13]. Advantages of staggered grid-based methods include their efficiency for combination with ADI methods [14], as well as the fact that no special numerical methods need to be used to prevent the formation of checkered board fields which often arise in subcritical flow [15]. An interesting example of application of staggered-grid scheme in dam-break modeling is the work of Stelling and Duinmeijer in 2003 in which they used an implicit method for modeling of rapidly varied flow using conservation equations for energy, mass, and momentum [13].Recently, fractional methods have been used in modeling shallow water equations [16][17][18]. Time splitting is one of these methods that allows the solution of each term with a different scheme that suits best for that particular term; it is even possible to solve certain terms implicitly while solving other terms explicitly.Here, a finite volume method has been used for solving shallow water equations. The staggeredgrid scheme, which in spite of its advantages, remains still a new approach in the study of the dam-break phenomenon, has been used in this study in conjunction with the Godunov-type shock capturing method. In the present model, a simple shock capturing technique is used that was introduced by Zia and Banihashemi [12] for dam-break modeling. Time-splitting method is also utilized that allows for implicit-explicit solving procedure, which in turn makes it possible to use larger time steps as well as paving the ground to create an implicit link between a 1-D and a 2-D model in integrated 1-D-2-D models without needing to implicitly solve the entire terms of equations. 1-D GOVERNING EQUATIONS

A finite element/volume method model of the depth‐averaged horizontally 2D shallow water equations

“…The time increment is set as 0.01 s. the computational results are diffusive compared with those of the recent high-resolution models [85,87]. Nevertheless, the results are less diffusive than those of the first-order models [88,89] and are comparable with some of the high-resolution models [90,91]. The computational results with the two numerical models are qualitatively different; the lumped model preserves the monotonic surface water profiles of the depression wave upstream of the dam, whereas the original model gives slightly dispersive nature due to the use of the standard Galerkin FEM scheme.…”

“…As expected, the computational results are diffusive compared with those of the recent high‐resolution models . Nevertheless, the results are less diffusive than those of the first‐order models and are comparable with some of the high‐resolution models .…”

A Finite Volume Method for a 2D Dam-Break Simulation on a Wet Bed Using a Modified HLLC Scheme

“…To evaluate the developed model, two experimental cases were considered. In the first case, the experimental test conducted by Fraccarollo and Toro [43] was employed and the data are taken from [45]. In this test, a reservoir sized 2 m × 1 m × 0.8 m was connected to a rectangular channel sized 2 m × 3 m through a rectangular gap sized 0.4 m × 0.8 m. Geometric characteristics of the experimental setup are presented in Figure 3 and Table 2.…”

“…Then, an adaptive graph-based comparison was made between the laboratory and the numerical results. The result of the model has also been compared to the WAF method developed in [38] and the semi-implicit method developed by Asadiani and Banihashemi [45], as seen in Figures 5 and 6 To evaluate and validate the results, the flow depth, and velocity values along the flow direction and perpendicular to it were recorded at predetermined points. Then, an adaptive graph-based comparison was made between the laboratory and the numerical results.…”

“…Then, an adaptive graph-based comparison was made between the laboratory and the numerical results. The result of the model has also been compared to the WAF method developed in [38] and the semi-implicit method developed by Asadiani and Banihashemi [45], as seen in Figures 5 and 6. In addition, a quantitative evaluation was made via calculation of the dimensionless error parameter between the current model, the semi-implicit model, and the WAF model at the specified points.…”