A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography

Aureli, Francesca; Maranzoni, Andrea; Mignosa, Paolo; Ziveri, Chiara

doi:10.1016/j.advwatres.2008.03.005

Cited by 113 publications

(89 citation statements)

References 26 publications

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“…In 226 contrast, the depth-gradient method is more stable and robust for bore front tracking. During this 227 work both methods were applied to transcritical flow over weirs; it was found that the surface 228 gradient method leads in some cases to unstable results in the tailwater supercritical portion of the 229 weir face, in agreement with the results of Aureli et al (2008). In contrast, the results using the 230 depth-gradient method were found accurate enough, and, thus, results based on that technique are 231 presented herein.…”

Section: Reconstruction Of Solution 209contrasting

confidence: 42%

“…To obtain second order accuracy in space, a piecewise linear reconstruction is made within each cell 211 Aureli et al (2008), the surface gradient method 225 may lead to oscillations and unphysical depths (even negative) for shallow supercritical flows. In 226 contrast, the depth-gradient method is more stable and robust for bore front tracking.…”

Section: Reconstruction Of Solution 209mentioning

confidence: 99%

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Minimum Specific Energy and Transcritical Flow in Unsteady Open-Channel Flow

Castro-Orgaz

Chanson

2016

J. Irrig. Drain Eng.

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Abstract:The study and computation of free surface flows is of paramount importance in 6 hydraulic and irrigation engineering. These flows are computed using mass and momentum 7 conservation equations whose solutions exhibit special features depending on whether the local 8Froude number is below or above unity, thereby resulting in wave propagation in the up-and 9 downstream directions or only in the downstream direction, respectively. This dynamic condition is 10 referred in the literature to as critical flow and is fundamental to the study of unsteady flows. 11Critical flow is also defined as the state at which the specific energy and momentum reach a 12 minimum, based on steady-state computations, and it is further asserted that the backwater equation

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Section: Reconstruction Of Solution 209contrasting

confidence: 42%

Section: Reconstruction Of Solution 209mentioning

confidence: 99%

Minimum Specific Energy and Transcritical Flow in Unsteady Open-Channel Flow

Castro-Orgaz

Chanson

2016

J. Irrig. Drain Eng.

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“…Several numerical and mathematical treatments have been proposed in the literature for balancing the flux gradient and the source terms (within the Godunov 2D FV framework), in order to properly compute stationary or almost stationary solutions. This property is known as well-balancing and is currently a very active subject of research, we refer for example to [7,8,17,22,30,33,42,51,64,70,73,82,88,91,94,99,111,117,119,121] among many others. The second important problem arising in engineering applications is the appearance of dry areas, due to initial conditions or as a result of the water motion, and as such the necessity to handle wetting and drying moving boundaries is a challenge that has attracted much attention from many researchers, see for example [15,24,26,28,29,36,37,39,40,42,49,75,87,96,102].…”

Section: Introductionmentioning

confidence: 99%

Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows

Delis

Nikolos

Καζολέα

2011

Arch Computat Methods Eng

108

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Finite volume (FV) methods for solving the two-dimensional (2D) nonlinear shallow water equations (NSWE) with source terms on unstructured, mostly triangular, meshes are known for some time now. There are mainly two basic formulations of the FV method: node-centered (NCFV) and cell-centered (CCFV). In the NCFV formulation the finite volumes, used to satisfy the integral form of the equations, are elements of the mesh dual to the computational mesh, while for the CCFV approach the finite volumes are the mesh elements themselves. For both formulations, details are given of the development and application of a second-order well-balanced Godunov-type scheme, developed for the simulation of unsteady 2D flows over arbitrary topography with wetting and drying. The popular approximate Riemann solver of Roe is utilized to compute the numerical fluxes, while second-order spatial accuracy is achieved with a MUSCL-type reconstruction technique. The Green-Gauss (G-G) formulation for gradient computations is implemented for both formulations, in order to maintain a common framework. Two different stencils for A.I. Delis ( ) the G-G gradient computations in the CCFV formulation are implemented and tested. An edge-based limiting procedure is applied for the control of the total variation of the reconstructed field. This limiting procedure is proved to be effective for the NCFV scheme but inadequate for the CCFV approach. As such, a simple but very effective modification to the reconstruction procedure is introduced that takes into account geometrical characteristics of the computational mesh. In addition, consistent well-balanced second-order discretizations for the topography source term treatment and the wet/dry front treatment are presented for both FV formulations, ensuring absolute mass conservation, along with a stable friction term treatment.Using a controlled environment for a fair comparison, a complete assessment of both FV formulations is attempted through rigorous individual and relative performance comparisons to the approximation of analytical benchmark solutions, as well as to experimental and field data. To this end, an extensive evaluation is performed using different time dependent and steady-state test cases that incorporate topography, wetting and drying process, different types of boundary conditions as well as friction. These test cases are chosen as to compare the performance and robustness of each formulation under certain conditions and evaluate the effectiveness of the proposed modifications. Emphasis to grid convergence studies is given, with the grids used to range from regular grids to irregular ones with random perturbations of nodes. The results indicate that the quality of the mesh has a major impact on the convergence performance of the CCFV method while the NCFV method exhibits a uniform behavior on the different grid types used. The proposed correction in the computation of the reconstructed values for the CCFV formulation greatly improves the convergence behavior to the formal order of accurac...

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“…Following many previous researches [22,24,25,28,32], numerical method developers are particularly interested in schemes that are able to maintain the still water surface of a lake at rest (h + z = constant and q = 0) over irregular topology, where extreme slopes are invoked (such as the complex features of the channel bed as illustrated in Figure 1 for the quiescent flow test). For this case of motionless water in a lake, the first (continuity) equation, of (1), is satisfied exactly for any consistent scheme given that the time derivative and the velocity (or discharge) terms automatically vanish.…”

Section: Natural Well-balancing In Second-ordermentioning

confidence: 99%

Well‐balancing issues related to the RKDG2 scheme for the shallow water equations

Kesserwani

Liang

Vázquez³

et al. 2009

Numerical Methods in Fluids

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SUMMARYDiscontinuous Galerkin (DG) finite element methods have salient features that are mainly highlighted by their locality, their easiness in balancing the flux and source term gradients and their component-wise structure. In the light of this, this paper aims to provide insights into the well-balancing property of a second-order Runge-Kutta Discontinuous Galerkin (RKDG2) method. For this purpose, a Godunov-type RKDG2 method is presented for solving the shallow water equations. The scheme is based on local DG linear approximations and does not entail any special treatment of the source terms in order to achieve well-balanced numerical results. The performance of the present RKDG2 scheme in reproducing conserved solutions for both free surface and discharge over strongly irregular topography is demonstrated by applying to several hydraulic benchmarks. Meanwhile, the effects of different slope limiting procedures on the well-balancing property are investigated and discussed. This work may provide useful guidelines for developing a well-balanced RKDG2 numerical scheme for shallow water flow simulation.

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A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography

Cited by 113 publications

References 26 publications

Minimum Specific Energy and Transcritical Flow in Unsteady Open-Channel Flow

Minimum Specific Energy and Transcritical Flow in Unsteady Open-Channel Flow

Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows

Well‐balancing issues related to the RKDG2 scheme for the shallow water equations

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