A well-balanced positivity-preserving central-upwind scheme for shallow water equations on unstructured quadrilateral grids

Shirkhani, Hamidreza; Mohammadian, Abdolmajid; Seidou, Ousmane; Kurganov, Alexander

doi:10.1016/j.compfluid.2015.11.017

Cited by 33 publications

(35 citation statements)

References 43 publications

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“…For instance, at this year's SIAM CSE conference 2 , there was a dedicated session on Positivity Preserving and Invariant Domain Preserving Methods, where many of the talks addressed these issues for high-order methods. Some of the more recent publications include [151,152,61,19,121]. Hence, simpler methods may be more convenient when we are dealing with very complex flows, such as multiphase flow models or flows with strong source terms.…”

Section: Motivationmentioning

confidence: 99%

“…Notable results include the paper by Einfeldt et al [34], where it is shown that the Godunov and HLLE schemes are positivity preserving, while the Roe scheme is not, the paper by Batten et al [6] where they showed that the HLLC [139] scheme is positivity preserving with an appropriate choice of wave velocity estimates, the work by Perthame and Shu [116] where they established a general framework to achieve high-order positivity preserving methods for the Euler equations in one and two dimensions, and the book by Bouchut [11] where the conditions on the wave velocities estimates are determined so that the HLLC scheme can also handle vacuum. Areas of interest include the Euler equations (Calgaro et al [14], Hu et al [61], Li et al [87], Zhang and Shu [150,151,152]), shallow water equations [121,71,147,3], magnetohydrodynamics [4,67,40], multiphase flows (Chen and Shu [18]), unstructured meshes (Berthon [8]) and flux-vector splitting methods (Gressier et al [46]), to name just a few. These papers consider standard methods and mostly tackle issues with positivity preserving that arise in high-order methods.…”

Section: Definition 3 (Einfeldt Et Al [34])mentioning

confidence: 99%

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Large Time Step Roe scheme for a common 1D two-fluid model

Prebeg

Flåtten

Müller

2017

Applied Mathematical Modelling

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Large Time Step (LTS) finite volume methods are presented, analyzed and applied to one-dimensional hyperbolic conservation laws.The HLL (Harten-Lax-van Leer) and HLLC (HLL-Contact) schemes are extended to LTS-HLL(C) schemes. The LTS-HLL-type schemes for scalar conservation laws are defined in the numerical viscosity, flux-difference splitting and wave propagation form, and TVD conditions on wave velocity estimates are determined. It is shown that the LTS-HLL-type schemes contain some already existing LTS methods such as the LTS-Roe and LTS-Lax-Friedrichs, and that they allow us to deduce LTS extensions of other methods, such as the Rusanov and Engquist-Osher schemes. The LTS-HLL(C) schemes for systems of conservation laws are developed by standard field-by-field decomposition. Numerical results suggest that the computational efficiency of LTS methods depends on the type of problem under consideration. In certain cases LTS methods for nonlinear systems of equations yield increased accuracy, efficiency and convergence rate compared to standard methods.Entropy stability of LTS-HLL-type schemes is analyzed. We use modified equation analysis to gain insight into numerical diffusion in LTS-HLLtype schemes and to conjecture about entropy stability of LTS methods. Theoretical results obtained through the modified equation analysis are in agreement with numerical results for both scalar conservation laws and the Euler equations.Positivity preservation in LTS methods is investigated. We show that the positivity preserving conditions for LTS methods are stronger than for standard methods. We describe different ways positivity can be lost in LTS-HLL-type schemes for the Euler equations, and we propose a simple way to increase the robustness of the LTS-HLL-type schemes by adding numerical diffusion. Numerical investigations show that LTS-HLL-type schemes successfully handle test cases for positivity preservation, and where the positivity is lost we can improve the robustness by adding numerical diffusion.In addition, we applied the LTS-Roe scheme to a two-fluid model and focused on the treatment of the boundary conditions and the source terms. It is shown that stability and accuracy of the solution can be greatly improved by appropriate treatment of boundary conditions and source terms.i

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Section: Motivationmentioning

confidence: 99%

Section: Definition 3 (Einfeldt Et Al [34])mentioning

confidence: 99%

Large Time Step Roe scheme for a common 1D two-fluid model

Prebeg

Flåtten

Müller

2017

Applied Mathematical Modelling

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“…These models are based on the twodimensional shallow water equations. Over the past few years, several 2D models have been developed using various numerical methods such as finite difference (FD) [2], finite element (FE) [3], and finite volume methods (FV) [4]. DELFT3D and MIKE21 are among the widely used 2D numerical models can be used for river hydrodynamics simulations.…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamic Simulation of an Irregularly Meandering Gravel-Bed River: Comparison of MIKE 21 FM and Delft3D Flow models

Parsapour-Moghaddam

Rennie

Slaney³

2018

E3S Web Conf.

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This study aims at hydrodynamic modelling of Bow River, which passes through the City of Calgary, Canada. Bow River has a mobile gravel bed. Erosion and deposition processes were exacerbated by a catastrophic flood in 2013. Channel banks were eroded at various locations, and large gravel bars formed, which could lead to water level changes and accordingly more flooding. This study investigates the performance of Delft3D-Flow and MIKE 21 FM to simulate the hydrodynamics of the river during the 2013 flood. MIKE 21FM employs unstructured triangular mesh while Delft3D-Flow model uses curvilinear structured grids. Performance of each model was evaluated by the available historical water levels. The results of this study demonstrated that, with approximately the same averaged grid resolution, MIKE 21 FM resulted in more accurate results with a higher computational cost compared to the Delft3DFlow model. It was shown that Delft3D-Flow model may require higher grid cell resolution to result in comparably same depth-averaged velocities throughout the study area. However, considering the balance between the computational cost and the accuracy of the results, both models were capable to adequately replicate the hydrodynamics of the river during the 2013 flood. Results of statistical KS and ANOVA test analysis showed that the model predictions were sensitive to the horizontal eddy viscosity and the Manning roughness. This confirms the necessity of an appropriate calibration of the generated numerical models. The findings of this study shed light on the Bow River flood modelling, which can guide flood management.

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“…For this second test case, we consider a hypothetical twodimensional dam break problem with nonsymmetric breach that is a typical validation made in many presented papers, for example, [30][31][32][33]. An illustration of this problem is shown in Figure 3 …”

Section: Two-dimensional Partial Dam Break Simulationmentioning

confidence: 99%

Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

Chaabelasri

2018

Modelling and Simulation in Engineering

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A simple radial basis function (RBF) meshless method is used to solve the two-dimensional shallow water equations (SWEs) for simulation of dam break flows over irregular, frictional topography involving wetting and drying. At first, we construct the RBF interpolation corresponding to space derivative operators. Next, we obtain numerical schemes to solve the SWEs, by using the gradient of the interpolant to approximate the spatial derivative of the differential equation and a third-order explicit Runge-Kutta scheme to approximate the temporal derivative of the differential equation. For the problems involving shock or discontinuity solutions, we use an artificial viscosity for shock capturing. en, we apply our scheme for several theoretical twodimensional numerical experiments involving dam break flows over nonuniform beds and moving wet-dry fronts over irregular bed topography. Promising results are obtained.

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A well-balanced positivity-preserving central-upwind scheme for shallow water equations on unstructured quadrilateral grids

Cited by 33 publications

References 43 publications

Large Time Step Roe scheme for a common 1D two-fluid model

Large Time Step Roe scheme for a common 1D two-fluid model

Hydrodynamic Simulation of an Irregularly Meandering Gravel-Bed River: Comparison of MIKE 21 FM and Delft3D Flow models

Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

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