A sign matrix based scheme for non-homogeneous PDE’s with an analysis of the convergence stagnation phenomenon

Mathematics and Computers in Simulation

Sari

Seaı̈d

2015

Self Cite

Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractA simple and accurate projection finite volume method is developed for solving shallow water equations in two space dimensions. The proposed approach belongs to the class of fractional-step procedures where the numerical fluxes are reconstructed using the method of characteristics, while an Eulerian method is used to discretize the conservation equations in a finite volume framework. The method is conservative and it combines advantages of the method of characteristics to accurately solve the shallow water flows with an Eulerian finite volume method to discretize the equations. Numerical results are presented for several applications in rotating shallow water problems. The aim of such a method compared to the conventional finite volume methods is to solve shallow water equations efficiently and with an appropriate level of accuracy.

Section: Numerical Resultsmentioning

confidence: 99%

Section: Projection Methods For Shallow Water Equationsmentioning

confidence: 99%

Section: Finite Volume Discretizationmentioning

confidence: 99%

See 1 more Smart Citation

Projection finite volume method for shallow water flows

Mathematics and Computers in Simulation

Sari

Seaı̈d

2015

Self Cite

“…When slopes are based on centroid values, the three points used to estimate the slopes are the centroid and the two neighboring centroids inside the computational domain. For further details on the implementation of boundary conditions for the proposed non-homogeneous Riemann solver we refer to [2,18].…”

Section: Second-order Reconstructionmentioning

confidence: 99%

“…The sign matrix of the Jacobian matrix is used in the reconstruction of the numerical fluxes. Most of these techniques have been recently investigated in [2][3][4][5]18] for solving the canonical shallow water models without accounting for porosity variation. The current study presents an extension of this method to transient flows involving porosity variation in the water flows.…”

Section: Introductionmentioning

confidence: 99%

A non-homogeneous Riemann solver for shallow water equations in porous media

Elmahi

Moumna

et al. 2015

Applicable Analysis

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The purpose of the current research is to develop an accurate and efficient solver for shallow water flows in porous media. The hydraulics is modeled by the two-dimensional shallow water flows with variable horizontal porosity. The variation of porosity in the water flows can be attributed to the variation of bed properties of the water system. As an example of porous shallow water flows is the passage of water discharge over vegetated areas in a river. Driving force of the phase separation and the mixing is the gradient of the porosity. For the numerical solution procedure we propose a non-homogeneous Riemann solver in the finite volume framework. The proposed method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe's scheme using the sign of the Jacobian matrix in the coupled system. A well-balanced discretization is used for the treatment of source terms. The efficiency of the solver is evaluated by several test problems for shallow water flows in porous media. The numerical results demonstrate high resolution of the proposed non-homogeneous Riemann solver and confirm its capability to provide accurate simulations for porous shallow water equations under flow regimes with strong shocks.

A two‐dimensional finite volume morphodynamic model on unstructured triangular grids

Numerical Methods in Fluids

Sahmim

Seaı̈d

2010

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SUMMARYWe discuss the application of a finite volume method to morphodynamic models on unstructured triangular meshes. The model is based on coupling the shallow water equations for the hydrodynamics with a sediment transport equation for the morphodynamics. The finite volume method is formulated for the quasi-steady approach and the coupled approach. In the first approach, the steady hydrodynamic state is calculated first and the corresponding water velocity is used in the sediment transport equation to be solved subsequently. The second approach solves the coupled hydrodynamics and sediment transport system within the same time step. The gradient fluxes are discretized using a modified Roe's scheme incorporating the sign of the Jacobian matrix in the morphodynamic system. A well-balanced discretization is used for the treatment of source terms. We also describe an adaptive procedure in the finite volume method by monitoring the bed-load in the computational domain during its transport process. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bed gradients that may form in the approximate solution. Numerical results are shown for a test problem in the evolution of an initially hump-shaped bed in a squared channel. For the considered morphodynamical regimes, the obtained results point out that the coupled approach performs better than the quasi-steady approach only when the bed-load rapidly interacts with the hydrodynamics.