On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas

Gallardo, José M.; Parés, Carlos; Castro, Manuel J.

doi:10.1016/j.jcp.2007.08.007

Cited by 190 publications

(144 citation statements)

References 36 publications

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“…The numerical schemes have been extended to high order by following the ideas developed in [7,20,12] by means of the use of the PHM third order reconstruction operator (see [18]). …”

Section: Discussionmentioning

confidence: 99%

“…Following [7,20,12] we have also derived some third order extensions of all the first order numerical schemes presented here. These extensions are based on the third order reconstruction operator PHM (piecewise hyperbolic method) introduced in [18].…”

Section: Numerical Testsmentioning

confidence: 99%

“…These extensions are based on the third order reconstruction operator PHM (piecewise hyperbolic method) introduced in [18]. For the one-layer and two-layer shallow water systems the implementation of these high order extensions can be done in such a way that the well-balanced properties of the first order schemes for the water at rest and vacuum solutions are preserved (see [12] for details). A third order TVD-Runge-Kutta discretization has been used here for the time-stepping.…”

Section: Numerical Testsmentioning

confidence: 99%

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On some fast well-balanced first order solvers for nonconservative systems

Castro¹,

Pardo²,

Parés³

et al. 2009

Math. Comp.

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Abstract. The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good wellbalanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.

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“…The numerical schemes have been extended to high order by following the ideas developed in [7,20,12] by means of the use of the PHM third order reconstruction operator (see [18]). …”

Section: Discussionmentioning

confidence: 99%

Section: Numerical Testsmentioning

confidence: 99%

Section: Numerical Testsmentioning

confidence: 99%

See 1 more Smart Citation

On some fast well-balanced first order solvers for nonconservative systems

Castro¹,

Pardo²,

Parés³

et al. 2009

Math. Comp.

Self Cite

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“…At the same time, this numerical scheme reduces computational times compared with other numerical schemes, while the amplitude of the first tsunami wave is preserved. Concerning the wet-dry fronts discretization, Tsunami-HySEA implements the numerical treatment described by Castro et al (2005) and Gallardo et al (2007) that consists of locally replacing the 1D Riemann solver used during the propagation step, by another 1D Riemann solver that takes into account Water level profiles during runup of the non-breaking wave in the case H/d = 0.019 on the 1:19.85 beach (at times t = 35 (d/g) 1/2 , t = 40 (d/ g) 1/2 , t = 45 (d/g) 1/2 , and t = 50 (d/g) 1/2 . Normalized root mean square deviation (NRMSD) and maximum wave amplitude error (ERR) are computed and shown for each time the presence of a dry cell.…”

Section: Numerical Solution Methodsmentioning

confidence: 99%

“…It has been thoroughly tested, and in particular has passed not only all tests by Synolakis et al (2008), but also other laboratory tests and proposed benchmark problems. Some of them can be found in the studies by Castro et al (2005Castro et al ( , 2006, Gallardo et al (2007), de la , and NTHMP (2016).…”

Section: The Tsunami-hysea Modelmentioning

confidence: 98%

Performance Benchmarking of Tsunami-HySEA Model for NTHMP’s Inundation Mapping Activities

Macías

Castro

Ortega

et al. 2017

Pure Appl. Geophys.

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Abstract-The Tsunami-HySEA model is used to perform some of the numerical benchmark problems proposed and documented in the ''Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop''. The final aim is to obtain the approval for Tsunami-HySEA to be used in projects funded by the National Tsunami Hazard Mitigation Program (NTHMP). Therefore, this work contains the numerical results and comparisons for the five benchmark problems (1, 4, 6, 7, and 9) required for such aim. This set of benchmarks considers analytical, laboratory, and field data test cases. In particular, the analytical solution of a solitary wave runup on a simple beach, and its laboratory counterpart, two more laboratory tests: the runup of a solitary wave on a conically shaped island and the runup onto a complex 3D beach (Monai Valley) and, finally, a field data benchmark based on data from the 1993 Hokkaido Nansei-Oki tsunami.

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Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

Cea¹,

Vázquez-Cendón²

2009

Numerical Methods in Fluids

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SUMMARYThis paper deals with the numerical discretization of two-dimensional depth-averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small-scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small-scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two-dimensional shallow water equations with porosity, both of them are high-order schemes. The numerical schemes proposed are well-balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high-order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed.

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On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas

Cited by 190 publications

References 36 publications

On some fast well-balanced first order solvers for nonconservative systems

On some fast well-balanced first order solvers for nonconservative systems

Performance Benchmarking of Tsunami-HySEA Model for NTHMP’s Inundation Mapping Activities

Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

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