Balancing the source terms in a SPH model for solving the shallow water equations

Xia, Xilin; Liang, Qiuhua; Pastor, Manuel; Zou, Wenming; Zhuang, Yan-Feng

doi:10.1016/j.advwatres.2013.05.004

Cited by 40 publications

(22 citation statements)

References 33 publications

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“…For this reason, meshless methods are, by design, suitable for problems that benefit from variable spatial resolution. Several meshless methods have been proposed already for the shallow-water equations, such as those based on smoothed particle hydrodynamics [9,43] or radial basis functions [5,10,11,24]. To the best of our knowledge, the shallow-water equations with variable bottom topography have not been considered extensively within the general meshless methodology, as it is quite challenging to preserve the inherent balance between the bottom topography source term and the flux term in the momentum equations.…”

Section: Introductionmentioning

confidence: 99%

A well-balanced meshless tsunami propagation and inundation model

Brecht

Bihlo

MacLachlan

et al. 2018

Advances in Water Resources

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We present a novel meshless tsunami propagation and inundation model. We discretize the nonlinear shallow-water equations using a well-balanced scheme relying on radial basis function based finite differences. The inundation model relies on radial basis function generated extrapolation from the wet points closest to the wet-dry interface into the dry region. Numerical results against standard one-and two-dimensional benchmarks are presented.

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Section: Introductionmentioning

confidence: 99%

A well-balanced meshless tsunami propagation and inundation model

Brecht

Bihlo

MacLachlan

et al. 2018

Advances in Water Resources

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“…To study the pressure distributions of the wave flow, the pressure contours are also shown in the same figure. It is worth mentioning that recently, quite a few SWEs-based SPH models have been developed with promising potentials in simulating the shallow water flows [4,33,34]. However, during the wave breaking at t = 7.2 s and wave impact at t = 13.8 s and so on, the local pressure patterns deviate remarkably from the hydrostatic law, and the pressure values are much larger than the hydrostatic ones.…”

Section: Ppe Source Term Errors In Isphmentioning

confidence: 99%

“…The findings imply that the hydrostatic assumption used in the SWEs models can be used for the wave propagation problems with enough accuracy up to the violent wave breaking and impacting points, but additional numerical algorithms must be included to address the local pressure variations around these zones; otherwise, relatively large prediction errors could be induced. It is worth mentioning that recently, quite a few SWEs-based SPH models have been developed with promising potentials in simulating the shallow water flows [4,33,34]. To evaluate the performance of different PPE source term treatments in ISPH for the wave force prediction, the density-invariant source term Equation (10) and the velocity divergencefree source term Equation (13) are separately used to recompute the time histories of wave impact forces on the right wall based on the same computational setting.…”

Section: Ppe Source Term Errors In Isphmentioning

confidence: 99%

Numerical study of PPE source term errors in the incompressible SPH models

Gui

Dong

Shao

2014

Numerical Methods in Fluids

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In a recent work by Gui et al. , an incompressible SPH model was presented that employs a mixed pressure Poisson equation (PPE) source term combining both the density‐invariant and velocity divergence‐free formulations. The present work intends to apply the model to a wider range of fluid impact situations in order to quantify the numerical errors associated with different formulations of the PPE source term in incompressible SPH (ISPH) models. The good agreement achieved between the model predictions and the documented data is taken as a further demonstration that the mixed source term formulation can accurately predict the fluid impact pressures and forces, both in the magnitude and in the spatial and temporal patterns. Furthermore, an in‐depth numerical analysis using either the pure density‐invariant or velocity divergence‐free formulation has revealed that the pure density‐invariant formulation can lead to relatively large divergence errors while the velocity divergence‐free formulation may cause relatively large density errors. As compared with these two approaches, the mixed source term formulation performs much better having the minimum total errors in all test cases. Although some recent studies found that the weakly compressible SPH models perform somewhat better than the incompressible SPH models in certain fluid impact problems, we have shown that this could be largely caused by the particular formulation of PPE source term in the previous ISPH models and a better formulation of the source term can significantly improve the accuracy of ISPH models. Copyright © 2014 John Wiley & Sons, Ltd.

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“…SPH was introduced independently by Lucy and Gingold and Monaghan in 1977 for astronomical modeling. Since then, it has been applied to model problems in hydrodynamics, flow trough porous media, shallow water flows, and avalanche propagation, just to mention a few representative cases. Good reviews can be found in the textbooks like …”

Section: Introductionmentioning

confidence: 99%

A two‐phase SPH model for debris flow propagation

Pastor

Yagüe

Stickle

et al. 2017

Num Anal Meth Geomechanics

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Summary This paper presents a model which can be used for fast landslides where coupling between solid and pore fluid plays a fundamental role. The proposed model is able to describe debris flows where the difference of velocities between solid grains and fluid is important. The approach is based on the mathematical model proposed by Zienkiewicz and Shiomi, which is similar to those of Pitman and Le and Pudasaini. The novelty of the present work is the numerical technique used, the smoothed particle hydrodynamics (SPH). We propose to use a double set of nodes for soil and water phases, the interaction between them being described by a suitable drag law. The paper presents both mathematical and numerical models, describing the main assumptions and their limitations. Then, the model is applied to (1) a simple case where shocks and expansion waves appear, (2) a dam break problem on a horizontal plane with a frictional soil phase, and (3) a debris flow which happened in Hong Kong. The main conclusions that can be drawn from the applications are: Debris flows having 2 phases with important relative mobility present a rich structure of shocks and rarefaction waves, which has to be properly modeled. Otherwise, the model will have numerical damping or dispersion. Dambreak exercises provide interesting information in simple and controlled situations. We can see how both phases move relative to each other. Real debris flows can be simulated with the proposed model, obtaining reasonable results.

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Balancing the source terms in a SPH model for solving the shallow water equations

Cited by 40 publications

References 33 publications

A well-balanced meshless tsunami propagation and inundation model

A well-balanced meshless tsunami propagation and inundation model

Numerical study of PPE source term errors in the incompressible SPH models

A two‐phase SPH model for debris flow propagation

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