Three-Layer Approximation of Two-Layer Shallow Water Equations

Chertock, Alina; Kurganov, Alexander; Qu, Zhuolin; Wu, Tong

doi:10.3846/13926292.2013.869269

Cited by 21 publications

(21 citation statements)

References 20 publications

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“…Thus, as concerns the two layers approximations, one can note for instance the Q-scheme proposed in [19], the recent relaxation approach [4] able to guarantee the preservation of motionless steady states, or the so called central-upwind scheme in [32]. Other splitting and upwind schemes can be found, with for instance in [18] (see also its extension to three layers proposed in [21] with a study of the hyperbolicity range), the f-wave propagation finite volume method in [38] handling dry states or the well-balancing and positivity-preserving results established in [10] within a splitting approach. At last, numerical methods for one-dimensional multilayer shallow water models with mass exchange are also proposed without density stratification in [6] and with in [5].…”

Section: Introductionmentioning

confidence: 99%

An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification

Couderc

Duran

Vila

2017

Journal of Computational Physics

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We present an explicit scheme for a two-dimensional multilayer shallow water model with density stratification, for general meshes and collocated variables. The proposed strategy is based on a regularized model where the transport velocity in the advective fluxes is shifted proportionally to the pressure potential gradient. Using a similar strategy for the potential forces, we show the stability of the method in the sense of a discrete dissipation of the mechanical energy, in general multilayer and non-linear frames. These results are obtained at first-order in space and time and extended using a simple second-order MUSCL extension. With the objective of minimizing the diffusive losses in realistic contexts, sufficient conditions are exhibited on the regularizing terms to ensure the scheme's linear stability at first and second-order in time and space. The other main result stands in the consistency with respect to the asymptotics reached at small and large time scales in low Froude regimes, which governs large-scale oceanic circulation. Additionally, robustness and well-balanced results for motionless steady states are also ensured. These stability properties tend to provide a very robust and efficient approach, easy to implement and particularly well suited for large-scale simulations. Some numerical experiments are proposed to highlight the scheme efficiency: an experiment of fast gravitational modes, a smooth surface wave propagation, an initial propagating surface water elevation jump considering a non trivial topography, and a last experiment of slow Rossby modes simulating the displacement of a baroclinic vortex subject to the Coriolis force.

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Section: Introductionmentioning

confidence: 99%

An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification

Couderc

Duran

Vila

2017

Journal of Computational Physics

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“…In these cases, the use of the proposed Riemann-problem-solver-free central-upwind scheme may be especially advantageous since the estimates on the one-sided local speeds a in jk and a out jk may be obtained using the Lagrange theorem [35] as it was done, for example, in [14,29,32] .…”

Section: Discussionmentioning

confidence: 99%

“…We then apply the midpoint rule to approximate the integrals on the right-hand side (RHS) of (14) to obtain the well-balanced quadrature for S (2) j :…”

Section: Well-balanced Discretization Of the Source Termmentioning

confidence: 99%

Well-balanced positivity preserving cell-vertex central-upwind scheme for shallow water flows

2016

Self Cite

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a b s t r a c tWe develop a new second-order two-dimensional central-upwind scheme on cell-vertex grids for approximating solutions of the Saint-Venant system with source terms due to bottom topography. Centralupwind schemes are developed based on the information about the local speeds of wave propagation. Compared to the triangular central-upwind schemes, the proposed cell-vertex one has an advantage of using more cell interfaces which provide more information on the waves propagating in different directions. We propose a new piecewise linear approximation of the bottom topography and a novel nonoscillatory reconstruction in which the gradient of each variable is computed using a modified minmodtype method to ensure the stability of the scheme. A new technique is proposed for the correction of the water surface elevation which guarantees the positivity of the water depth. The well-balanced property of the proposed central-upwind scheme is ensured using a special discretization for the cell averages of the topography source terms. The proposed scheme is tested on a number of numerical examples, among which we consider steady-state solutions with almost dry areas and their perturbations and solutions with rapidly varying flows over discontinuous bottom topography. Our numerical experiments confirm stability, well-balanced, positivity preserving properties and second-order accuracy of the proposed method. This scheme can be applied to shallow water models when the bed topography is discontinuous and/or highly oscillatory, and on complicated domains where the use of unstructured grids is advantageous.

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“…Our modification of the SPH-method consists in the generalization of Eq. (7). In case of the traditional SPH-approach for the SWE [27]…”

Section: The Lagrangian's Stagementioning

confidence: 99%

“…The SWM modifications are effective for studying various geophysical problems such as the dynamics of the pyroclastic flows [6] and the riverbed processes including the sediment dynamics and the diffusion of pollutant particles in reservoirs. The multilayer models utilization significantly expands the opportunities of the shallow water approach [7]. The tasks associated with flooding research of river valleys or interfluves [1] are stood out among the hydrological problems.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Simulation of the Shallow Water Flow on a Complex Topography

Хоперсков¹,

Khrapov²

2018

Numerical Simulations in Engineering and Science

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In current chapter, we have thoroughly described a numerical integration scheme of nonstationary 2D equations of shallow water. The scheme combines the smoothed particle hydrodynamics (SPH) and the total variation diminishing (TVD) methods, which are sequentially used at various steps of the combined SPH-TVD algorithm. The method is conservative and well balanced. It provides stable through calculations in presence of nonstationary "water-dry bottom" boundaries on complex irregular bottom topography including the transition of such a boundary between wet and dry bottom through the computational boundary. Multifarious tests demonstrate the effectiveness of the combined SPH-TVD scheme application for a solution of diverse problems of the engineering hydrology.

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Three-Layer Approximation of Two-Layer Shallow Water Equations

Cited by 21 publications

References 20 publications

An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification

An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification

Well-balanced positivity preserving cell-vertex central-upwind scheme for shallow water flows

A Numerical Simulation of the Shallow Water Flow on a Complex Topography

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