Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization

Bonaventura, Luca; Fernández-Nieto, Enrique D.; Garres-Díaz, José; Narbona-Reina, Gladys

doi:10.1016/j.jcp.2018.03.017

Cited by 33 publications

(48 citation statements)

References 37 publications

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“…∞ (t), according to the approximations (10), (17), respectively. Among the many possible formulations for the parabolic terms, for definiteness we choose that corresponding to the Symmetric Interior Penalty Galerkin method (SIPG), see, e.g., [3,31] and the review in [22].…”

Section: The Affine Local Mapsmentioning

confidence: 99%

“…Approximating c h using (10) and (17), and taking v = φ m j and v = φ ∞ k , one obtains a set of equations for the discrete degrees of freedom c…”

Section: The Affine Local Mapsmentioning

confidence: 99%

“…4. For the non-linear problems, a second order implicit-explicit (IMEX) method is used, that is described, e.g., in [10,15]. As the terms associated with the diffusion process can entail rather restrictive stability constraints on the time step size if discretized explicitly, they are discretized implicitly, while the terms associated with the hyperbolic conservation law are treated explicitly.…”

Section: The Affine Local Mapsmentioning

confidence: 99%

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A Seamless, Extended DG Approach for Advection–Diffusion Problems on Unbounded Domains

2021

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We propose and analyze a seamless extended Discontinuous Galerkin (DG) discretization of advection-diffusion equations on semi-infinite domains. The semi-infinite half line is split into a finite subdomain where the model uses a standard polynomial basis, and a semiunbounded subdomain where scaled Laguerre functions are employed as basis and test functions. Numerical fluxes enable the coupling at the interface between the two subdomains in the same way as standard single domain DG interelement fluxes. A novel linear analysis on the extended DG model yields unconditional stability with respect to the Péclet number. Errors due to the use of different sets of basis functions on different portions of the domain are negligible, as highlighted in numerical experiments with the linear advection-diffusion and viscous Burgers' equations. With an added damping term on the semi-infinite subdomain, the extended framework is able to efficiently simulate absorbing boundary conditions without additional conditions at the interface. A few modes in the semi-infinite subdomain are found to suffice to deal with outgoing single wave and wave train signals more accurately than standard approaches at a given computational cost, thus providing an appealing model for fluid flow simulations in unbounded regions.

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Section: The Affine Local Mapsmentioning

confidence: 99%

“…Approximating c h using (10) and (17), and taking v = φ m j and v = φ ∞ k , one obtains a set of equations for the discrete degrees of freedom c…”

Section: The Affine Local Mapsmentioning

confidence: 99%

Section: The Affine Local Mapsmentioning

confidence: 99%

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A Seamless, Extended DG Approach for Advection–Diffusion Problems on Unbounded Domains

2021

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“…This is done through the incorporation of the mass and momentum transfer terms that could be written as non-conservative terms appearing in the equations. Some recent works on multilayer shallow water systems even consider a variable number of layers, where the number of layers locally adapts depending on the presence of relevant vertical effects (see [9]).…”

Section: Journal Of Scientific Computingmentioning

confidence: 99%

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

et al. 2021

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In this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.

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“…A drawback of this approach is that many layers should be employed if very complex profiles of velocity have to be recovered, leading to a high computational cost (although much lower than solving the full 3D Navier-Stokes system). In [3], an application to bedload transport problem is simulated by using a multilayer model and the Grass formula, which allows the authors to use the velocity near the bottom and not the depth-averaged velocity as in the Shallow Water model. It should be noted that the resulting multilayer model seems to be hyperbolic based on numerical simulations, although this question remains open for arbitrary numbers of layers.…”

Section: Introductionmentioning

confidence: 99%

Shallow Water Moment Models for Bedload Transport Problems

sci¹

2021

CiCP

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In this work a simple but accurate shallow model for bedload sediment transport is proposed. The model is based on applying the moment approach to the Shallow Water Exner model, making it possible to recover the vertical structure of the flow. This approach allows us to obtain a better approximation of the fluid velocity close to the bottom, which is the relevant velocity for the sediment transport. A general Shallow Water Exner moment model allowing for polynomial velocity profiles of arbitrary order is obtained. A regularization ensures hyperbolicity and easy computation of the eigenvalues. The system is solved by means of an adapted IFCP scheme proposed here. The improvement of this IFCP type scheme is based on the approximation of the eigenvalue associated to the sediment transport. Numerical tests are presented which deal with large and short time scales. The proposed model allows to obtain the vertical structure of the fluid, which results in a better description on the bedload transport of the sediment layer.

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Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization

Cited by 33 publications

References 37 publications

A Seamless, Extended DG Approach for Advection–Diffusion Problems on Unbounded Domains

A Seamless, Extended DG Approach for Advection–Diffusion Problems on Unbounded Domains

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

Shallow Water Moment Models for Bedload Transport Problems

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