A combined finite volume - finite element scheme for a dispersive shallow water system

Aïssiouene, Nora; Bristeau, Marie-Odile; Godlewski, Edwige; Sainte-Marie, Jacques

doi:10.3934/nhm.2016.11.1

Cited by 17 publications

(35 citation statements)

References 36 publications

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“…In particular, thanks to the formulation (), the advection (left‐hand side) of the vertical velocity is solved by conventional passive transport scheme. This strategy has already been used in the work of Aissiouene et al and entropy‐satisfying schemes to solve it can be used . However, in the cited works, the dispersion step is solved using a prediction‐correction strategy based on an implicit system on the hydrodynamic pressure q B .…”

Section: Numerical Schemes For the Reduced Modelsmentioning

confidence: 99%

“…More precisely, by neglecting the dispersal operator D NH , the first two unknowns of U NH are estimated using , whereas the third is simply advected to the flow as a passive pollutant . This strategy was already used for dispersive model in the work of Aissiouene et al We write

U_{k}^{NH, n ⋆} = U_{k}^{NH, n} - \frac{δ_{t}^{n}}{(||, k)} \sum_{f \in {double-struckF}_{k}} ((), {scriptF}_{f}^{NH, n} \cdot {scriptN}_{k}^{{\underline{k}}_{f}} + {scriptS}_{f, k}^{NH, n}) (||, f),

with the state vector

U_{k}^{NH, n} = {((), U_{k}^{SW, n}, h_{k}^{n} {\overset{w}{true}}_{k}^{n})}^{t}

, the source term

{scriptS}_{f, k}^{NH, n} = {((), {scriptS}_{f, k}^{SW, n}, 0)}^{t}

, the numerical flux

F_{f}^{NH, n} \cdot N_{k}^{{\underset{true_}{k}}_{f}} = (\begin{array}{c} {scriptF}_{f}^{SW, n} \cdot {scriptN}_{k}^{{\underline{k}}_{f}} \\ {scriptF}_{δ}^{u p} ((), scriptF <...>) \end{array})

…”

Section: Numerical Schemes For the Reduced Modelsmentioning

confidence: 99%

“…Despite the splitting strategy, third‐order derivatives still required to be discretized. Aissiouene et al propose a developed form of (), where the vertical‐averaged vertical velocity is introduced. This new unknown added an equation to the hyperbolic step and limited the dispersive step to a second‐order elliptical pressure‐based equation.…”

Section: Introductionmentioning

confidence: 99%

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Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Parisot

2019

Numerical Methods in Fluids

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Summary This work is devoted to the numerical resolution in the multidimensional framework of a hierarchy of reduced models of the water wave equations, such as the Serre‐Green‐Naghdi model. A particular attention is paid to the dissipation of mechanical energy at the discrete level, which acts as a stability argument of the scheme, even with source terms such space and time variation of the bathymetry. In addition, the analysis leads to a natural way to deal with dry areas without leakage of energy. To illustrate the accuracy and the robustness of the strategy, several numerical experiments are carried out. In particular, the strategy is capable of treating dry areas without special treatment.

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Section: Numerical Schemes For the Reduced Modelsmentioning

confidence: 99%

U_{k}^{NH, n ⋆} = U_{k}^{NH, n} - \frac{δ_{t}^{n}}{(||, k)} \sum_{f \in {double-struckF}_{k}} ((), {scriptF}_{f}^{NH, n} \cdot {scriptN}_{k}^{{\underline{k}}_{f}} + {scriptS}_{f, k}^{NH, n}) (||, f),

with the state vector

U_{k}^{NH, n} = {((), U_{k}^{SW, n}, h_{k}^{n} {\overset{w}{true}}_{k}^{n})}^{t}

, the source term

{scriptS}_{f, k}^{NH, n} = {((), {scriptS}_{f, k}^{SW, n}, 0)}^{t}

, the numerical flux

F_{f}^{NH, n} \cdot N_{k}^{{\underset{true_}{k}}_{f}} = (\begin{array}{c} {scriptF}_{f}^{SW, n} \cdot {scriptN}_{k}^{{\underline{k}}_{f}} \\ {scriptF}_{δ}^{u p} ((), scriptF <...>) \end{array})

…”

Section: Numerical Schemes For the Reduced Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Parisot

2019

Numerical Methods in Fluids

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“…For this system, our scheme is second order accurate in time and, if we use a reconstruction algorithm [3] in the hyperbolic step, it is formally second order accurate in space [3,2] . In the application, we use a kinetic solver for its good mathematical properties.…”

Section: The Averaged Euler Systemmentioning

confidence: 99%

“…To do so, we consider the equations (14)-(15) as a mixed problem [2] and, starting with an appropriate variational formulation of the problem, we apply the finite element method to obtain the pressure p n+1 which is solution of an elliptic equation and the velocity u n+1 . The elliptic equation of the pressure can be written under the form:…”

Section: The Averaged Euler Systemmentioning

confidence: 99%

Application of a Combined Finite Element—Finite Volume Method to a 2D Non-hydrostatic Shallow Water Problem

Aïssiouene¹,

Bristeau

Godlewski

et al. 2017

Springer Proceedings in Mathematics &Amp; Statistics

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We propose a numerical method for a two-dimensional non-hydrostatic shallow water system with topography [6]. We use a prediction-correction scheme initially introduced by Chorin-Temam [13], and which has been applied previously to the one dimensional problem in [1]. The prediction part leads to solving a shallow water system for which we use finite volume methods [3], while the correction part leads to solving a mixed problem in velocity/pressure using a finite element method. We present an application of the method with a comparison between a hydrostatic and a non-hydrostatic model.

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Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations

et al. 2021

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In some configurations, dispersion effects must be taken into account to improve the simulation of complex fluid flows. A family of free-surface dispersive models has been derived in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018). The hierarchy of models is based on a Galerkin approach and parameterised by the number of discrete layers along the vertical axis. In this paper we propose some numerical schemes designed for these models in a 1D open channel. The cornerstone of this family of models is the Serre – Green-Naghdi model which has been extensively studied in the literature from both theoretical and numerical points of view. More precisely, the goal is to propose a numerical method for the $$LDNH_2$$ L D N H 2 model that is based on a projection method extended from the one-layer case to any number of layers. To do so, the one-layer case is addressed by means of a projection-correction method applied to a non-standard differential operator. A special attention is paid to boundary conditions. This case is extended to several layers thanks to an original relabelling of the unknowns. In the numerical tests we show the convergence of the method and its accuracy compared to the $$LDNH_0$$ L D N H 0 model.

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A combined finite volume - finite element scheme for a dispersive shallow water system

Cited by 17 publications

References 36 publications

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Application of a Combined Finite Element—Finite Volume Method to a 2D Non-hydrostatic Shallow Water Problem

Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations

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