On the well-balanced numerical discretization of shallow water equations on unstructured meshes

Duran, Arnaud; Liang, Qiuhua; Marche, Fabien

doi:10.1016/j.jcp.2012.10.033

Cited by 58 publications

(64 citation statements)

References 45 publications

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“…Before describing in details the dG discretization of (12), let us reformulate the model under the pre-balanced formulation, in order to adapt the FVM well-balanced discretization introduced in [20,48]. The main ideas are, firstly, to use the free surface elevation instead of h as a flow variable, and secondly to introduce an alternative splitting and redistributing of the free surface gradient term gh∂ x ζ, exploiting the deviations from the system's unforced equilibrium.…”

Section: Pre-balanced Formulationmentioning

confidence: 99%

“…Simple and robust well-balanced schemes for the NSW equations relying on this pre-balanced formulation are proposed for instance in [20,74]. Using dG approximations for NSW equations, we show in the next section that this formulation also provides a natural balance between fluxes and topography source term, provided that the corresponding integral terms are computed exactly.…”

Section: Pre-balanced Formulationmentioning

confidence: 99%

“…To ensure the preservation of motionless steady states, we adapt to the dG framework the method introduced in [20,47] for the FVM discretization of the pre-balanced SW equations. The following scheme can also be regarded as the adaptation to the pre-balanced formulation of the ideas introduced in [77].…”

Section: Preservation Of Motionless Steady Statesmentioning

confidence: 99%

“…An arbitrary order of accuracy in space is obtained through the use of the Legendre polynomials hierarchical basis, and low storage strong-stability preserving Runge-Kutta methods (SSP-RK in the following) are used for the time discretization. The whole model is shown to exactly preserve the motionless steady states, thanks to the pre-balanced reformulation of the surface gradient term and suitable interface fluxes [20], and a robust treatment is implemented for the moving shoreline, based on the enforcement of an element-wise water height positivity preservation property, borrowing the recent accuracy-preserving method introduced for the dGM in [79,81]. We lastly introduce an efficient way of handling broken waves, relying on the GN-NSW switching method.…”

Section: Introductionmentioning

confidence: 99%

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Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Duran

Marche

2015

Commun. Comput. Phys.

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Abstract. We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using an expansion basis of arbitrary order. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to handle broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.. . IntroductionDepth-averaged equations are widely used in coastal engineering for the simulation of nonlinear waves propagation and transformations in nearshore areas. The full description of surface water waves in an incompressible, homogeneous, inviscid fluid, is provided by the free surface Euler (or water waves) equations but this problem remains mathematically and numerically challenging. As a consequence, the use of depth averaged equations helps to reduce the three-dimensional problem to a two-dimensional problem, while keeping a good level of accuracy in many configurations. Many Boussinesq-like models are used nowadays and a detailed review can be found in [42] and the recent monograph [41]. Denoting by λ the typical horizontal scale of the flow and h 0 the typical depth, the shallow water regime usually corresponds to the configuration where µ := [52,54,57] for instance, an additional smallness amplitude assumption on the typical wave amplitude a is classically performed: ε := a h0 = O(µ). This assumption often appears as too restrictive for many applications in coastal oceanography. Removing the small amplitude assumption while still keeping all the O(µ) terms, we obtain the so-called Green-Naghdi equations (GN equations in the following) [34], also referred to as Serre equations [62] or fully non-linear Boussinesq equations [76]. A large number of numerical methods have been developed in the past few years for the BT equations. Let us mention for instance some Finite-Difference (FDM in the following) approaches [49,54,65,75] As far as flexibility is concerned, the use of discontinuous-Galerkin methods (dG methods in the following) would appear as a natural choice. Indeed, this class of method provides several appealing features, like compact discretization stencils and hp-adaptivity, flexibility with a natural handling of unstructured meshes, easy parallel computation and local conservation properties in the approximation of conservation laws. A general review of dG methods for convection dominated problems is performed in [14]. Concerning the approximation of more general problems, involving higher-order derivatives, several methods a...

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Section: Pre-balanced Formulationmentioning

confidence: 99%

Section: Pre-balanced Formulationmentioning

confidence: 99%

Section: Preservation Of Motionless Steady Statesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Duran

Marche

2015

Commun. Comput. Phys.

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“…First order FV schemes are widely found in the literature [2,3,4,5,6,7,8]. The conceptual simplicity of first order FV schemes and its rather straightforward implementation and ease for parallelization have made it very popular.…”

Section: Introductionmentioning

confidence: 99%

Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations

Gerhard

Caviedes‐Voullième

Müller

et al. 2015

Journal of Computational Physics

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We provide an adaptive strategy for solving shallow water equations with dynamic grid adaptation including a sparse representation of the bottom topography. A challenge in computing approximate solutions to the shallow water equations including wetting and drying is to achieve the positivity of the water height and the well-balancing of the approximate solution. A key property of our adaptive strategy is that it guarantees that these properties are preserved during the refinement and coarsening steps in the adaptation process.The underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. Furthermore we use the multiresolution analysis of the underlying data as an additional indicator whether the limiter has to be applied on a cell or not. By this the number of cells where the limiter is applied is reduced without spoiling the accuracy of the solution.By means of well-known 1D and 2D benchmark problems, we verify that multiwavelet-based grid adaptation can significantly reduce the computational cost by sparsening the computational grids, while retaining accuracy and keeping well-balancing and positivity.

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A faster numerical scheme for a coupled system modeling soil erosion and sediment transport

Cordier

Lucas

2015

Water Resources Research

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Overland flow and soil erosion play an essential role in water quality and soil degradation.Such processes, involving the interactions between water flow and the bed sediment, are classically described by a well-established system coupling the shallow water equations and the Hairsine-Rose model. Numerical approximation of this coupled system requires advanced methods to preserve some important physical and mathematical properties; in particular, the steady states and the positivity of both water depth and sediment concentration. Recently, finite volume schemes based on Roe's solver have been proposed by Heng et al. (2009) and Kim et al. (2013) for one and two-dimensional problems. In their approach, an additional and artificial restriction on the time step is required to guarantee the positivity of sediment concentration. This artificial condition can lead the computation to be costly when dealing with very shallow flow and wet/dry fronts. The main result of this paper is to propose a new and faster scheme for which only the CFL condition of the shallow water equations is sufficient to preserve the positivity of sediment concentration. In addition, the numerical procedure of the erosion part can be used with any well-balanced and positivity preserving scheme of the shallow water equations. The proposed method is tested on classical benchmarks and also on a realistic configuration.

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On the well-balanced numerical discretization of shallow water equations on unstructured meshes

Cited by 58 publications

References 45 publications

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations

A faster numerical scheme for a coupled system modeling soil erosion and sediment transport

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