DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations

Engsig-Karup, Allan Peter; Hesthaven, Jan S.; Bingham, Harry B.; Warburton, Tim

doi:10.1016/j.coastaleng.2007.09.005

Cited by 46 publications

(37 citation statements)

References 35 publications

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“…Indeed, the use of such fully nonlinear equations appear as a reasonable compromise between the weakly non-linear equations studied in [30,31] and the highly-dispersive (and computationally costly) three-variables equations investigated in [22,23]. Additionally, it is easily possible to extend the range of validity of the GN equations to moderately deep water by the introduction of some optimization parameters, as shown in [9,43].…”

Section: Introductionmentioning

confidence: 99%

“…The most common one is to add an artificial viscous term to the momentum equation, whose role is to account for the energy dissipation that occurs during wave breaking (see for instance [10,12,39] for a related approach). This is also the method chosen in [23], embedded in the nodal-dG method. One of the drawbacks of this approach, as mentioned in [13], is that the computation remains very sensitive to the calibrations of the dissipation parameters.…”

Section: Introductionmentioning

confidence: 99%

“…This formulation is further investigated in [31], accounting for variable depth, and in [32] with the study of the enhanced equations of Madsen and Sorensen [51]. In [22,23], an arbitrary order nodal dG-FEM method is developed for the set of highly-dispersive BT equations introduced in [50], respectively in 1d and 2d on unstructured meshes. These equations have a larger range of validity and can theoretically model fully nonlinear waves transformation, but they are also more complex, introducing a dependence in the vertical velocity, and consequently additional degrees of freedoms in the discrete approximations.…”

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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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…Such a comparison is the objective of the present paper. The platform for this comparison are the SWEs for a variety of reasons: a) the relative facility of their spatial and temporal discretization with respect to more complex partial differential equations, such as the Navier-Stokes equations, b) their capability for non-dissipative propagation of highly non-linear waves, which renders them an ideal experimentation tool for testing numerical schemes for nonlinear advection, the primary source of the aliasing-driven instabilities mentioned above and c) their role as a predictive tool of ocean wave phenomena for the purpose of coastal engineering applications [33] and tsunami propagation [34]. We specifically aim to compare the two methods in terms of formulation (with a focus on subdomain communication), accuracy, conservation properties, numerical stability and computational cost in the framework of specific linear and non-linear test-cases.…”

Section: Introductionmentioning

confidence: 99%

High-order discontinuous element-based schemes for the inviscid shallow water equations: Spectral multidomain penalty and discontinuous Galerkin methods

Escobar-Vargas

Diamessis

Giraldo

2012

Applied Mathematics and Computation

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“…Levy, Yan and Shu in [29] developed an LDG method for nonlinear dispersive equations for the simulation of compactons. Later, Eskilsson and Sherwin [21,22] and Engsig-Karup et al [20] proposed DG schemes to solve Boussinesq-type equations for the simulation of water waves. Alternative methods for the simulation of non-hydrostatic free surface flows have been proposed and analyzed in [6][7][8].…”

Section: Discontinuous Galerkin Methodsmentioning

confidence: 99%