Finite volume schemes for dispersive wave propagation and runup

Dutykh, Denys; Katsaounis, Theodoros; Mitsotakis, Dimitrios

doi:10.1016/j.jcp.2011.01.003

Cited by 86 publications

(94 citation statements)

References 70 publications

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“…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], Finite-Element methods (FEM in the following) [46,59,66,73], Finite-Volume discretizations (FVM in the following) for 1d equations [21], hybrid FDM / FVM [24,25,55,64,69], or even a purely 2d FVM discretization on unstructured recently been performed in [43]. Let us also mention [10] and more recently [63] for a 2D cartesian numerical model based on fully non-linear BT equations of [76].…”

Section: Introductionmentioning

confidence: 99%

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%

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

Duran

Marche

2015

Commun. Comput. Phys.

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“…The numerical scheme presented above has already been validated in several studies even in the case of dispersive waves [24,25]. Consequently, we do not present here the standard convergence tests which can be found in references cited above.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…This goal is achieved by various so-called reconstruction procedures, such as MUSCL TVD [38,59], UNO [35], ENO [34], WENO [64] and many others. In our previous study on Boussinesq-type equations [24], the UNO2 scheme showed a good performance with low dissipation in realistic propagation and runup simulations. Consequently, we retain this scheme for the discretization of the modified Saint-Venant equations.…”

Section: High-order Reconstructionmentioning

confidence: 88%

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Modified shallow water equations for significantly varying seabeds

Dutykh

Clamond²

2016

Applied Mathematical Modelling

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Abstract. In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.

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“…The extension to more realistic bi-directional wave propagation models such as Boussinesq type equations [40,36,32,16].…”

Section: Discussionmentioning

confidence: 99%

Finite volume methods for unidirectional dispersive wave models

Dutykh

Katsaounis

Mitsotakis

2012

Numerical Methods in Fluids

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Abstract. We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves and their various interactions.

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Finite volume schemes for dispersive wave propagation and runup

Cited by 86 publications

References 70 publications

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

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

Modified shallow water equations for significantly varying seabeds

Finite volume methods for unidirectional dispersive wave models

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