Boussinesq-type modelling using an unstructured finite element technique

Sørensen, Ole R.; Schäffer, Hemming A.; Sørensen, Lene Tolstrup

doi:10.1016/j.coastaleng.2003.10.005

Cited by 89 publications

(32 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.

View full text Add to dashboard Cite

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...

show abstract

Section: Introductionmentioning

confidence: 99%

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

Duran

Marche

2015

Commun. Comput. Phys.

View full text Add to dashboard Cite

show abstract

“…Li, Liu, Yu and Lai (1999) solved the improved Boussinesq equations by Beji and Nadaoka (1996). The extended Boussinesq equations by Madsen and Sørensen (1992) were solved using a FEM model by Sørensen, Schäffer and Sørensen (2004), while the equations of Nwogu (1993) were solved by Berzins (1999,2002), and Sherwin (2003,2005). Recently, Sherwin (2002,2005) solved the same set of extended equations using a DG-FEM method.…”

Section: Introductionmentioning

confidence: 99%

Nodal DG-FEM solution of high-order Boussinesq-type equations

et al. 2006

View full text Add to dashboard Cite

Abstract. We present a discontinuous Galerkin finite element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one and two horizontal dimensions. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are verified for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar; and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in a future paper.

show abstract

“…This test has been used, for example in [47,14,42,25], to test the ability of a model to reproduce wave propagation from deep water to the shore, including the surf zone, and breaking included nearshore circulation. More precisely, the nearshore circulation induced by breakers in the presence of a beach with a rip channel is numerically investigated.…”

Section: Rip Currentmentioning

confidence: 99%

Irregular wave propagation with a 2DH Boussinesq-type model and an unstructured finite volume scheme

Καζολέα¹,

Delis

2018

European Journal of Mechanics - B/Fluids

View full text Add to dashboard Cite

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. Project-Teams CardamomResearch Report n°9008 -December 2016 -24 pages Abstract:The application and validation, with respect to the transformation, breaking and run-up of irregular waves, of an unstructured high-resolution finite volume (FV) numerical solver for the 2D extended Boussinesq-type (BT) equations of Nwogu (1993) is presented. The numerical model is based on the combined FV approximate solution of the BT model and that of the nonlinear shallow water equations (NSWE) when wave breaking emerges. The FV numerical scheme satisfies the desired properties of wellbalancing, for flows over complex bathymetries and in presence of wet/dry fronts, and shock-capturing for an intrinsic representation of wave breaking, that is handled as a shock by the NSWE. Several simulations and comparisons with experimental data show that the model is able to simulate wave height variations, mean water level setup, wave run-up, swash zone oscillations and the generation of near-shore currents with satisfactory accuracy.

show abstract

Boussinesq-type modelling using an unstructured finite element technique

Cited by 89 publications

References 27 publications

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

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

Nodal DG-FEM solution of high-order Boussinesq-type equations

Irregular wave propagation with a 2DH Boussinesq-type model and an unstructured finite volume scheme

Contact Info

Product

Resources

About