Numerical modeling of Boussinesq equations by finite element method

Li, Y.S.; Liu, S.-X.; Yu, Yang; Lai, Guozhang

doi:10.1016/s0378-3839(99)00014-9

Cited by 86 publications

(47 citation statements)

References 22 publications

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“…with η n j = 1 |E j | E j η h (x, t n ) dx and the interface fluxes q given by (46). In the framework of the pre-balanced formulation, as we work with η instead of h, we have to ensure that the first order scheme associated with (53) preserve the positivity of the water height.…”

Section: Robustnessmentioning

confidence: 99%

“…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%

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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: Robustnessmentioning

confidence: 99%

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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“…A set of weakly dispersive Boussinesq-type equations was solved and studied by Langtangen and Pedersen (1998). 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).…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2006

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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.

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“…A set of weakly dispersive Boussinesqtype equations was solved and studied by Langtangen and Pedersen (1998). Li et al (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 et al (2004), while the equations of Nwogu (1993) were solved by Walkley and Berzins (1999) and Walkley and Berzins (2002), and Eskilsson and Sherwin, Eskilsson and Sherwin (2003).…”

Section: Introductionmentioning

confidence: 99%

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

Engsig-Karup

Hesthaven

Bingham

et al. 2008

Coastal Engineering

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Abstract. We present a high-order nodal Discontinuous Galerkin Finite Element Method (DG-FEM) solution based on a set of highly accurate Boussinesq-type equations for solving general water-wave problems in complex geometries. A nodal DG-FEM is used for the spatial discretization to solve the Boussinesq equations in complex and curvilinear geometries which amends the application range of previous numerical models that have been based on structured Cartesian grids. The Boussinesq method provides the basis for the accurate description of fully nonlinear and dispersive water waves in both shallow and deep waters within the breaking limit. To demonstrate the current applicability of the model both linear and nonlinear test cases are considered where the water waves interact with bottom-mounted fully reflecting structures. It is established that by simple symmetry considerations combined with a mirror principle it is possible to impose weak slip boundary conditions for both structured and general curvilinear wall boundaries while maintaining the accuracy of the scheme. As is standard for current high-order Boussinesq-type models, arbitrary waves can be generated and absorbed in the interior of the computational domain using a flexible relaxation technique applied on the free surface variables.

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Numerical modeling of Boussinesq equations by finite element method

Cited by 86 publications

References 22 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

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

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