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

Engsig-Karup, Allan Peter; Hesthaven, Jan S.; Bingham, Harry B.; Madsen, Peter Hauge

doi:10.1007/s10665-006-9064-z

Cited by 61 publications

(66 citation statements)

References 35 publications

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“…As expected, we obtain similar orders of accuracy for both GNO and GNC models. Concerning the impact of the numerical flux choice on the convergence rates, we can observe on Table (5) that, for both models and as mentioned in previous studies [15,22,30,31], the use of BR fluxes may lead to sub-optimal convergence rates for odd values of N , while the LDG fluxes lead to optimal convergence rates. The corresponding convergence orders are reported on the last column of Table (5). Let us now investigate the computational improvements obtained with the new GNC model.…”

Section: Accuracy and Convergence Analysis In The Presence Of Non-flamentioning

confidence: 57%

“…Using these global differentiation matrices, we are now able to approximate all the derivatives occurring in (24a)-(24c). The nonlinear products are treated directly, in a collocation manner, inspired from [22]. We can also build the global square N d ×N e matrix of the discrete version of 1+αT[h b ].…”

Section: High-order Derivatives and Dispersive Terms Computationmentioning

confidence: 99%

“…The use of direct interpolation/collocation methods for the computation of the nonlinear products in the dispersive source terms reduces the computational cost but can generate some aliasing which deteriorates the solution quality and can lead to instabilities. To amend this, we use the stabilization filtering method (mild nodal filter), as suggested in [22].…”

Section: High-order Derivatives and Dispersive Terms Computationmentioning

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.…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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: Accuracy and Convergence Analysis In The Presence Of Non-flamentioning

confidence: 57%

Section: High-order Derivatives and Dispersive Terms Computationmentioning

confidence: 99%

Section: High-order Derivatives and Dispersive Terms Computationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Duran

Marche

2015

Commun. Comput. Phys.

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Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

Bassi

Franchina

Ghidoni

et al. 2010

Numerical Methods in Fluids

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SUMMARYDiscontinuous Galerkin (DG) methods are very well suited for the construction of very high-order approximations of the Euler and Navier-Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high-order spectral element DG approximation of the Navier-Stokes equations coupled with a p-multigrid solution strategy based on a semi-implicit Runge-Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p-multigrid scheme with non-spectral elements and a spectral element DG approach with an implicit time integration scheme.

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A wave generation toolbox for the open‐source CFD library: OpenFoam®

Jacobsen

Fuhrman

Fredsøe

2011

Numerical Methods in Fluids

806

521

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SUMMARY The open‐source CFD library OpenFoam® contains a method for solving free surface Newtonian flows using the Reynolds averaged Navier–Stokes equations coupled with a volume of fluid method. In this paper, it is demonstrated how this has been extended with a generic wave generation and absorption method termed ‘wave relaxation zones’, on which a detailed account is given. The ability to use OpenFoam for the modelling of waves is demonstrated using two benchmark test cases, which show the ability to model wave propagation and wave breaking. Furthermore, the reflection coefficient from outlet relaxation zones is considered for a range of parameters. The toolbox is implemented in C++, and the flexibility in deriving new relaxation methods and implementing new wave theories along with other shapes of the relaxation zone is outlined. Subsequent to the publication of this paper, the toolbox has been made freely available through the OpenFoam‐Extend Community. Copyright © 2011 John Wiley & Sons, Ltd.

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

Cited by 61 publications

References 35 publications

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

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

Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

A wave generation toolbox for the open‐source CFD library: OpenFoam®

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