Space–time discontinuous Galerkin finite element method for shallow water flows

Ambati, V.R.; Bokhove, Onno

doi:10.1016/j.cam.2006.01.047

Cited by 29 publications

(28 citation statements)

References 4 publications

(13 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The algorithms and codes used have been verified against rotating and nonrotating exact solutions and validated against experiments and bore-vortex interactions in Refs. [16][17][18][19]. We predominantly used grids of 175ϫ 40 elements and ran a few cases with double resolution as a verification.…”

Section: A Existence Of 2d Oblique Hydraulic Jumpsmentioning

confidence: 99%

Hydraulic flow through a channel contraction: Multiple steady states

Akers

Bokhove

2008

Physics of Fluids

View full text Add to dashboard Cite

We have investigated shallow water flows through a channel with a contraction by experimental and theoretical means. The horizontal channel consists of a sluice gate and an upstream channel of constant width b 0 ending in a linear contraction of minimum width b c . Experimentally, we observe upstream steady and moving bores/shocks, and oblique waves in the contraction, as single and multiple ͑steady͒ states, as well as a steady reservoir with a complex hydraulic jump in the contraction occurring in a small section of the b c / b 0 and Froude number parameter plane. One-dimensional hydraulic theory provides a comprehensive leading-order approximation, in which a turbulent frictional parametrization is used to achieve quantitative agreement. An analytical and numerical analysis is given for two-dimensional supercritical shallow water flows. It shows that the one-dimensional hydraulic analysis for inviscid flows away from hydraulic jumps holds surprisingly well, even though the two-dimensional oblique hydraulic jump patterns can show large variations across the contraction channel.

show abstract

Section: A Existence Of 2d Oblique Hydraulic Jumpsmentioning

confidence: 99%

Hydraulic flow through a channel contraction: Multiple steady states

Akers

Bokhove

2008

Physics of Fluids

View full text Add to dashboard Cite

show abstract

“…Significant progress in the application of DG methods to the SWE has been achieved in the last few years [15,16,17,18,19,20,21,22,23,24,25]. However, two issues relevant in many applications, namely preserving steady-states at rest with variable bathymetry and properly handling flooding and drying, have not been addressed in previous work, with the exception of [17] where a moving mesh was used to deal with dry areas in a one-dimensional setting; the extension to two space dimensions does not seem to be straightforward.…”

Section: Introductionmentioning

confidence: 99%

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

Ern

Piperno

Djadel

2007

Numerical Methods in Fluids

111

110

View full text Add to dashboard Cite

SUMMARYWe build and analyze a Runge-Kutta Discontinuous Galerkin method to approximate the one-and two-dimensional Shallow-Water Equations. We introduce a flux modification technique to derive a wellbalanced scheme preserving steady-states at rest with variable bathymetry and a slope modification technique to deal satisfactorily with flooding and drying. Numerical results illustrating the performance of the proposed scheme are presented.

show abstract

“…A recent review is performed in [80] and a unified analysis can be found in [3], and [26,27], respectively for elliptic problems and both 1 st and 2 nd order problems in the framework of Friedrichs' systems. The application of dG methods to the Saint-Venant equations (also called Nonlinear Shallow Water equations, NSW in the following) has recently lead to several improvements, see for instance [2,28,78,79] and the recent review [19]. However, dG methods for BT equations have been under-investigated.…”

mentioning

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

Space–time discontinuous Galerkin finite element method for shallow water flows

Cited by 29 publications

References 4 publications

Hydraulic flow through a channel contraction: Multiple steady states

Hydraulic flow through a channel contraction: Multiple steady states

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

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

Contact Info

Product

Resources

About