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

Ern, Alexandre; Piperno, Serge; Djadel, Karim

doi:10.1002/fld.1674

Cited by 114 publications

(120 citation statements)

References 39 publications

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“…However, this Riemann solver is an ideal choice to be employed within RKDG methods for simulating shallow water flows when the model merely considers frictionless and horizontal channel flows [36]. The two articles of Ambati and Bokhove [51,52] are among the very few papers that deal with DG methods over dry beds [53,54]. In their work, the authors presented a second-order DG space-time FE discretization combined with an improvement to the HLLC [4] flux.…”

Section: Introductionmentioning

confidence: 97%

A practical implementation of high‐order RKDG models for the 1D open‐channel flow equations

Kesserwani

Mosé²,

Vázquez³

et al. 2008

Numerical Methods in Fluids

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SUMMARYThis paper comprises an implementation of a fourth-order Runge-Kutta discontinuous Galerkin (RKDG4) scheme for computing the open-channel flow equations. The main features of the proposed methodology are simplicity and easiness in the implementation, which may be of possible interest to water resources numerical modellers. A version of the RKDG4 is blended with the Roe Riemann solver, an adaptive high-order slope limiting procedure, and high-order source terms approximations. A comparison of the performance of the proposed method with lower-order RKDG models is performed showing a benefit of considering the RKDG4 model. The scheme is applied to computerize the Saint Venant system by considering benchmark tests that have exact solutions. Finally, numerical results are illustrated discussing the performance of the proposed high-order model.

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Section: Introductionmentioning

confidence: 97%

A practical implementation of high‐order RKDG models for the 1D open‐channel flow equations

Kesserwani

Mosé²,

Vázquez³

et al. 2008

Numerical Methods in Fluids

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“…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.…”

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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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“…The success of Godunov-type flood models can be attributed to approximate Riemann solvers [37,75] which are also embedded in discontinuous Galerkin finite element schemes [4,26,45] and Boussinesq models that account for non-hydrostatic flow effects [47]. Godunov-type models have generally assumed either a structured mesh of quadrilateral cells [3,13,29,35,42,86] or an unstructured mesh of triangular cells [7,15,41,70,84].…”

Section: Introductionmentioning

confidence: 99%

Mesh type tradeoffs in 2D hydrodynamic modeling of flooding with a Godunov-based flow solver

Kim

Sanders

Schubert

et al. 2014

Advances in Water Resources

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a b s t r a c tThe effect of mesh type on the accuracy and computational demands of a two-dimensional Godunov-type flood inundation model is critically examined. Cartesian grids, constrained and unconstrained triangular grids, constrained quadrilateral grids, and mixed meshes are considered, with and without local time stepping (LTS), to determine the approach that maximizes computational efficiency defined as accuracy relative to computational effort. A mixed-mesh numerical scheme is introduced so all grids are processed by the same solver. Analysis focuses on a wide range of dam-break type test cases, where Godunov-type flood models have proven very successful. Results show that different mesh types excel under different circumstances. Cartesian grids are 2-3 times more efficient with relatively simple terrain features such as rectilinear channels that call for a uniform grid resolution, while unstructured grids are about twice as efficient in complex domains with irregular terrain features that call for localized refinements. The superior efficiency of locally refined, unstructured grids in complex terrain is attributable to LTS; the locally refined unstructured grid becomes less efficient using global time stepping. These results point to mesh-type tradeoffs that should be considered in flood modeling applications. A mixed mesh model formulation with LTS is recommended as a general purpose solver because the mesh type can be adapted to maximize computational efficiency.

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

Cited by 114 publications

References 39 publications

A practical implementation of high‐order RKDG models for the 1D open‐channel flow equations

A practical implementation of high‐order RKDG models for the 1D open‐channel flow equations

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

Mesh type tradeoffs in 2D hydrodynamic modeling of flooding with a Godunov-based flow solver

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