Numerical resolution of well-balanced shallow water equations with complex source terms

Liang, Qiuhua; Marche, Fabien

doi:10.1016/j.advwatres.2009.02.010

Cited by 440 publications

(344 citation statements)

References 39 publications

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“…Before describing in details the dG discretization of (12), let us reformulate the model under the pre-balanced formulation, in order to adapt the FVM well-balanced discretization introduced in [20,48]. The main ideas are, firstly, to use the free surface elevation instead of h as a flow variable, and secondly to introduce an alternative splitting and redistributing of the free surface gradient term gh∂ x ζ, exploiting the deviations from the system's unforced equilibrium.…”

Section: Pre-balanced Formulationmentioning

confidence: 99%

“…The main ideas are, firstly, to use the free surface elevation instead of h as a flow variable, and secondly to introduce an alternative splitting and redistributing of the free surface gradient term gh∂ x ζ, exploiting the deviations from the system's unforced equilibrium. This method was first introduced in [61] for the NSW equations in the framework of FV methods and then extended to handle the occurrence of dry areas in [47,48], where the following splitting is introduced, relying on the use of the total free surface elevation η = h + z (see Fig. 1):…”

Section: Pre-balanced Formulationmentioning

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: Pre-balanced Formulationmentioning

confidence: 99%

Section: Pre-balanced Formulationmentioning

confidence: 99%

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

Duran

Marche

2015

Commun. Comput. Phys.

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“…This treatment can balance the source term and the flux gradient, which preserves the well balanced property of the model at static water condition (Liang and Marche [27]). For the friction source term f S , the splitting implicit method is used with limited friction force, which can avoid nonphysical velocity as the depth is very small [27]. In eq.…”

Section: Numerical Schemementioning

confidence: 99%

“…Notably, there is no clipping treatment to modify flow and sediment variables or fluxes in order to maintain numerical stability. This procedure essentially ensures absolute mass conservation [27].…”

Section: Numerical Schemementioning

confidence: 99%

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Coupled flood and sediment transport modelling with adaptive mesh refinement

Huang

Cao

Pender

et al. 2015

Sci. China Technol. Sci.

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Coupled flood and sediment transport modelling in large-scale domains has for long been hindered by the high computational cost. Adaptive mesh refinement is one of the viable ways to solving this problem without degrading the accuracy. This goal can be accomplished through mesh adaptation, e.g., mesh coarsening and refining based on the dynamic regime of the flow and sediment transport along with bed evolution. However, previous studies in this regard have been limited to cases either without involving sediment transport or featuring flow-sediment-bed decoupling and the assumption of sediment transport capacity, which are not generally justified. Here, a coupled hydrodynamic and non-capacity sediment transport model is developed on adaptive non-uniform rectangular mesh. The proposed model is validated against experimental tests and numerical results based on the fixed meshes. It is demonstrated that the proposed model can properly capture shock waves, resolve the wetting/drying transition and reproduce morphological evolution. Compared with models based on the fixed meshes, the proposed model features great advantage in computational efficiency and holds promise for wide applications.shallow water flow, sediment transport, adaptive mesh Citation:Huang W, Cao Z X, Pender G, et al. Coupled flood and sediment transport modelling with adaptive mesh refinement.

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Discontinuous Galerkin flood model formulation: Luxury or necessity?

Kesserwani

Wang

2014

Water Resources Research

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The finite volume Godunov-type flood model formulation is the most comprehensive amongst those currently employed for flood risk modeling. The local Discontinuous Galerkin method constitutes a more complex, rigorous, and extended local Godunov-type formulation. However, the practical merit associated with such an increase in the level of complexity of the formulation is yet to be decided. This work makes the case for a second-order Runge-Kutta Discontinuous Galerkin (RKDG2) formulation and contrasts it with the equivalently accurate finite volume (MUSCL) formulation, both of which solve the Shallow Water Equations (SWE) in two space dimensions. The numerical complexity of both formulations are presented and their capabilities are explored for wide-ranging diagnostic and real-scale tests, incorporating all challenging features relevant to flood inundation modeling. Our findings reveal that the extra complexity associated with the RKDG2 model pays off by providing higher-quality solution behavior on very coarse meshes and improved velocity predictions. The practical implication of this is that improved accuracy for flood modeling simulations will result when terrain data are limited or of a low resolution.

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Numerical resolution of well-balanced shallow water equations with complex source terms

Cited by 440 publications

References 39 publications

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

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

Coupled flood and sediment transport modelling with adaptive mesh refinement

Discontinuous Galerkin flood model formulation: Luxury or necessity?

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