High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations

Li, Gang; Song, Lina; Gao, Jinmei

doi:10.1016/j.cam.2017.10.027

Cited by 29 publications

(21 citation statements)

References 54 publications

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“…DG and ADER-DG have been successfully applied to geophysical flows in the shallow water framework. See for instance the works for the one-layer shallow water model [62,63,81,82] among others. Application of this technique to immiscible two-layer models can be found in [28,39,57].…”

Section: Journal Of Scientific Computingmentioning

confidence: 99%

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

et al. 2021

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In this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.

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Section: Journal Of Scientific Computingmentioning

confidence: 99%

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

et al. 2021

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“…The ADER technique was first developed by Toro and Titarev for finite volume methods [61][62][63][64] and later extended to DG methods [55,65,66]. Since then, it has been widely used in order to solve many hyperbolic systems of conservation laws see [67][68][69][70][71][72]. The recent ADER approach for DG methods is based on a predictor-corrector scheme that computes an approximated predictor solution through a fixed point algorithm locally in each element.…”

Section: Introductionmentioning

confidence: 99%

Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

2021

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This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

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“…Several numerical methods can solve this equation. These schemes are finite-difference [9]- [12], finite element [13]- [16], and finite volume method [17]- [21]. Of all these methods, the finite difference method is the most frequently used.…”

Section: Introductionmentioning

confidence: 99%

2D Numerical Model of Sediment Transport Under Dam-break Flow Using Finite Element

Hafiyyan¹,

Harlan²,

Adityawan³

et al. 2021

International Journal on Advanced Science, Engineering and Information Technology

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The potential hazard of dam construction is the possibility of dam failure. Dam failure will cause damage to property and the environment, as well as loss of human life. In addition, the dam-break flow also causes erosion and sediment transport which can affect the morphology of rivers around the dam. Dam-break flow analysis is needed to minimize the potential hazards of dam construction. Dam-break flow analysis can be done by performing numerical modeling. This study develops a numerical model using the Taylor Galerkin method. The Taylor Galerkin model is used in simulating the dam-break flow along with the sediment transport that occurs. Mathematically, this flow is generally expressed by the shallow water-Exner equations. The shallow water equations describe the movement of water, and the Exner equation describes the movement of sediment. The model will use the Galerkin method for spatial derivatives and the Taylor series approach for time derivatives in this study. A numerical filter by Hansen was also added to the model to overcome the instability of the model due to numerical oscillations. To determine the performance of the Taylor Galerkin model, simulation results were compared with experimental data and other numerical results from previous studies. The Taylor Galerkin model can simulate the dam-break flow with sediment movement over a movable bed well based on this study. Studies like this are needed to reduce the high risk of dam failure.

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High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations

Cited by 29 publications

References 54 publications

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

2D Numerical Model of Sediment Transport Under Dam-break Flow Using Finite Element

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