Galerkin-finite element methods for the shallow water equations with characteristic boundary conditions

Antonopoulos, D. C.; Dougalis, Vassilios A.

doi:10.1093/imanum/drw017

Cited by 11 publications

(16 citation statements)

References 17 publications

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“…We now estimate the various terms in the right-hand side of (2.33) for 0 ≤ t ≤ t h . As in the proof of Proposition 2.1 of [11] adapted in the case of a variable β(x) ∈ C 1 and using an appropriate variable-β superapproximation property to estimate P 0 γ − γ . We finally obtain from (2.33) and the fact that ω ≥ 0, that for 0 ≤ t ≤ t h it holds that d dt…”

Section: Semidiscretization Of An Ibvp With Absorbing (Characteristicmentioning

confidence: 99%

Galerkin finite element methods for the Shallow Water equations over variable bottom

Kounadis

Dougalis

2020

Journal of Computational and Applied Mathematics

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We consider the one-dimensional shallow water equations (SW) in a finite channel with variable bottom topography. We pose several initialboundary-value problems for the SW system, including problems with transparent (characteristic) boundary conditions in the supercritical and the subcritical case. We discretize these problems in the spatial variable by standard Galerkin-finite element methods and prove L 2 -error estimates for the resulting semidiscrete approximations. We couple the schemes with the 4 th order-accurate, explicit, classical Runge-Kutta time stepping procedure and use the resulting fully discrete methods in numerical experiments of shallow water wave propagation over variable bottom topographies with several kinds of boundary conditions. We discuss issues related to the attainment of a steady state of the simulated flows, including the good balance of the schemes.

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Section: Semidiscretization Of An Ibvp With Absorbing (Characteristicmentioning

confidence: 99%

Galerkin finite element methods for the Shallow Water equations over variable bottom

Kounadis

Dougalis

2020

Journal of Computational and Applied Mathematics

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“…The boundary conditions are transmissive at x = −200 × 10 3 and impermeable otherwise. Transmissive boundaries for this sub-critical flow were implemented following [2] using a standard approach based on Riemann invariants.…”

Section: Idealised Hurricane Approaching a Linearly Sloping Coastmentioning

confidence: 99%

An Adaptive Discontinuous Galerkin Method for the Simulation of Hurricane Storm Surge

Beisiegel¹,

Vater²,

Behrens³

et al. 2019

Preprint

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Numerical simulations based on solving the 2D shallow water equations using a Discontinuous Galerkin (DG) discretisation have evolved to be a viable tool for many geophysical applications. In the context of flood modelling, however, they have not yet been methodologically studied to a large extent. On geographic scale, hurricane storm surge can be interpreted as a localised phenomenon making it ideally suited for adaptive mesh refinement (AMR). Past studies employing dynamic AMR have exclusively focused on nested meshes. For that reason we have developed a DG storm surge model on a triangular and dynamically adaptive mesh. In order to increase computational efficiency, the refinement is driven by physics-based refinement indicators capturing major model sensitivities. Using idealised numerical test cases, we demonstrate the model's ability to correctly represent all source terms and reproduce known variability of coastal flooding with respect to hurricane characteristics such as size and approach speed. Finally, the unstructured mesh significantly reduces computing time with no effect on storm waves measured at discrete wave gauges just off the coast which shows the model's capability for use as a robust simulation tool for real-time predictions.

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“…However, there are no artificial dissipation or numerical filter for shock capturing. A Galerkin FEM with a characteristic method to absorb shock wave was employed in [9]. Nevertheless, the application of a numerical filter or artificial dissipation is favorable due to its…”

Section: Introductionmentioning

confidence: 99%

Application of Numerical Filter to a Taylor Galerkin Finite Element Model for Movable Bed Dam Break Flows

Harlan¹

2019

GEOMATE

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The dam break-induced propagation over a movable bed has given strong interest in numerical modeling. The shock wave, generally found in the wavefront, is a challenge in numerical modeling since this condition often leads to numerical instability. Additionally, the scouring beneath the wave requires coupling of hydraulic and sediment transport models. The objective of this research is to develop a one-dimensional model based on a finite element method to simulate an experimental case of a dam break-induced flow and the bed scouring beneath it. The two-step Taylor Galerkin scheme is used to solve the governing equations. In addition, Hansen numerical filter is used to handle shock wave and to increase the numerical stability of the developed model. The results show good agreement to the experimental data. Additionally, the applied numerical filter is able to reduce oscillation and improve the stability of the model.

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Galerkin-finite element methods for the shallow water equations with characteristic boundary conditions

Cited by 11 publications

References 17 publications

Galerkin finite element methods for the Shallow Water equations over variable bottom

Galerkin finite element methods for the Shallow Water equations over variable bottom

An Adaptive Discontinuous Galerkin Method for the Simulation of Hurricane Storm Surge

Application of Numerical Filter to a Taylor Galerkin Finite Element Model for Movable Bed Dam Break Flows

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