A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: One-dimensional case

Vater, Stefan; Beisiegel, Nicole; Behrens, Jörn

doi:10.1016/j.advwatres.2015.08.008

Cited by 46 publications

(64 citation statements)

References 26 publications

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“…The second test case, which goes back to the work of Thacker, is also defined in a parabolic bowl but describes a circular flow with a linear surface elevation in the wet part of the domain. It is the 2D analog of the 1D test case described in the work of Vater et al Here, we follow the particular setup of the work of Gallardo et al In a square domain Ω=[−2,2] 2 with bottom topography b = b ( x )=0.1( x 2 + y 2 ), an analytical solution of the shallow‐water equations is given by

\begin{matrix} alignleft \\ align-1 h (x, t) & align-2 = \max \{0, 0.1 (x \cos (ω t) + y \sin (ω t) + \frac{3}{4}) - b (x)\} \\ align-1 (u, v) (x, t) & align-2 = \{\begin{matrix} \frac{ω}{2} false(- \sin false(ω t false), \cos false(ω t false) false) & if h false(bold-italicx, t false) > 0 \\ bold0 & otherwise, \end{matrix} \end{matrix}

with

ω = \sqrt{0.2 g}

.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…A standard benchmark problem to evaluate wetting and drying behavior of a numerical scheme is the wave runup onto a plane beach. We perform this quasi-one-dimensional test case 38 to compare the results to the ones already obtained with the one-dimensional version of the scheme in the work of Vater et al 21 The test case admits an exact solution following a technique developed in the work of Carrier et al 39 In a rectangular domain Ω = [−400, 50 000]×[0, 400] with linearly sloping bottom topography b(x) = 5000− x, = 0.1, and initial velocity u(x, 0) = 0, an initial surface elevation is prescribed in nondimensional variables by…”

Section: Tsunami Runup Onto a Linearly Sloping Beachmentioning

confidence: 99%

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A limiter‐based well‐balanced discontinuous Galerkin method for shallow‐water flows with wetting and drying: Triangular grids

Vater

Beisiegel

Behrens

2019

Numerical Methods in Fluids

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Summary A novel wetting and drying treatment for second‐order Runge‐Kutta discontinuous Galerkin methods solving the nonlinear shallow‐water equations is proposed. It is developed for general conforming two‐dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a nondestructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well balancing with an innovative velocity‐based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme, even on unstructured grids, and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near‐realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, superlinear convergence, mass conservation, well balancedness, and stability are verified.

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\begin{matrix} alignleft \\ align-1 h (x, t) & align-2 = \max \{0, 0.1 (x \cos (ω t) + y \sin (ω t) + \frac{3}{4}) - b (x)\} \\ align-1 (u, v) (x, t) & align-2 = \{\begin{matrix} \frac{ω}{2} false(- \sin false(ω t false), \cos false(ω t false) false) & if h false(bold-italicx, t false) > 0 \\ bold0 & otherwise, \end{matrix} \end{matrix}

with

ω = \sqrt{0.2 g}

.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Tsunami Runup Onto a Linearly Sloping Beachmentioning

confidence: 99%

A limiter‐based well‐balanced discontinuous Galerkin method for shallow‐water flows with wetting and drying: Triangular grids

Vater

Beisiegel

Behrens

2019

Numerical Methods in Fluids

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“…Whenever the fluid depth h is below this tolerance, the velocity is set to zero. In [47] it was shown that the specific value of this parameter does not affect the stability of the scheme. It rather influences the accuracy of the wetting and drying computation.…”

Section: Numerical Modelmentioning

confidence: 99%

“…The method is mass-conservative and preserves the steady state at rest (i.e., it is well-balanced). A comprehensive presentation of the scheme including its validation with respect to the shallow water equations is given in Vater et al [47].…”

Section: Numerical Modelmentioning

confidence: 99%

“…Its stability is governed by a Courant-Friedrichs-Lewy (CFL) time step restriction [50] with ∆t ≤ cfl∆x/c max , where ∆t and ∆x are the time step and grid cell size, respectively, cfl is the CFL number and c max = max{|u| + gh} is the maximum speed at which information propagates. To obtain linear stability and positivity of the water depth in the RKDG method, the validity of cfl ≤ 1/3 has to be ensured [47,51].…”

Section: Numerical Modelmentioning

confidence: 99%

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An Experimental and Numerical Study of Long Wave Run-Up on a Plane Beach

Drähne

Goseberg

Vater

et al. 2015

JMSE

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This research is to facilitate the current understanding of long wave dynamics at coasts and during on-land propagation; experimental and numerical approaches are compared against existing analytical expressions for the long wave run-up. Leading depression sinusoidal waves are chosen to model these dynamics. The experimental study was conducted using a new pump-driven wave generator and the numerical experiments were carried out with a one-dimensional discontinuous Galerkin non-linear shallow water model. The numerical model is able to accurately reproduce the run-up elevation and velocities predicted by the theoretical expressions. Depending on the surf similarity of the generated waves and due to imperfections of the experimental wave generation, riding waves are observed in the experimental results. These artifacts can also be confirmed in the numerical study when the data from the physical experiments is assimilated. Qualitatively, scale effects associated with the experimental setting are discussed. Finally, shoreline velocities, run-up and run-down are determined and shown to largely agree with analytical predictions.

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Development of an efficient wetting and drying treatment for shallow‐water modeling using the quadrature‐free Runge‐Kutta discontinuous Galerkin method

Zhang

2020

Numerical Methods in Fluids

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SummaryTo develop a robust, well‐balanced and quadrature‐free Runge‐Kutta discontinuous Galerkin (RKDG) shallow water solver, we introduce an efficient wetting and drying (WD) treatment in this paper. The main feature of this WD treatment is the use of vertex‐based linear reconstructed solutions in transition (partially wet) regions and high‐order solutions in smooth wet areas. To preserve the positivity of water depth, we also propose a modified time step size with the quadrature‐free scheme. The advantages of the WD treatment include robustness and the capability to address arbitrary high‐order RKDG methods in wet regions, thus making the quadrature‐free scheme more accurate and applicable to various flooding and drying problems. Two numerical test cases are used to validate the numerical method, and the results indicate that the WD method is accurate and robust for different flow regimes with dry areas.

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A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: One-dimensional case

Cited by 46 publications

References 26 publications

A limiter‐based well‐balanced discontinuous Galerkin method for shallow‐water flows with wetting and drying: Triangular grids

A limiter‐based well‐balanced discontinuous Galerkin method for shallow‐water flows with wetting and drying: Triangular grids

An Experimental and Numerical Study of Long Wave Run-Up on a Plane Beach

Development of an efficient wetting and drying treatment for shallow‐water modeling using the quadrature‐free Runge‐Kutta discontinuous Galerkin method

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