A semi-implicit multiscale scheme for shallow water flows at low Froude number

Vater, Stefan; Klein, Rupert

doi:10.2140/camcos.2018.13.303

Cited by 14 publications

(4 citation statements)

References 39 publications

(62 reference statements)

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“…For this convergence study, a quasi‐stationary vortex is transported along the diagonal on the unit square (with the domain

\Omega ={\left[0\kern0.3em \mathrm{m},1\kern0.3em \mathrm{m}\right]}^2

). This test case is similar to vortex test cases in Vater 15 and Benacchio 16 . Periodic boundary conditions are used on all sides of the domain.…”

Section: Verification and Validation Of The Dealii And Stormflash Imp...mentioning

confidence: 99%

A linear low effort stabilization method for the Euler equations using discontinuous Galerkin methods

Bänsch,

Behrens,

Vater

2023

Numerical Methods in Fluids

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SummaryWe present a novel and simple yet intuitive approach to the stabilization problem for the numerically solved Euler equations with gravity source term relying on a low‐order nodal Discontinuous Galerkin Method (DGM). Instead of assuming isothermal or polytropic solutions, we only take a hydrostatic balance as a given property of the flow and use the hydrostatic equation to calculate a hydrostatic pressure reconstruction that replaces the gravity source term. We compare two environments that both solve the Euler equations using the DGM: deal.II and StormFlash. We utilize StormFlash as it allows for the use of the novel stabilization method. Without stabilization, StormFlash does not yield results that resemble correct physical behavior while the results with stabilization for StormFlash, as well as deal.II model the fluid flow more accurately. Convergence rates for deal.II do not match the expected order while the convergence rates for StormFlash with the stabilization scheme (with the exceptions for the L errors for momentum) meet the expectation. The results from StormFlash with stabilization also fit reference solutions from the literature much better than those from deal.II. We conclude that this novel scheme is a low cost approach to stabilize the Euler equations while not limiting the flow in any way other than it being in hydrostatic balance.

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“…For this convergence study, a quasi‐stationary vortex is transported along the diagonal on the unit square (with the domain

\Omega ={\left[0\kern0.3em \mathrm{m},1\kern0.3em \mathrm{m}\right]}^2

). This test case is similar to vortex test cases in Vater 15 and Benacchio 16 . Periodic boundary conditions are used on all sides of the domain.…”

Section: Verification and Validation Of The Dealii And Stormflash Imp...mentioning

confidence: 99%

A linear low effort stabilization method for the Euler equations using discontinuous Galerkin methods

Bänsch,

Behrens,

Vater

2023

Numerical Methods in Fluids

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“…until the vortex reaches its starting position again. The analytical solution for this test can be found in [30].…”

Section: A Strongly Localisedmentioning

confidence: 99%

Metrics for Performance Quantification of Adaptive Mesh Refinement

2021

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Non-uniform, dynamically adaptive meshes are a useful tool for reducing computational complexities for geophysical simulations that exhibit strongly localised features such as is the case for tsunami, hurricane or typhoon prediction. Using the example of a shallow water solver, this study explores a set of metrics as a tool to distinguish the performance of numerical methods using adaptively refined versus uniform meshes independent of computational architecture or implementation. These metrics allow us to quantify how a numerical simulation benefits from the use of adaptive mesh refinement. The type of meshes we are focusing on are adaptive triangular meshes that are non-uniform and structured. Refinement is controlled by physics-based indicators that capture relevant physical processes and determine the areas of mesh refinement and coarsening. The proposed performance metrics take into account a number of characteristics of numerical simulations such as numerical errors, spatial resolution, as well as computing time. Using a number of test cases we demonstrate that correlating different quantities offers insight into computational overhead, the distribution of numerical error across various mesh resolutions as well as the evolution of numerical error and run-time per degree of freedom.

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“…Possible splittings have been introduced in, e.g., [6,12,16,18,24,31,40], see also the references in the cited papers. For asymptotic analysis and/or alternative methods such as staggered or relaxation schemes, we refer to the recent works [1,4,[9][10][11]14,25,29,36,37].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic properties of a class of linearly implicit schemes for weakly compressible Euler equations

et al. 2021

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In this paper we derive and analyse a class of linearly implicit schemes which includes the one of Feistauer and Kučera (J Comput Phys 224:208–221, 2007) as well as the class of RS-IMEX schemes (Schütz and Noelle in J Sci Comp 64:522–540, 2015; Kaiser et al. in J Sci Comput 70:1390–1407, 2017; Bispen et al. in Commun Comput Phys 16:307–347, 2014; Zakerzadeh in ESAIM Math Model Numer Anal 53:893–924, 2019). The implicit part is based on a Jacobian matrix which is evaluated at a reference state. This state can be either the solution at the old time level as in Feistauer and Kučera (2007), or a numerical approximation of the incompressible limit equations as in Zeifang et al. (Commun Comput Phys 27:292–320, 2020), or possibly another state. Subsequently, it is shown that this class of methods is asymptotically preserving under the assumption of a discrete Hilbert expansion. For a one-dimensional setting with some limitations on the reference state, the existence of a discrete Hilbert expansion is shown.

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A semi-implicit multiscale scheme for shallow water flows at low Froude number

Cited by 14 publications

References 39 publications

A linear low effort stabilization method for the Euler equations using discontinuous Galerkin methods

A linear low effort stabilization method for the Euler equations using discontinuous Galerkin methods

Metrics for Performance Quantification of Adaptive Mesh Refinement

Asymptotic properties of a class of linearly implicit schemes for weakly compressible Euler equations

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