A New Approach for Designing Moving-Water Equilibria Preserving Schemes for the Shallow Water Equations

Cheng, Yuanzhen; Chertock, Alina; Herty, Michaël; Kurganov, Alexander; Wu, Tong

doi:10.1007/s10915-019-00947-w

Cited by 46 publications

(37 citation statements)

References 39 publications

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“…In this section, a well-balanced WENO-ADER scheme using the Augmented Roe solver, which preserves the jets in (56) and (57), is proposed. The keystone of this scheme is the treatment of the Coriolis source terms as geometric sources in order to discretize them in the same way than the bed elevation source term.…”

Section: Formulation Of a Well-balanced Scheme For The Swe In The Rotmentioning

confidence: 99%

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2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame

Navas-Montilla

Murillo

2018

Journal of Computational Physics

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In this work, an arbitrary order augmented WENO-ADER scheme for the resolution of the 2D Shallow Water Equations (SWE) with geometric source term is presented and its application to other shallow water models involving non-geometric sources is explored. This scheme is based in the 1D Augmented Roe Linearized-ADER (ARL-ADER) scheme, presented by the authors in a previous work and motivated by a suitable compromise between accuracy and computational cost. It can be regarded as an arbitrary order version of the Augmented Roe solver, which accounts for the contribution of continuous and discontinuous geometric source terms at cell interfaces in the resolution of the Derivative Riemann Problem (DRP). The main novelty of this work is the extension of the ARL-ADER scheme to 2 dimensions, which involves the design of a particular procedure for the integration of the source term with arbitrary order that ensures an exact balance between flux fluctuations and sources. This procedure makes the scheme preserve equilibrium solutions with machine precision and capture the transient waves accurately. The scheme is applied to the SWE with bed variation and is extended to handle non-geometric source terms such as the Coriolis source term. When considering the SWE with bed variation and Coriolis, the most relevant equilibrium states are the still water at rest and the geostrophic equilibrium. The traditional well-balanced property is extended to satisfy the geostrophic equilibrium. This is achieved by means of a geometric reinterpretation of the Coriolis source term. By doing this, the formulation of the source terms is unified leading to a single geometric source regarded as an apparent topography. The numerical scheme is tested for a broad variety of situations, including some cases where the first order scheme ruins the solution.

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Section: Formulation Of a Well-balanced Scheme For The Swe In The Rotmentioning

confidence: 99%

“…For a comparison of different well-balanced methods in the framework of DG schemes, see [50]. The well-balanced property can still be enhanced by considering energy conservation criteria in the numerical scheme [51,52,53,54,55,56,57,58,28,29,55].…”

Section: Introductionmentioning

confidence: 99%

2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame

Navas-Montilla

Murillo

2018

Journal of Computational Physics

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“…Central-upwind schemes have been originally developed for hyperbolic systems of conservation laws [27,29,31] and then extended and applied to hyperbolic systems of balance laws arising in modeling shallow water flows; see, e.g., [6,13,25,26,28,30]. Here, we proceed along the lines of previous works in [8,[10][11][12] and develop a central-upwind scheme, which is well-balanced in the sense that it exactly preserves the geostrophic equilibria (1.11). This is achieved by performing a piecewise linear reconstruction of the equilibrium variables followed by well-balanced evolution, which uses modified central-upwind fluxes similar to those presented in [10].…”

Section: C)mentioning

confidence: 98%

“…is applied in (3.10) in a component-wise manner. Next, using the piecewise polynomial reconstruction (3.9), we obtain the one-sided point values of V at the cell interfaces: Equations (3.11) and (3.12) are solved as in [8]. For example, we describe how to solve (3.11) (the solution of (3.12) can be obtained in a similar way, we omit here):…”

Section: Equipped With the Valuesmentioning

confidence: 99%

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A well-balanced central-upwind scheme for the thermal rotating shallow water equations

Kurganov

Liu

Zeitlin

2020

Journal of Computational Physics

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We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients-the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-upwind scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion when the computed solution is near (at) thermo-geostrophic equilibria.The designed scheme is successfully tested on a series of numerical examples. Motivated by future applications to large-scale motions in the ocean and atmosphere, the model is considered on the tangent plane to a rotating planet both in mid-latitudes and at the Equator. The numerical scheme is shown to be capable of quite accurately maintaining the equilibrium states in the presence of nontrivial topography and rotation. Prior to numerical simulations, an analysis of the TRSW model based on the use of Lagrangian variables is presented, allowing one to obtain criteria of existence and uniqueness of the equilibrium state, of the wave-breaking and shock formation, and of instability development out of given initial conditions. The established criteria are confirmed in the conducted numerical experiments.

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Moving water equilibria preserving nonstaggered central scheme for open‐channel flows

Dong

Luo

et al. 2022

Math Methods in App Sciences

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In this paper, we investigate a well‐balanced and positive‐preserving nonstaggered central scheme, which has second‐order accuracy on both time and spatial scales, for open‐channel flows with the variable channel width and the nonflat bottom. We perform piecewise linear reconstructions of the conserved variables and energy as well as discretize the source term using the property that the energy remains constant, so that the complex source term and the flux can be precisely balanced so as to maintain the steady state. The scheme also ensures that the cross‐sectional wet area is positive by introducing a draining time‐step technique. Numerical experiments demonstrate that the scheme is capable of accurately maintaining both the still steady‐state solutions and the moving steady‐state solutions, simultaneously. Moreover, the scheme has the ability to accurately capture small perturbations of the moving steady‐state solution and avoids generating spurious oscillations. It is also capable of showing that the scheme is positive‐preserving and robust in solving the dam‐break problem.

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A New Approach for Designing Moving-Water Equilibria Preserving Schemes for the Shallow Water Equations

Cited by 46 publications

References 39 publications

2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame

2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame

A well-balanced central-upwind scheme for the thermal rotating shallow water equations

Moving water equilibria preserving nonstaggered central scheme for open‐channel flows

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