Well-balanced schemes for the shallow water equations with Coriolis forces

Chertock, Alina; Dudzinski, Michael; Kurganov, Alexander; Lukáčová-Medviďová, Mária

doi:10.1007/s00211-017-0928-0

Cited by 49 publications

(54 citation statements)

References 44 publications

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“…One steady-state constraint for the shallow water equations is the "lake at rest" condition [4,5,6], since the relevant waves in a flow can be viewed as small perturbations of the lake at rest, see [5]. A good numerical method for the shallow water equations should accurately capture both steady states and their small perturbations (quasi-steady flows) so as to diminish the appearance of unphysical waves with magnitude proportional to the mesh size (a so-called "numerical storm" [7]), that are normally present for numerical schemes that cannot preserve the "lake at rest" condition. A numerical method that exactly preserves the "lake at rest" steady state property is said to be well-balanced, see e.g.…”

Section: Introductionmentioning

confidence: 99%

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

Wintermeyer

Winters

Gassner

et al. 2017

Journal of Computational Physics

102

114

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We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

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Section: Introductionmentioning

confidence: 99%

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

Wintermeyer

Winters

Gassner

et al. 2017

Journal of Computational Physics

102

114

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“…In the third example, we numerically investigate the Rossby adjustment problem with the constant Coriolis parameter f ≡ 1, which was studied previously in [4,11]. We consider the following initial conditions, which correspond to a jet over a flat h: h(y, 0) ≡ 1, v(y, 0) ≡ 0, u(y, 0) = 2(1 + tanh(2y + 2))(1 − tanh(2y − 2)) (1 + tanh(2)) 2 , which are prescribed in the computational domain [−250, 250].…”

Section: Example 3 -Rossby Adjustment In An Open Domain In the F -Planementioning

confidence: 99%

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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“…For instance, the reader is referred to works devoted to the well-known shallow-water system [5][6][7][8] and some related model [9][10][11][12] (see also [13] for isentropic steady states associated with (1)). For instance, the reader is referred to works devoted to the well-known shallow-water system [5][6][7][8] and some related model [9][10][11][12] (see also [13] for isentropic steady states associated with (1)).…”

Section: Lemmamentioning

confidence: 99%

“…This PDE system can be easily solved as soon as the pressure law only depends on the density. For instance, the reader is referred to works devoted to the well-known shallow-water system [5][6][7][8] and some related model [9][10][11][12] (see also [13] for isentropic steady states associated with (1)). Unfortunately, because the pressure function p depends on both and e, we cannot exhibit algebraic relations satisfied by the solutions of (14), except under restrictive assumptions.…”

Section: Lemmamentioning

confidence: 99%

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A well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity

Desveaux

Zenk

Berthon

et al. 2015

Numerical Methods in Fluids

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Summary This paper describes a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear partial differential equation, whose solutions are called hydrostatic equilibria. We present a well‐balanced method, meaning that besides discretizing the complete equations, the method is also able to maintain all hydrostatic equilibria. The method is a finite volume method, whose Riemann solver is approximated by a so‐called relaxation Riemann solution that takes all hydrostatic equilibria into account. Relaxation ensures robustness, accuracy, and stability of our method, because it satisfies discrete entropy inequalities. We will present numerical examples, illustrating that our method works as promised. Copyright © 2015 John Wiley & Sons, Ltd.

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Well-balanced schemes for the shallow water equations with Coriolis forces

Cited by 49 publications

References 44 publications

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

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

A well‐balanced scheme to capture non‐explicit steady states in the Euler equations with gravity

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