High Order Still-Water and Moving-Water Equilibria Preserving Discontinuous Galerkin Methods for the Ripa Model

Britton, Jolene; Xing, Yulong

doi:10.1007/s10915-020-01134-y

Cited by 15 publications

(7 citation statements)

References 33 publications

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“…We choose this example to verify the ability of our well-balanced MM-DG scheme to capture small perturbations over the lake-at-rest water surface and temperature field. Similar examples have been used by a number of researchers, e.g., [6,31]. The bottom topography in this example is taken as…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In recent years studies have been made on the development of well-balanced numerical schemes for the Ripa model. The first work seems to be [8] [31] (a second-order positivity preserving finite volume scheme on rectangular meshes), Sánchez-Linares et al [27] (a second-order positivity preserving HLLC scheme based in path-conservative approximate Riemann solvers, for the onedimensional Ripa model), Han and Li [13] (a high-order finite difference weighted essentially non-oscillatory (WENO) scheme), Saleem et al [26] (a kinetic flux vector splitting scheme on rectangular meshes), Thanh et al [30] (a high-order scheme of van Leer's type for the one-dimensional SWEs with temperature gradient), Rehman et al [24] (a fifth-order finite volume multi-resolution WENO scheme on rectangular meshes), Britton and Xing [6] (a DG scheme for the one-dimensional Ripa model), Qian et al [23] (a DG method based on a source term approximation technique), and Li et al [20] (a DG method based on hydrostatic reconstruction on rectangular meshes). Fixed meshes are employed in the above mentioned works.…”

Section: Introductionmentioning

confidence: 99%

“…Generally speaking, the above scheme does not preserve the lake-at-rest steady-state solution and thus is not well-balanced. Following the idea of hydrostatic reconstruction [2,6,20], after computing the boundary values U int h,K | e K and U ext h,K | e K , we define (2.5)…”

Section: Introductionmentioning

confidence: 99%

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A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

Huang¹,

Li²,

Qiu³

et al. 2022

Preprint

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A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of the Ripa model -a generalization of the shallow water equations that accounts for effects of water temperature variations. Thermodynamic processes are important particularly in the upper layers of the ocean where the variations of sea surface temperature play a fundamental role in climate change. The well-balance property which requires numerical schemes to preserve the lake-at-rest steady state is crucial to the simulation of perturbation waves over that steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance, positivity-preserving, and high-order properties, a DG-interpolation scheme (with or without scaling positivity-preserving limiter) and special treatments pertaining to the Ripa model are employed in the transfer of both the flow variables and bottom topography from the old mesh to the new one and in the TVB limiting process. Mesh adaptivity is realized using an MMPDE moving mesh approach and a metric tensor based on an equilibrium variable and water depth. A motivation is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution (bottom structure). Numerical examples in one and two dimensions are presented to demonstrate the well-balance, high-order accuracy, and positivitypreserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

Huang¹,

Li²,

Qiu³

et al. 2022

Preprint

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“…These stationary solutions are well-explained in the articles. 5,41 One of them is described as H = h + z = constant, u = 0, u = constant: ð5Þ…”

Section: The Ripa Systemsmentioning

confidence: 99%

Well-balanced finite volume multi-resolution schemes for solving the Ripa models

Rehman

Ali

Zia

et al. 2021

Advances in Mechanical Engineering

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In this article, fifth order well-balanced finite volume multi-resolution weighted essentially non-oscillatory (FV MR-WENO) schemes are constructed for solving one-dimensional and two-dimensional Ripa models. The Ripa system generalizes the shallow water model by incorporating horizontal temperature gradients. The presence of temperature gradients and source terms in the Ripa models introduce difficulties in developing high order accurate numerical schemes which can preserve exactly the steady-state conditions. The proposed numerical methods are capable to exactly preserve the steady-state solutions and maintain non-oscillatory property near the shock transitions. Moreover, in the procedure of derivation of the FV MR-WENO schemes unequal central spatial stencils are used and linear weights can be chosen any positive numbers with only restriction that their total sum is one. Various numerical test problems are considered to check the validity and accuracy of the derived numerical schemes. Further, the results obtained from considered numerical schemes are compared with those of a high resolution central upwind scheme and available exact solutions of the Ripa model.

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“…Additional, more recent terminology for the IL 0 is "thermal rotating shallow-water model" [66,71], which emphasizes the ability of the IL 0 to include (horizontal) gradients of temperature. The IL 0 is also being called the "Ripa model" in the literature [15,18,19,41,47,59], in recognition of Pedro Ripa's contribution to its understanding [49-51, 53, 55]. We will reserve that to refer to the model generalized here, which was introduced in Ref.…”

Section: Introductionmentioning

confidence: 99%

Multilayer shallow-water model with stratification and shear

Beron-Vera

2021

Rev. Mex. Fís.

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The purpose of this paper is to present a shallow-water-type model with multiple inhomogeneous layers featuring variable linear velocity vertical shear and startificaion in horizontal space and time. This is achieved by writing the layer velocity and buoyancy fields as linear functions of depth, with coefficients that depend arbitrarily on horizontal position and time. The model is a generalization of Ripa's (1995) single-layer model to an arbitrary number of layers. Unlike models with homogeneous layers the present model is able to represent thermodynamics processes driven by heat and freshwater fluxes through the surface or mixing processes resulting from fluid exchanges across contiguous layers. By contrast with inhomogeneous-layer models with depth-independent velocity and buoyancy, the model derived here can sustain explicitly at low frequency a current in thermal wind balance (between the vertical vertical shear and the horizontal density gradient) within each layer. In the absence of external forcing and dissipation, energy, volume, mass, and buoyancy variance constrain the dynamics; conservation of total zonal momentum requires in addition the usual zonal symmetry of the topography and horizontal domain. The inviscid, unforced model admits a formulation suggestive of a generalized Hamiltonian structure, which enables the classical connection between symmetries and conservation laws via Noether's theorem. A steady solution to a system involving one Ripa-like layer and otherwise homogeneous layers can be proved formally (or Arnold) stable using the above invariants. A model configuration with only one layer has been shown previously to provide: a very good representation of the exact vertical normal modes up to the first internal mode; an exact representation of long-perturbation (free boundary) baroclinic instability; and a very reasonable representation of short-perturbation (classical Eady) baroclinic instability. Here it is shown that substantially more accurate overall results with respect to single-layer calculations can be achieved by considering a stack of only a few layers. A similar behavior is found in ageostrophic (classical Stone) baroclinic instability by describing accurately the dependence of the solutions on the Richardson number with only two layers.

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High Order Still-Water and Moving-Water Equilibria Preserving Discontinuous Galerkin Methods for the Ripa Model

Cited by 15 publications

References 33 publications

A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

Well-balanced finite volume multi-resolution schemes for solving the Ripa models

Multilayer shallow-water model with stratification and shear

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