Well-balanced central finite volume methods for the Ripa system

Touma, R.; Klingenberg, Christian

doi:10.1016/j.apnum.2015.07.001

Cited by 38 publications

(27 citation statements)

References 23 publications

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“…This scheme is well-balanced, positivity preserving and entropy dissipative in the case of flat or continuous bottom topography Z. For several other recently developed well-balanced schemes for the TSW model, we refer the reader to [17,23,24,42]. If both the Coriolis force and buoyancy are taken into account, the situation is even more complicated and it is quite challenging to design an accurate and robust well-balanced scheme for the studied system.…”

Section: C)mentioning

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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Section: C)mentioning

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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“…Our last one-dimensional flat bottom experiment is a dam break over a flat waterbed. The initial conditions for this problem are defined as [ 6 ] The numerical solution is calculated at t = 0.2 in the computational domain [−1, 1] using the KFVS scheme and its results are compared with those obtained from the central NT scheme. The computational domain is divided into 200 grid points.…”

Section: Numerical Case Studiesmentioning

confidence: 99%

“…Our last one-dimensional problem of variable bottom topography is a discontinuous waterbed problem which is taken from [ 6 ] for the case of Ripa system. The waterbed for the rectangular bump is defined as and the initial data are given by Numerical solutions are obtained in the computational domain [0, 600] at time t = 12 using the well-balanced KFVS and central NT schemes.…”

Section: Numerical Case Studiesmentioning

confidence: 99%

“…The present work deals with the numerical approximation of shallow water equations which have temperature fluctuations of prime interest [ 4 ]. This set of shallow water equations, containing horizontal temperature gradient terms, is called as Ripa system and is used to investigate ocean currents [ 5 , 6 ]. The main difference between the well known shallow water model and the Ripa system is the definition of steady states at rest.…”

Section: Introductionmentioning

confidence: 99%

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A kinetic flux vector splitting scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

et al. 2018

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This paper is concerned with the derivation of a well-balanced kinetic scheme to approximate a shallow flow model incorporating non-flat bottom topography and horizontal temperature gradients. The considered model equations, also called as Ripa system, are the non-homogeneous shallow water equations considering temperature gradients and non-uniform bottom topography. Due to the presence of temperature gradient terms, the steady state at rest is of primary interest from the physical point of view. However, capturing of this steady state is a challenging task for the applied numerical methods. The proposed well-balanced kinetic flux vector splitting (KFVS) scheme is non-oscillatory and second order accurate. The second order accuracy of the scheme is obtained by considering a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. The scheme is applied to solve the model equations in one and two space dimensions. Several numerical case studies are carried out to validate the proposed numerical algorithm. The numerical results obtained are compared with those of staggered central NT scheme. The results obtained are also in good agreement with the recently published results in the literature, verifying the potential, efficiency, accuracy and robustness of the suggested numerical scheme.

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“…There are also well-balanced schemes for related models, like e.g. the Ripa model [13,37]. More recently, well-balanced schemes for Euler equations with gravitational source term have been developed.…”

Section: Introductionmentioning

confidence: 99%

Second Order Finite Volume Scheme for Euler Equations with Gravity which is Well-Balanced for General Equations of State and Grid Systems

Berberich¹

2019

CICP

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We develop a second order well-balanced finite volume scheme for compressible Euler equations with a gravitational source term. The well-balanced property holds for arbitrary hydrostatic solutions of the corresponding Euler equations without any restriction on the equation of state. The hydrostatic solution must be known a priori either as an analytical formula or as a discrete solution at the grid points. The scheme can be applied on curvilinear meshes and in combination with any consistent numerical flux function and time stepping routines. These properties are demonstrated on a range of numerical tests. † By the term time-stepping routines we refer to ODE solvers which are applied to evolve the semi-discrete scheme obtained after spatial discretization by a finite volume method.

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Well-balanced central finite volume methods for the Ripa system

Cited by 38 publications

References 23 publications

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

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

A kinetic flux vector splitting scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

Second Order Finite Volume Scheme for Euler Equations with Gravity which is Well-Balanced for General Equations of State and Grid Systems

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