Well-balanced schemes to capture non-explicit steady states: Ripa model

Desveaux, Vivien; Zenk, Markus; Berthon, Christophe; Klingenberg, Christian

doi:10.1090/mcom/3069

Cited by 43 publications

(35 citation statements)

References 37 publications

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“…Different relaxation models were tested in order to include the collisional source term, but, because of their complexity, they lead to configurations where a Riemann invariant is missing and the problem remains unclosed. In a recent work [31], the same issue is encountered and an additional relation is arbitrarily imposed. In the present situation, this strategy leads to particularly inconvenient solutions and the admissibility conditions are lost.…”

Section: Limit Of the Relaxation Approachmentioning

confidence: 99%

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

et al. 2017

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Abstract. This work is devoted to the derivation of an asymptotic-preserving scheme for the electronic M1 model in the diffusive regime. The case without electric field and the homogeneous case are studied. The derivation of the scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modified to enable the derivation of a numerical scheme which also satisfies the admissible conditions and is well-suited for capturing steady states. Moreover, it enjoys asymptotic-preserving properties and handles the diffusive limit recovering the correct diffusion equation. Numerical tests cases are presented, in each case, the asymptotic-preserving scheme is compared to the classical HLL [A. Harten, P.D. Lax and B. Van Leer, SIAM Rev. 25 (1983) 35-61.] scheme usually used for the electronic M1 model. It is shown that the new scheme gives comparable results with respect to the HLL scheme in the classical regime. On the contrary, in the diffusive regime, the asymptotic-preserving scheme coincides with the expected diffusion equation, while the HLL scheme suffers from a severe lack of accuracy because of its unphysical numerical viscosity.Mathematics Subject Classification. 65C20, 65M12.

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Section: Limit Of the Relaxation Approachmentioning

confidence: 99%

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

et al. 2017

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“…Following the ideas introduced in the companion paper [22] devoted to the Ripa model, we propose to design a Godunov-type scheme. To address such an issue, the key point stays in the derivation of an approximate Riemann solver that contains the source term in order to preserve the steady states defined by (14) ( [13,14,23]).…”

Section: Lemmamentioning

confidence: 99%

“…In the previous work [22], a relaxation model was developed in the framework of the shallowwater model with horizontal temperature gradients, also known as the Ripa model. Its particularity is that the associated Riemann problem is under-determined, and there is one relation missing.…”

Section: The Relaxation Modelmentioning

confidence: 99%

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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“…(14) To ensure the constant-temperature lake at rest steady state (4) property of the Ripa system one should pay special attention to the way the predicted values are computed at time t n+1/2 in equation (12), as well as to the back and forth projection steps in equations (10) and (14). Note that in the steady state case (h + Z = constant, u = 0, and θ = constant) the Ripa system reduces to: (15) and therefore specific discretizations of the momentum's flux component and its corresponding source term component are required in order to ensure well-balancing in the steady state case.…”

Section: One-dimensional Well-balanced Central Scheme For the Ripa Symentioning

confidence: 99%

“…Chertok, Kurganov and Liu [6] build a central scheme coupled with an interface tracking method. In [10], the authors design a finite volume method that utilizes a new relaxation Riemann solver which is able to well balance the discretization.…”

Section: Introductionmentioning

confidence: 99%

Well-balanced central finite volume methods for the Ripa system

Touma

Klingenberg

2015

Applied Numerical Mathematics

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Well-balanced schemes to capture non-explicit steady states: Ripa model

Cited by 43 publications

References 37 publications

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

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

Well-balanced central finite volume methods for the Ripa system

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Well-balanced schemes to capture non-explicit steady states: Ripa model

Cited by 43 publications

References 37 publications

Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: Particular cases

Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: Particular cases

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

Well-balanced central finite volume methods for the Ripa system

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Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases