A simple 1D model of inviscid fluid–solid interaction

Lagoutìère, Frédéric; Seguin, Nicolas; Takahashi, Takéo

doi:10.1016/j.jde.2008.03.011

Cited by 22 publications

(91 citation statements)

References 27 publications

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“…A transmission map γ tends to prescribe the value of u r as a monotone (possibly multivalued) function 8 of u l : u r ∈ γ(u l ). Many well-studied situations enter the framework of transmission maps (e.g.,the vanishing capillarity solutions for Buckley-Leverett equation, see § 3.2.2 and the non-conservative one-dimensional fluid-particle interaction model of [55], see § 3.2.3). As a result, new simple finite volume schemes for these models are derived, see § 3.3.…”

Section: Towards More Convenient Admissibility Criteriamentioning

confidence: 99%

“…Let us provide one example of such non-conservative coupling with a transmission map γ and a dissipation map ψ. The following equation introduced in [55] models the resistance of a small particle located at x = h(t) to motion of a one-dimensional fluid governed by the Burgers equation:…”

Section: The Non-conservative Case and The Burgers-particle Modelmentioning

confidence: 99%

“…the fluid) velocity (h (t)−u). Actually, in the coupled Burgers-particle model of [55], also the retroaction of the fluid on the particle is present; in this case, the particle evolution is governed by the drag force…”

Section: The Non-conservative Case and The Burgers-particle Modelmentioning

confidence: 99%

“…The meaning of (31), (32) is not obvious because of the possible discontinuity of u at (t, h(t)); the precise meaning, derived from regularization of is explained in [16,55]. A well-posedness theory of (31), (32) was constructed in [16,17] on the basis of the "germ" formalism (the extension of germs to the non-conservative coupling is straightforward).…”

Section: The Non-conservative Case and The Burgers-particle Modelmentioning

confidence: 99%

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New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux

Andreïanov

2015

ESAIM: Proc.

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Abstract. Hyperbolic conservation laws of the form ut + div f(t, x; u) = 0 with discontinuous in (t, x) flux function f attracted much attention in last 20 years, because of the difficulties of adaptation of the classical Kruzhkov approach developed for the smooth case. In the discontinuous-flux case, non-uniqueness of mathematically consistent admissibility criteria results in infinitely many different notions of solution. A way to describe all the resulting L 1 -contractive solvers within a unified approach was proposed in the work [Andreianov, Karlsen, Risebro, 2011]. We briefly recall the ideas and results developed there for the model one-dimensional case with f(t, x; u) = f l (u)1 1x<0 + fr(u)1 1x>0 and highlight the main hints needed to address the multi-dimensional situation with curved interfaces.Then we discuss two recent developments in the subject which permit to better understand the issue of admissibility of solutions in relation with specific modeling assumptions; they also bring useful numerical approximation strategies. A new characterization of limits of vanishing viscosity approximation proposed in [Andreianov and Mitrović, 2014] permits to encode admissibility in singular but intuitively appealing entropy inequalities. Transmission maps introduced in ([Andreianov and Cancès, 2015]) have applications in modeling flows in strongly heterogeneous porous media and lead to a simple algorithm for numerical approximation of the associated solutions.Moreover, in order to embed all the aforementioned results into a natural framework, we put forward the concept of interface coupling conditions (ICC) which role is analogous to the role of boundary conditions for boundary-value problems. We link this concept to known examples and techniques.

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Section: Towards More Convenient Admissibility Criteriamentioning

confidence: 99%

Section: The Non-conservative Case and The Burgers-particle Modelmentioning

confidence: 99%

Section: The Non-conservative Case and The Burgers-particle Modelmentioning

confidence: 99%

Section: The Non-conservative Case and The Burgers-particle Modelmentioning

confidence: 99%

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New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux

Andreïanov

2015

ESAIM: Proc.

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“…One can for example use the theory developed in [8]. Here it is possible to use the simpler approach developed in [1,15] that gives a path-independent definition to the nonconservative products. Furthermore, it is well known that such a nonconservative system may lead to resonant solutions of the Riemann problem [7,12,14].…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

Aguillon¹,

Audusse²,

Godlewski³

et al. 2018

SIAM J. Math. Anal.

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Some shallow water type models describing the vertical profile of the horizontal velocity with several degrees of freedom have been recently proposed. The question addressed in the current work is the hyperbolicity of a shallow water model with two velocities. The model is written in a nonconservative form and the analysis of its eigenstructure shows the possibility that two eigenvalues coincide. A definition of the nonconservative product is given which enables us to analyse the resonance and coalescence of waves. Eventually, we prove the well-posedness of the two dimensional Riemann problem with initial condition constant by half-plane.

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Parameter‐uniform numerical methods for singularly perturbed linear transport problems

Gracia

Navas-Montilla

O’Riordan

2022

Math Methods in App Sciences

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Pointwise accurate numerical methods are constructed and analysed for three classes of singularly perturbed first order transport problems. The methods involve piecewise-uniform Shishkin meshes and the numerical approximations are shown to be parameter-uniformly convergent in the maximum norm. A transport problem from the modelling of fluid-particle interaction is formulated and used as a test problem for these numerical methods. Numerical results are presented to illustrate the performance of the numerical methods and to confirm the theoretical error bounds established in the paper.

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A simple 1D model of inviscid fluid–solid interaction

Cited by 22 publications

References 27 publications

New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux

New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

Parameter‐uniform numerical methods for singularly perturbed linear transport problems

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