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September  2010, 5(3): 385-404. doi: 10.3934/nhm.2010.5.385

Small solids in an inviscid fluid

Boris Andreianov 1, , Frédéric Lagoutière 2, , Nicolas Seguin 3, and  Takéo Takahashi 4,

1. 

Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 25030 Besançon Cedex, France
2. 

Laboratoire de mathématiques, Université Paris-Sud, 91405 Orsay cedex, France
3. 

UMR 7598 Laboratoire J.-L. Lions, UPMC Univ Paris 06, Paris, F-75005, France
4. 

Institut Élie Cartan UMR 7502, INRIA, Nancy-Université, CNRS, 54506 Vandoeuvre-lès-Nancy Cedex, France

Received  January 2010 Revised  June 2010 Published  July 2010

We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.
Keywords: Burgers equation, random-choice method., adapted entropy, Solid-fluid interaction, singular source term, well-balanced scheme.
Mathematics Subject Classification: Primary: 35F25, 35L80, 65M9.
Citation: Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi. Small solids in an inviscid fluid. Networks and Heterogeneous Media, 2010, 5 (3) : 385-404. doi: 10.3934/nhm.2010.5.385
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