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.
Citation: Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi. Small solids in an inviscid fluid[J]. Networks and Heterogeneous Media, 2010, 5(3): 385-404. doi: 10.3934/nhm.2010.5.385
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Abstract
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.