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Convergence of finite volume schemes for the coupling between the inviscid Burgers equation and a particle


Authors: Nina Aguillon, Frédéric Lagoutière and Nicolas Seguin
Journal: Math. Comp. 86 (2017), 157-196
MSC (2010): Primary 35R37, 65M12, 35L65
DOI: https://doi.org/10.1090/mcom/3082
Published electronically: September 6, 2016
MathSciNet review: 3557797
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Abstract: The convergence of a class of finite volume schemes for a model of coupling between a Burgers fluid and a pointwise particle is proved. In this model, introduced by Lagoutière, Seguin and Takahashi in 2008, the particle is seen as a moving point through which an interface condition is imposed, which links the velocity of the fluid on the left and on the right of the particle and the velocity of the particle (the three quantities are all not equal in general). The total momentum of the system is conserved through time.

The proposed schemes are consistent with a ``large enough'' part of the interface conditions. The proof of convergence is an extension of the one of Andreianov and Seguin (2012) to the case where the particle moves under the influence of the fluid (two-way coupling). This extension contains two new main difficulties: first, the fluxes and interface conditions are time-dependent, and second, the coupling between an ODE and a PDE.


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Additional Information

Nina Aguillon
Affiliation: Département de Mathématiques, Université Paris Sud, 91405 Orsay Cedex Paris, France
Email: nina.aguillon@math.u-psud.fr

Frédéric Lagoutière
Affiliation: Département de Mathématiques, Université Paris Sud, 91405 Orsay Cedex Paris, France
Email: frederic.lagoutiere@math.u-psud.fr

Nicolas Seguin
Affiliation: Laboratoire Jacques Louis Lions, Université Pierre et Marie Curie, 75005 Paris, France
Email: nicolas.seguin@upmc.fr

DOI: https://doi.org/10.1090/mcom/3082
Keywords: Fluid-particle interaction, Burgers equation, Nonconservative coupling, moving interface, convergence of finite volume schemes, PDE-ODE coupling
Received by editor(s): October 24, 2014
Received by editor(s) in revised form: April 11, 2015
Published electronically: September 6, 2016
Article copyright: © Copyright 2016 American Mathematical Society