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

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.