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

TL;DR

This paper proves the convergence of finite volume schemes for a coupled system of Burgers fluid and a moving particle, addressing challenges of time-dependent flux and fluid-particle interaction.

Contribution

It extends existing convergence proofs to include a moving particle influenced by the fluid, handling time-dependent interface conditions and coupling of ODE and PDE.

Findings

Convergence of the proposed schemes is established.

The schemes are consistent with key interface conditions.

The proof manages the coupling between fluid dynamics and particle motion.

Abstract

In this paper, we prove the convergence of a class of finite volume schemes for the model of coupling between a Burgers fluid and a pointwise particle introduced in [LST08]. In this model, the particle is seen as a moving interface 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 impulsion 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 [AS12] to the case where the particle moves under the influence of the fluid. It yields two main difficulties: first, we have to deal with time-dependent flux and interface condition, and second with the coupling between and ODE and a PDE.