The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles

We propose a finite difference method based on the Lax-Friedrichs scheme for a model of interaction between multiple solid particles and an inviscid fluid. The single-particle version has been studied extensively during the past decade. The model studied here consists of the inviscid Burgers equation with multiple nonconservative moving source terms that are singular and account for drag force interaction between the fluid and the particles. Each particle trajectory satisfies a differential equation that ensures conservation of momentum of the entire system. To deal with the singular source terms we discretize a model that associates with each particle an advection PDE whose solution is a shifted Heaviside function. This alternative model is well known but has not previously been used in numerical methods. We propose a definition of entropy solution which directly generalizes the previously defined single-particle notion of entropy solution. We prove convergence (along a subsequence) of the Lax-Friedrichs approximations, and also prove that if the set of times where the particle paths intersect has Lebesgue measure zero, then the limit is an entropy solution. We also propose a higher resolution version of the scheme, based on MUSCL processing, and present the results of numerical experiments.