Finite volume approximations of the Euler system with variable congestion

Abstract

We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure [15]. This model describes for instance the macroscopic motion of a crowd with individual congestion preferences. We propose an asymptotic preserving (AP) scheme based on a conservative formulation of the system in terms of density, momentum and density fraction. A second order accuracy version of the scheme is also presented. We validate the scheme on one-dimensionnal test-cases and compare it with a scheme previously proposed in [15] and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics.

Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint

“…Again, the impact law is not integrated in the global limit system, but this approach natively recovers the elastic setting (e = 1). In [11], a similar approach is carried out in the case of a variable congestion (the constraint ρ ≤ 1 is replaced by ρ ≤ ρ ⋆ , where ρ ⋆ is a given, non-uniform, barrier density). We also refer to [28] for an analysis of a similar system with additional memory effects induced by the presence of an underlying viscous fluid.…”

“…Let us add that the system is commonly written without any collision law, the actual choice being usually made in an implicit way, depending on the approach which is followed. For instance, in [7], particular solutions are built by means of sticky blocks with a purely inelastic collision law, whereas in [10,11], the approach is based on compressible Euler equation with a barrier-like pressure with respect to the density, natively leading to a purely elastic behavior.…”

An overview on congestion phenomena in fluid equations

“…A good introduction to the modeling of congestion phenomena both from microscopic and macroscopic viewpoints is proposed in [49]. Among the possible general extensions, let us mention the case of heterogeneous maximal density constraints ρ ≤ ρ * (t, x) (see for instance [24], [38], [60] or [7] for applications to traffic flows). Depending on the applications, other reference fluid systems can be preferred to the classical Navier-Stokes equations: non-linear shallow water or Boussinesq equations for wave-structure interactions (see [38], [9], [36]), gradient flow formulations in the modeling of crowds [50], porous media equations and Hele-Shaw free boundary problems for tissue growth modeling (see for instance [35], [61]), Bingham equations for complex geophysical flows [21], etc.…”

“…If the limit hard congestion system may be difficult to handle due to the (unknown) sharp interface between the free and the congested domains, one can instead try to simulate numerically solutions of the approximate soft congestion system for which it is possible to adapt classical compressible numerical schemes. The interested reader is referred to the works of Degond and collaborators in [23,25,24] for more details on the subject.…”

One-Dimensional Granular System with Memory Effects

“…Numerical simulations are have been studied in [11,12] with applications to crowd dynamics. This type of heterogeneous maximal constraint may be also relevant for the dynamics of floating structures; see for instance Lannes [14].…”