Abstract
We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure (Degond et al., 2016). 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 Degond et al. (2016) and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics.
Highlights
In this work we study two phase compressible/incompressible Euler system with variable congestion:
We are interested in the numerical simulation of the Euler system with a singular pressure modeling variable congestion
As the stiffness of the pressure increases (ε tends to 0), the model tends to a free boundary transition between compressible and incompressible dynamics
Summary
In this work we study two phase compressible/incompressible Euler system with variable congestion:. The main purpose of this work is to analyze (1) numerically, i.e. to propose the numerical scheme capturing the phase transition To this end we use the fact that (1) can be obtained as a limit when ε → 0 of the compressible Euler system with the congestion pressure π replaced by a singular approximation πε α. The diffusion is solved by means of cell-centered finite volume scheme, and the transport of the congested density is resolved with the upwind scheme The extension of this method to two-dimensions is one of the main results of the present paper.
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