Abstract:
In this talk, we develop two efficient numerical methods for a multiscale kinetic equation in the context of crowd dynamics with emotional contagion. In the continuum setting, the mesoscopic kinetic equation produces a natural Eulerian limit with nonlocal interactions. However, such limit ceases to be valid when the underlying microscopic particle characteristics cross, corresponding to the blow up of the solution in the Eulerian system. One method is to couple these two situations -- using Eulerian dynamics for regions without characteristic crossing and kinetic evolution for regions with characteristic crossing. The other method is based on a level set formulation for the continuum system. The so-derived level set equation shares similar forms as the kinetic equation, and it successfully captures the multi-valued solution in velocity, which implies that the multi-valued solution other than the viscosity solution should be the physically relevant ones for the continuum system. |