| Abstract: |
| We present a numerical scheme for a Hughes-type model where agents optimize based only on the current population distribution $m(t)$. Unlike classical Mean Field Games where agents anticipate future crowd evolution, at each time step a new Hamilton-Jacobi-Bellman equation is solved with the instantaneous density.
Our method combines a semi-Lagrangian discretization for the value function with a Lagrange-Galerkin approximation for the continuity equation. We prove well-posedness and convergence of the scheme. We also introduce a hybrid system coupling both MFG and Hughes-type populations. Various numerical tests are presented for both models, showing distinct behavioral patterns in congestion and target accumulation. |
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