Simone Cacace, Fabio Camilli, Claudio Marchi
We propose a numerical method for stationary Mean Field Games defined on a network. In this framework a correct approximation of the transition conditions at the vertices plays a crucial role. We prove existence, uniqueness and convergence of the scheme and we also propose a least squares method for the solution of the discrete system. Numerical experiments are carried out.
Fabio Camilli, Elisabetta Carlini, Claudio Marchi
We consider the Cauchy problem \[\partial_t u+H(x,Du)=0 \quad (x,t)\inΓ\times (0,T),\quad u(x,0)=u_0(x) \quad x\inΓ\] where $Γ$ is a network and $H$ is a convex and positive homogeneous Hamiltonian which may change from edge to edge. In the former part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a flame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.