A semi-Lagrangian scheme for first-order mean field games based on monotone operators

We construct a semi-Lagrangian scheme for first-order, time-dependent and nonlocal mean field games. The convergence of the scheme to a weak solution of the system is analyzed by exploiting a key monotonicity property. To solve the resulting discrete problem, we implement a learning value algorithm, prove its convergence and propose an acceleration strategy based on a policy iteration method. Finally, we present numerical experiments that validate the effectiveness of the proposed schemes and show that the accelerated version significantly improves performance.