Ricerc@Sapienza

In this paper, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation.

We consider a class of “filtered” schemes for first order time dependent Hamilton–Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε= ε(Δ t, Δ x) > 0 which goes to 0 as the time and space steps (Δ t, Δ x) are going to 0 and does not depend on the time tn, for each iteration n.

We discuss convergence results for a class of finite difference schemes approximating Mean Field Games systems either on the torus or a network. We also propose a quasi-Newton method for the computation of discrete solutions, based on a least squares formulation of the problem. Several numerical experiments are carried out including the case with two or more competing populations.