Abstract. We investigate the application of weighted essentially nonoscillatory (WENO) reconstructions to a class of semi-Lagrangian schemes for first order time-dependent Hamilton-Jacobi equations. In particular, we derive a general form of the scheme, study sufficient conditions for its convergence with high-order reconstructions, and perform numerical tests to study its efficiency. In addition, we prove that the weights of the WENO interpolants are positive for any order.

International audienceIn this work we propose a fully discrete semi-Lagrangian scheme for a first order mean field game system. We prove that the resulting discretization admits at least one solution and, in the scalar case, we prove a convergence result for the scheme. Numerical simulations and examples are also discussed

International audienceIn this paper we study a fully discrete Semi-Lagrangian approximation of a second order Mean Field Game system, which can be degenerate. We prove that the resulting scheme is well posed and, if the state dimension is equals to one, we prove a convergence result. Some numerical simulations are provided, evidencing the convergence of the approximation and also the difference between the numerical results for the degenerate and non-degenerate cases

We present a new Fast Marching algorithm for an eikonal equation with a velocity changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the Fast Marching Method in two respects. The first is that the new scheme can deal with a time-dependent velocity and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in R 2 .

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