A Fast Marching Method for Hamilton-Jacobi Equations Modeling Monotone Front Propagations

In this paper we present a generalization of the Fast Marching method introduced by J.A. Sethian in 1996 to solve numerically the eikonal equation. The new method, named Buffered Fast Marching (BFM), is based on a semi-Lagrangian discretization and is suitable for Hamilton-Jacobi equations modeling monotonically advancing fronts, including Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations which arise in the framework of optimal control problems and differential games. We also show the convergence of the algorithm to the viscosity solution. Finally we present several numerical tests comparing the BFM method with other existing methods. This research was partially supported by the MIUR Project 2006 “Modellistica Numerica per il Calcolo Scientifico ed Applicazioni Avanzate” and by INRIA–Futurs and ENSTA, Paris, France.