We examine convergence of finite difference schemes approximating lower semicontinuous (Barron-Jensen [11]) viscosity solutions for Cauchy problems of Hamilton-Jacobi type. Even for merely bounded and lower semicontinuous initial data, we identify general conditions on a finite difference scheme which guarantee that discrete approximations given by the scheme are uniformly close to the lower semicontinuous viscosity solution of the associated PDE, on compact sets where the solution is continuous. We also provide examples of admissible schemes and initial conditions for which our main results apply. Convergence results for finite difference schemes have recently been used to establish limit theorems for discrete random processes [1, 2]. As an application of our finite difference scheme convergence results, we prove a new limit theorem for a random process called the “cooperative leader random walk.”
