Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data

Abstract

We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton–Jacobi–Bellman equation of the form v t + max α (f(x, α)v x ) = 0, v(0, x) = v 0(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1–33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L 1-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.