L1-error estimates for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1

The goal of this paper is to study some numerical approximations of particular HamiltonJacobi-Bellman equations in dimension 1 and with possibly discontinuous initial data. We investigate two anti-diffusive numerical schemes, the first one is based on the Ultra-Bee scheme and the second one is based on the Fast Marching Method. We prove the convergence and derive L-error estimates for both schemes. We also provide numerical examples to validate their accuracy in solving smooth and discontinuous solutions.