We are interested to the approximation of first order Hamilton-Jacobi-Bellman equations in the case of discontinuous initial data. Such data may come from optimal control problems. We study here a scheme based on an extention of the Ultra-Bee scheme of Despres-Lagoutiere for linear advection. We show an L^1 error estimate for a particular HJB equation of the form u_t + max_a [f(x,a)u_x] = 0, in one space dimension. Since the proposed scheme is completely non-monotone, we derive a new type of proof for this particular case. It uses a representation of the solution into max/min of elementary monotonous solutions (decreasing or non-deacreasing in the space variable). Interesting anti-diffusive properties of the scheme will also be discussed, as well as possible extentions to two or more dimensions.