Section: New Results

Antidiffusive schemes for first order HJB equations

Participants : N. Megdich, H. Zidani.

In the framework of the thesis of N. Megdich, we have continued the study of numerical schemes for HJB equations coming from optimal control problems with state constraints (RDV problems, target problems, minimal time). When some controlability assumptions are not satisfied, the solution of the HJB equation is discontinuous and the classical schemes, relying on finite diffrences or/and interpolation technics, provide poor quality approximation. In fact, these schemes lead to an increasing loss of precision around the discontinuities as well as for long time approximations. Hence, the numerical solution is unsatisfying for long time approximations even in the continuous case.

We prove the convergence of a non-monotone scheme for one-dimensional Hamilton-Jacobi-Bellman equations of the form u_t + max_a(f(x, a)u_x) = 0 , u(0, x) = u₀(x) . The scheme is related to the HJB-Ultra-Bee scheme suggested in [1] , [14] , which has an anti-diffusive behavior, but where the convergence was not proved. In our approach we can consider discontinous initial data u₀ . In particular we show a first-order convergence of the scheme, in L¹ -norm (i.e. an error bounded by a constant times x where x is the mesh size) towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on some numerical examples. A corresponding preprint is under preparation.

Let us stress on that our scheme is explicit and is non-monotone (neither -monotone in the sense of R. Abgrall [28] ). As far as we know, there are few non-monotone scheme that has been proved to converge for HJ equations (see also Lions and Souganidis [37] for an implicit non-monotone scheme).