Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellmann equations

We propose two new antidiffusive schemes for advection (or linear transport), one of them being a mixture of Roe's Super-Bee scheme and of the "Ultra-Bee" scheme. We show how to apply these schemes to treat time-dependent first order Hamilton-Jacobi-Bellman equations with discontinuous initial data, possibly infinitely-valued. Numerical tests are proposed, in one and two space dimensions, in order to validate the methods.