  A new algorithm for time dependent Hamilton Jacobi equations on networks,
based on semi Lagrangian scheme, is proposed. It is based on the definition of
viscosity solution for this kind of problems recently given in. A thorough
convergence analysis, not requiring weak semilimits, is provided. In
particular, the check of the supersolution property at the vertices is
performed through a dynamical technique which seems new. The scheme is
efficient, explicit, allows long time steps, and is suitable to be implemented
in a parallel algorithm. We present some numerical tests, showing the advantage
in terms of computational cost over the one proposed in [7]
