Numerical Schemes for Conservation Laws via Hamilton - Jacobi Equations

We present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux. The numerical methods described here take their origin from approximation schemes for Hamilton-Jacobi-Bellman equations related to optimal control problems and exhibit several interesting features: the convergence result still holds for quite arbitrary time steps, the main assumption for convergence can be interpreted as a discrete analogue of Oleinik's entropy condition, numerical diffusion around the shocks is very limited. Some tests are included in order to compare the performances of these methods with other classical methods (Godunov, TVD).