I will present some results illustrated in my PhD thesis. I will discuss numerical schemes for some first order non-linear Hamilton-Jacobi (HJ) equations and their applications. We introduce a new class of ``filtered" schemes for some first order non-linear Hamilton-Jacobi equations. The proposed schemes are not monotone butstill satisfy some weak monotone property. A general convergence result together with a precise error estimate is given, of the order of dx^1/2 where dx is the mesh size. Numerical tests on several examples are presented to validate the approach. We also propose the coupling of two schemes with different properties. We will introduce the indicator parameter for the coupling, show how to couple the two schemes and prove some properties of the resulting coupled scheme. Finally, I will briefly illustrate some new results on traffic flow on a road network, studying the connections between microscopic follow-the-leader and macroscopic traffic flow models on road networks.