We use an Eulerian network model of the airspace to simulate air traffic in congested areas of airspace. The model relies on a set of coupled first order hyperbolic partial differential equations (PDEs), obtained from the original Lighthill-Whitham-Richards (LWR) traffic model. The Jameson-Schmidt-Turkel (JST) is selected among other numerical schemes to perform simulations, and evidence of numerical convergence is assessed against known analytical solutions. Linear numerical schemes are discarded because of their poor performance, thus prohibiting the use of linear optimization for controlling the network. Instead, the adjoint problem of the linearized network control problem is computed. The constraints of the problem are enforced using a logarithmic barrier method. Simulations are run with real air traffic data to demonstrate the applicability of the method for traffic management. Scenarios involving several airports between Chicago and the East Coast are investigated.