Interactions occur between vehicle density, velocity, traffic demand and roadway capacity, creating complex, nonlinear traffic flow dynamics. This research proposes a mathematical modelling and simulation approach to traffic flow analysis based on non-linear differential equations. Basic relations between traffic density, flow and velocity are derived together with the fundamental conservation law for traffic moving along road networks. Different initial and boundary conditions are used to simulate traffic states numerically, using solution techniques. The model goes on to study the propagation of a shock wave and stop and go traffic conditions resulting from obstruction and congestion and changes in traffic density. The simulation results presented are used to show the temporal and spatial changes of traffic characteristics, and to identify the change from free-flow to congested traffic. The framework proposed here is an effective mathematical basis for traffic prediction, traffic congestion analysis, and transportation management applications.