Flödesmekaniska modeller av trafik

Sammanfattning: When planning for road building and infrastructure, simulations of tra-c 　ow are useful tools for further investigation. Two models are analyzed in this report, both supported by the assumption that tra-c 　ow behaves as a continous 　uid on a macroscopic scale. This means that the vehicle density will be considered instead of observing the vehicles as discrete objects. Conservational laws along with piecewise constant initial data are called Riemann problems. Tra-c 　ow in one dimension at a tra-c light can be modeled as such a problem with the continuity equation and a scaled Heaviside function as initial data. Two special cases of the general problem are solved, in the rst model the velocity of the cars is a linear function of the density while in the second model the velocty is a function of the denisty and its derivative. The rst case is solved analytically with a similarity solution. The second case, where diusion is included, is solved numerically and the method utilized is proved to be stable. Simulations are presented in order to see how the density of vehicles develop in front of and after the tra-c light. The results are used to determine a time for the tra-c light to be green, given a xed time for red, that optimzes the 　ow of vehicles. 1