Intelligent driver model
In traffic flow modeling, the intelligent driver model (IDM) is a time-continuous car-following model for the simulation of freeway and urban traffic. It was developed by Treiber, Hennecke and Helbing in 2000 to improve upon results provided with other "intelligent" driver models such as Gipps' model, which loses realistic properties in the deterministic limit. Model definition. As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles. For vehicle formula_1, formula_2 denotes its position at time formula_3, and formula_4 its velocity. Furthermore, formula_5 gives the length of the  vehicle. To simplify notation, we define the "net distance" formula_6, where formula_7 refers to the vehicle directly in front of vehicle formula_1, and the velocity difference, or "approaching rate", formula_9. For a simplified version of the model, the dynamics of vehicle formula_1 are then described by the following two ordinary differential equations: formula_14, formula_15, formula_16, formula_17, and formula_18 are model parameters which have the following meaning: The exponent formula_25 is usually set to 4. Model characteristics. The acceleration of vehicle formula_1 can be separated into a "free road term" and an "interaction term": This leads to a driving behavior that  compensates velocity differences while trying not to brake much harder than the comfortable braking deceleration formula_18. Solution example. Let's assume a ring road with 50 vehicles. Then, vehicle 1 will follow vehicle 50. Initial speeds are given and since all vehicles are considered equal, vector ODEs are further simplified to: For this example, the following values are given for the equation's parameters. The two ordinary differential equations are solved using Runge–Kutta methods of orders 1, 3, and 5 with the same time step, to show the effects of computational accuracy in the results. This comparison shows that  the IDM does not show extremely irrealistic properties such as negative velocities or vehicles sharing the same space even for from a low order method such as with the Euler's method (RK1). However, traffic wave propagation is not as accurately represented as in the higher order methods, RK3 and RK 5. These last two methods show no significant differences, which lead to conclude that a solution for IDM reaches acceptable results from RK3 upwards and no additional computational requirements would be needed. Nonetheless, when introducing heterogeneous vehicles and both jam distance parameters, this observation could not suffice. Intelligent driver