page_title: Intelligent driver model
section_title: Solution example
hierarchical_section_title: Intelligent driver model / Solution example
caption_attribution_description: English: Comparison of IDM model simulation for increasing orders in RK methods.
context_page_description: In traffic flow modeling, the intelligent driver model 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.
context_section_description: 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.
caption: Comparison of differential equation solutions for intelligent driver model using RK1,3,5
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