Multirate solver with speed and gap error control for vehicular traffic simulation

Nowadays computer simulation of vehicular traffic on the real road network can be used as a tool for solving actual and practical problems. The microscopic approach and large number of vehicles to simulate (tens of thousands) lead to tremendous systems of ordinary differential systems. The vehicles dynamics can vary sufficiently from vehicle to vehicle. As a result the corresponding differential equations system has different time scales, which are localized over the components. In other words, the temporal variations have different time scales for different components, which are in this case vehicles speeds and distances between them (gaps). In this paper we suggest the numerical integration scheme, which exploits an individual time step for each component (microstep) within one macrostep. The microstep value of a particular system component is determined by the local temporal variation of the solution, instead of using a single step size for the whole system. This time stepping strategy is obtained both for vehicles speeds and gaps. What is more, the local error estimation for the gaps is derived one order higher than for the speeds, because drivers assess first of all the distance, not the speed. Comparison with the corresponding single-rate scheme demonstrates substantial gains in CPU times.