Multirate Solver with Speed and Gap Control

Abstract

Computer simulation of vehicular traffic on the real road network can be used to solve a whole range of relevant and practical problems. The microscopic approach and tens of thousands of vehicles used in simulation lead to large systems of ordinary differential equations. The vehicle dynamics can vary significantly. As a result, the corresponding systems of differential equations have a certain feature, i.e., the rate of change in unknown vector components, which in this case are the speeds of vehicles and the distances (gaps) between them, varies significantly. In this paper, we propose a multirate numerical integration scheme, in which an individual microstep is used for each component of the vector of unknowns within each macrostep. The values of the steps are determined using the obtained local error estimate of the given numerical scheme. The corresponding time-stepping strategy is obtained both for vehicle speeds and for distances between vehicles. Moreover, the multirate solver’s local error for gaps is estimated one order of accuracy higher than for speeds because drivers estimate the gap primarily rather than the speed. The developed numerical method shows a significant computational speedup in comparison with the corresponding single-rate method.