Comparing numerical integration schemes for time-continuous car-following models

Abstract

h i g h l i g h t s• We propose novel performance metrics for numerical integration schemes. • For car-following models, the ballistic scheme is always superior to Euler's scheme.• The standard RK4 scheme is only efficient for unperturbed single-lane traffic.• Heun's scheme is generally the best for simple situations. • The ballistic scheme prevails for complex situations with stops and lane changes. a b s t r a c tWhen simulating trajectories by integrating time-continuous car-following models, standard integration schemes such as the fourth-order Runge-Kutta method (RK4) are rarely used while the simple Euler method is popular among researchers. We compare four explicit methods both analytically and numerically: Euler's method, ballistic update, Heun's method (trapezoidal rule), and the standard RK4. As performance metrics, we plot the global discretization error as a function of the numerical complexity. We tested the methods on several time-continuous car-following models in several multi-vehicle simulation scenarios with and without discontinuities such as stops or a discontinuous behavior of an external leader. We find that the theoretical advantage of RK4 (consistency order 4) only plays a role if both the acceleration function of the model and the trajectory of the leader are sufficiently often differentiable. Otherwise, we obtain lower (and often fractional) consistency orders. Although, to our knowledge, Heun's method has never been used for integrating car-following models, it turns out to be the best scheme for many practical situations. The ballistic update always prevails over Euler's method although both are of first order.

Comparing numerical integration schemes for a car-following model with real-world data

“…Similarly to the authors of [1] we will investigate four different integration schemes. The first three are well-known numerical methods for integrating ODEs, namely…”

“…Besides the three standard approaches outlined above, the authors of [1] use an alternative first-order integration scheme called ballistic update, which can be used only in special cases where Eq. (1) represents Newtonian dynamic acceleration equations.…”

“…In their recent article [1], Treiber and Kanagaraj compare different numerical integration schemes for intelligent driver model (IDM) [2]. The IDM is a time-continuous car-following model for the simulation of freeway and urban traffic, developed in part by the first author of [1].…”

“…The IDM is a time-continuous car-following model for the simulation of freeway and urban traffic, developed in part by the first author of [1]. The model assumes that drivers maintain certain minimal time and space gap from their leading vehicle, and at the same time try to keep their preferred speed and acceleration and deceleration profile.…”

“…Reference [1] contains almost complete information needed to re-implement the original experiments of the authors. We will now briefly overview the important parts, adding one detail that has been omitted from the original paper.…”