  When simulating trajectories by integrating time-continuous car-following
models, standard integration schemes such as the forth-order Runge-Kutta method
(RK4) are rarely used while the simple Euler's method is popular among
researchers. We compare four explicit methods: Euler's method, ballistic
update, Heun's method (trapezoidal rule), and the standard forth-order 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 external data of the simulation scenario 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 Euler's method
although both are of first order.
