# Comparing Numerical Integration Schemes for Time-Continuous Car-Following Models

arXiv: https://arxiv.org/abs/1403.4881 · HTML: https://mlsanity.com/paper/1403.4881/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2014.

## Built on by (most cited first, 8 of 8)

- [FollowNet: A Comprehensive Benchmark for Car-Following Behavior Modeling](https://mlsanity.com/paper/2306.05381/) (2023)
- [Formulation and validation of a car-following model based on deep reinforcement learning](https://mlsanity.com/paper/2109.14268/) (2021)
- [Spatial-temporal recurrent reinforcement learning for autonomous ships](https://mlsanity.com/paper/2211.01004/) (2022)
- [Vessel-following model for inland waterways based on deep reinforcement learning](https://mlsanity.com/paper/2207.03257/) (2022)
- [2-Level Reinforcement Learning for Ships on Inland Waterways: Path Planning and Following](https://mlsanity.com/paper/2307.16769/) (2023)
- [Towards Robust Car Following Dynamics Modeling via Blackbox Models: Methodology, Analysis, and Recommendations](https://mlsanity.com/paper/2402.07139/) (2024)
- [Deep reinforcement learning-based longitudinal control strategy for automated vehicles at signalised intersections](https://mlsanity.com/paper/2505.08896/) (2025)
- [Macroscopic Characteristics of Mixed Traffic Flow with Deep Reinforcement Learning Based Automated and Human-Driven Vehicles](https://mlsanity.com/paper/2603.25328/) (2026)

## Abstract (arXiv)

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.