Numerical Solution of Traffic Flow Model via an Adaptive IMEX–Crank–Nicolson Hybrid Scheme with MATLAB

Abstract

This paper describes an Adaptive IMEX-Crank-Nicolson (A-IMEX-CN) hybrid scheme for the numerical solution of the Lighthill-Whitham-Richards (LWR) traffic flow model. The proposed method couples a two-step explicit Godunov predictor with an implicit Crank-Nicolson corrector, enhanced with an adaptive time-stepping strategy based on the local Courant-Friedrichs-Lewy (CFL) condition and a defect estimator. With this kind of hybrid approach, there is a possibility to avoid problems that traditional methods suffer from: explicit methods have strict stability bounds while implicit methods involve very expensive nonlinear solves and may be affected by numerical diffusion. The A-IMEX-CN reaches second-order time accuracy, allowing far larger time steps with respect to stable explicit schemes while reducing computational costs compared to fully implicit methods. Numerical tests with smooth and discontinuous initial conditions are given, which confirm robustness and efficiency, enabling the realization of real-time and large-scale traffic flow simulations.