A Diffusively Corrected Multiclass Lighthill-Whitham-Richards Traffic Model with Anticipation Lengths and Reaction Times (Advances in Applied Mathematics and Mechanics, 2013)
Multiclass Lighthill-Whitham-Richards traffic models [Benzoni-Gavage andColombo, Euro. J. Appl. Math., 14 (2003), pp. 587–612; Wong and Wong, Transp. Res.A, 36 (2002), pp. 827–841] give rise to first-order systems of conservation laws that arehyperbolic under usual conditions, so that their associated Cauchy problems are well-posed. Anticipation lengths and reaction times can be incorporated into these modelsby adding certain conservative second-order terms to these first-order conservationlaws. These terms can be diffusive under certain circumstances, thus, in principle, ensuring the stability of the solutions. The purpose of this paper is to analyze the stability of these diffusively corrected models under varying reaction times and anticipationlengths. It is demonstrated that instabilities may develop for high reaction times andshort anticipation lengths, and that these instabilities may have controlled frequenciesand amplitudes due to their nonlinear nature.