The transition of Riemann solutions with composite waves for the improved Aw-Rascle-Zhang model in dusty gas

We study the cavitation and concentration of the Riemann solutions for the improved Aw-Rascle-Zhang (IARZ) model in dusty gas with a non-genuinely nonlinear field. The Riemann solutions containing composite waves are constructed by Liu-entropy condition first. Second, we investigate the limits of the inflection point and tangent point along the 1-family wave curve and find that the composite waves tend to elementary waves as pressure vanishes. Third, we obtain the limiting behavior of the Riemann solutions and observe the formation of d-shock wave and vacuum as pressure vanishes. We conclude that the limit of Riemann solutions of the IARZ model is not the Riemann solutions of the limit of the IARZ model. The phenomenon is consistent with the work of C. Shen and M. Sun [Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Aw-Rascle model, J. Differ. Equations 249, 3024-3051 (2010)]. Finally, we perform some numerical simulations to verify our theoretical analysis.

Automated summary

We study the cavitation and concentration of Riemann solutions for the improved Aw-Rascle-Zhang model in dusty gas with a non-genuinely nonlinear field. We construct the Riemann solutions containing composite waves using Liu-entropy condition, investigate the limits of inflection points and tangent points along the 1-family wave curve, and observe the formation of d-shock wave and vacuum as pressure vanishes. We conclude that the limit of Riemann solutions of the IARZ model is not the Riemann solutions of the limit of the IARZ model, consistent with the work of C. Shen and M. Sun.