## Preprint 2018-38

## Raimund Bürger, Hector Torres, Carlos A. Vega:

### An entropy stable scheme for the multiclass Lighthill-Whitham-Richards traffic model

### Abstract:

An entropy conservative (EC) numerical flux for the multiclass Lighthill- Whitham-Richards (MCLWR) kinematic traffic model based on the general framework by Tadmor [E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I, Math. Comp., 49 (1987), pp. 91–103] is proposed. The approach exploits the existence of an entropy pair for a particular form of this model. The construction of EC fluxes is of interest since in combination with numerical diffusion terms they allow one to design entropy stable schemes for the MCLWR model. In order to obtain a higher-order accurate scheme and control oscillations near discontinuities, a third-order WENO reconstruction recently pro- posed by Ray [D. Ray, Third-order entropy stable scheme for the compressible Euler equations, in C. Klingenberg and M. Westdickenberg (eds.), Springer Proc. in Math. and Stat. 237, pp. 503–515] is used. Numerical experiments for different classes of drivers are presented to test the performance of the entropy stable scheme constructed with the entropy conservative flux proposed.