## Preprint 2013-14

## Raimund Bürger, Christophe Chalons, Luis M. Villada:

### Anti-diffusive and random-sampling Lagrangian-remap schemes for the multi-class Lighthill-Whitham-Richards traffic model

### Abstract:

The multi-class Lighthill-Whitham-Richards (MCLWR) traffic model, which distinguishes N classes of drivers differing in preferential velocity, gives rise to a system of N strongly coupled, nonlinear first-order conservation laws for the car densities as a function of distance and time. The corresponding velocities involve a hindrance function that depends on the local total density of cars. Since the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic-wise numerical schemes for the MCLWR model become involved. Alternative simple schemes for this model directly utilize that the velocity functions are non-negative and strictly decreasing, which allows one to construct a new class of schemes by splitting the system of conservation laws into two different first-order quasilinear systems, which are solved successively for each time iteration, namely the Lagrangian and remap steps. The new schemes are addressed as Lagrangian-remap (LR) schemes. One version of LR~schemes incorporates recent anti-diffusive techniques for transport equations. The corresponding subclass of LR schemes are named Lagrangian-anti-diffusive-remap (L-AR) schemes. Alternatively, the remap step can be handled by Glimm-like random sampling, which gives rise to a statistically conservative Lagrangian-random sampling (L-RS) scheme that is less diffusive than other remap techniques. The LR schemes for the MCLWR model are supported by a partial analysis of the L-AR schemes for N=1, which are total variation diminishing (TVD) under a suitable CFL condition and therefore converge to a weak solution, and by numerical examples for both L-AR and L-RS subclasses of schemes.

This preprint gave rise to the following definitive publication(s):

**Raimund BüRGER, Christophe CHALONS, Luis M. VILLADA: ***Anti-diffusive and random-sampling Lagrangian-remap schemes for the multi-class Lighthill-Whitham-Richards traffic model*. SIAM Journal on Scientific Computing, vol. 35, 6, pp. 1341-1368, (2013)