An approximate Riemann solver for the Aw-Rascle trafficflow model and the extension of Greenberg is constructed on the algebraic level. A discrete diffusion operator is added to the (oscillatory) high-order finite difference/element discretization to enforce a vectorial LED criterion. This yields a nonoscillatory but diffusive low-order scheme. To increase the accuracy an antidiffusion operator is added.The amount of antidiffusion is controlled using TVD flux limiters. Practical algorithms are presented for the derivation of the low-order scheme, construction of the antidiffusion operator, and the solution of the arising nonlinear algebraic system. Another option considered in this paper is based on a segregated treatment of the equations at hand.It is shown that scalar upwinding/limiting techniques are inappropriate, since even the low-order scheme produces oscillatory solutions to the coupled AR model, although it is monotone for each single equation. Hence, the strongly coupled algorithm developed in this paper yields superior results, as demonstrated by numerical examples