The solutions of Riemann problems of a particular higher-order model - the Payne-Whitham (PW) model - are studied using Roe's flux splitting scheme as presented by Leo and Pretty. Despite numerous works on higher-order models, little is known about Riemann solutions of these models and how relaxation and anticipation affect these solutions. Riemann solutions of the PW model are computed and compared with those of the Lighthill-Whitham-Richards (LWR) model having the same initial (density) data. It was found that faster relaxation forces the PW model to behave much like the LWR model, that strong anticipation has a stabilizing effect on traffic, and that shock waves travel at different speeds in the PW model than they do in the LWR model. These findings provide a basic checklist for experimental validation of PW-like higher models.