This paper is concerned with the numerical prediction of the extrudate swell behaviour of branched polymer melts in a planar configuration. The multi-mode extended pom-pom (XPP) model is used to describe the polymer dynamics. A second-order operator-integration-factor splitting scheme is used for the temporal discretisation of the problem, whilst a spectral element scheme is used in space. The free surface is evolved in a Lagrangian manner using the third-order conditionally stable Adams-Bashforth method. A thorough mesh convergence study is performed with respect to the temporal and spatial discretisation parameters. The influence of the nature of the discrete relaxation spectrum on the swelling ratio and as an indicator of polydispersity is investigated. The predictions of numerical simulations are also compared with a selection of experimental results from the literature. The parameters in the XPP model are determined from rheological data. Good agreement is obtained for branched low-density polyethylenes. The ability to model a melt with a high molecular weight tail using a discrete relaxation spectrum for which the largest relaxation time is isolated from the others is also investigated.