This study is concerned with the higher order macroscopic traffic flow modeling and simulation of traffic flow on junctions. More precisely, we deal with a generic class of second order models, known in the literature as the GSOM family which embeds a large variety of traffic models. While classical approaches focus on the Eulerian point-of-view, here we recast the model using its Lagrangian coordinates and we treat the junction as a specific discontinuity in Lagrangian framework. We propose a complete numerical methodology based on a finite difference scheme for solving such a model which is compatible with micro and macro description, and satisfy the constraints of the invariance principle.